ENTRY ARTIFICIAL INTELLIGENCE - Department of Mathematics ...

ENTRY ARTIFICIAL INTELLIGENCE 

[ENTRY ARTIFICIAL INTELLIGENCE] Authors: Oliver Knill: March 2000 Literature: Peter Norvig, 

Paradigns of Artificial Intelligence Programming Daniel Juravsky and James Martin, Speech and Language 

Processing 

Adaptive Simulated Annealing 

[Adaptive Simulated Annealing] A language interface to a neural net simulator. 

artificial intelligence 

[artificial intelligence] (AI) is a field of computer science concerned with the concepts and methods of symbolic 

knowledge representation. AI attempts to model aspects of human thought on computers. Aspectrs of AI: 

• computer vision 

• language processing 

• pattern recognition 

• expert systems 

• problem solving 

• roboting 

• optical character recognition 

• artificial life 

• grammars 

• game theory 

[Babelfish] Online translation system from Systran. 

Babelfish 

Chomsky 

[Chomsky] Noam Chomsky is a pioneer in formal language theory. He is MIT Professor of Linguistics, Linguistic 

Theory, Syntax, Semantics and Philosophy of Language. 

Eliza 

[Eliza] One of the first programs to feature English output as well as input. It was developed by Joseph 

Weizenbaum at MIT. The paper appears in the January 1966 issue of the ”Communications of the Association 

of Computing Machinery”.

Google 

[Google] A search engine emerging at the end of the 20’th century. It has AI features, allows not only to answer 

questions by pointing to relevant webpages but can also do simple tasks like doing arithmetic computations, 

convert units, read the news or find pictures with some content. 

GPS 

[GPS] General Problem Solver. A program developed in 1957 by Alan Newell and Herbert Simon. The aim was 

to write a single computer program which could solve any problem. One reason why GPS was destined to fail 

is now at the core of computer science. There are a large set of problems which are NP hard and where finding 

a solution becomes exponentially hard in dependence of the size of the problem. Nonetheless, GPS has been a 

useful tool for exploring AI programming. 

HAL 

[HAL] The HAL 9000 computer was the main character in Stanley Kurbrick’s film 2001: a Space Odyssey. HAL 

is an AI agent capable to understand advanced language processing behavior as speaking and understanding 

language and even reading lips. 

Lisp 

[Lisp] Lisp is one of the oldest programming languages still in widespread use today. ”Common Lisp” is the most 

widely accepted standard. Other dialects like ”Franz Lisp” MacLisp, InterLisp, ZetaLisp or ”Standard Lisp” 

are considered obsolete. Lisp is the most popular language for AI programming. Lisp programs are concise and 

are uncluttered by low-level detail. 

Loebner Prize 

[Loebner Prize] A competition attempted to put various computer programs to the Turing test. A consistent 

result over the years has been that even the crudest programs can fool some of the judges some of the time. 

MIT ai lab 

[MIT ai lab] Massachusetts Institute of Technology AI laboratory.

neural network 

[neural network] Artificial neural networks try to simulate biological neural networks as found in the brain. 

Such a network consists of many simple processors called neurons, each possibly having some local memory. 

These neurons are connected and evolve depending to their local data and on the inputs they receive via 

the connections. A neural network can either be an algorithm, or be realized as actual hardware. Neural 

networks typically allow training. They learn by adjusting the weights of the connections on the basis of 

presented patterns. The individual neurons are elementary non-linear signal processors. Neural networks are 

distinguished from other computing devices by a high degree of interconnection allowing parallelism. There is 

no idle memory containing data and programs. Each neuron is pre-programmed and continuously active. 

pattern recognition 

[pattern recognition] A branch of artificial intelligence concerned with the classification or description of observations. 

The classification uses either statistical, syntactic or neural aproches. 

[pilot] Programmed Inquiry Learning Or Teaching. 

pilot 

prolog 

[prolog] A popular AI programming language used in Europe and Japan. Prolog shares most of Lisp’s advantages 

in terms of flexibility and conciseness. 

regular expression 

[regular expression] is a language for specifying text search strings. It is used in UNIX programs like vi, perl, 

emacs or grep. It is also used in Microsoft word or web search engines. 

scheme 

[scheme] A dialect of Lisp which is gaining popularity, primarily for teaching and experimenting with programming 

language design and techniques. 

Shrdlu 

[Shrdlu] Terry Winograd’s SHRDLU system of 1972 simulated a robot embedded in a world of toy blocks. The 

program was able to accept natural language text commands.

Student 

[Student] Student was an early language understanding program written by Daniel Bubrow in 1964. It was 

designed to read and solve the kind of word problems found in high school algebra books. Unlike Eliza, 

”Student” must process and understand a great deal of input as well as be able to solve algebraic equations. 

toy problem 

[toy problem] A deliberately oversimplified case of a challenging problem used to investigate, prototype, or test 

algorithms for a real problem. 

Turing test 

[Turing test] A test introduced in 1950 by Alan Turing. There are three participants. Two people and a 

computer. One person plays the role of an interrogator who has to find out, which of the two others is a 

machine. This interrogator is connected to the two other participants through teletype. The task of the 

machine is to fool the interrogator into believing it is a person. The task of the other participant is to convince 

the interrogator that he is human. Turing predicted that in 2000 a machine with 10 Gig memory would have a 

30 percent change of fooling a human interrogator after 5 minutes of questioning. 

Weizenbaum 

[Weizenbaum] Joseph Weizenbaum was the principal developer of Eliza, one of the first programs to feature 

English output as well as input. 

This file is part of the Sofia project sponsored by the Provost’s fund for teaching and learning at Harvard 

university. There are 22 entries in this file.

Index 

Adaptive Simulated Annealing, 1 

artificial intelligence, 1 

Babelfish, 1 

Chomsky, 1 

Eliza, 1 

Google, 2 

GPS, 2 

HAL, 2 

Lisp, 2 

Loebner Prize, 2 

MIT ai lab, 2 

neural network, 3 

pattern recognition, 3 

pilot, 3 

prolog, 3 

regular expression, 3 

scheme, 3 

Shrdlu, 3 

Student, 4 

toy problem, 4 

Turing test, 4 

Weizenbaum, 4 

5

ENTRY ABSTRACT ALGEBRA 

[ENTRY ABSTRACT ALGEBRA] Authors: started Oliver Knill: September 2003 Literature: Lecture notes 

additive 

A function f : G → H from a semigroup G to a semigroup H is [additive] if f(a + b) = f(a) + f(b). A 

group-valued function on sets is additive if f(Y ∪ Z) = f(Y ) + f(Z) if Y and Z are disjoint. 

algebra 

An [algebra] over a field K is a ring with 1 which is also a vector space over K and whose multiplication is 

bilinear with respect to K. Examples: 

• the complex numbers C is an algebra over the field of real numbers K = R. 

• The quaternion algebra H is an algebra over the field of complex numbers. 

• The matrix algebra M(n, R) is an algebra over the field R. 

An algebraic number field 

[An algebraic number field] is a subfield of the complex numbers that arises as a finite degree algebraic extension 

field over the field of rationals. 

alternating group 

The [alternating group] G is the subgroup of the symmetric group of n objects given by the elements which can 

be written as a product of an even number of transpositions. 

Artinian module 

An [Artinian module] is a module which satisfies the descending chain condition. Every Artinian module is a 

Noetherian module but the integers for example are a Noetherian module which is not an Artinian module. 

Artinian ring 

An [Artinian ring] is a ring which when considered as a R-module is an Artinian module. 

Artinian ring 

Two elements of an integral domain that are unit-multipliers of each other are called [associate numbers].

Cayley’s theorem 

[Cayley’s theorem] assures that every finite group is isomorphic to a permutation group. 

center 

The [center] of a group (G, ∗) is the set of all elements g which satisfy gh = hg for all h in G. The center is a 

subgroup of G. 

commutator 

The [commutator] of two elements g, h in a group (G, ∗) is defined as [g, h] = g ∗ h ∗ g −1 ∗ h −1 . 

commutator subgroup 

The [commutator subgroup] of a group (G, ∗) is the set of all commutators [g, h] in G. It is a subgroup of G. 

factor group 

A [factor group] G/N is defined when N is a normal subgroup of the group G. It is the group, where the 

elements are equivalent classes gN and operation (gN)(hN) = (gh)N which is defined because N was assumed 

to be normal. For example, if G is the group of additive integers and N = kN with an integer k, then G/N = Zk 

is finite group of integers modulo k. 

finite group 

A group is called a [finite group] if G is a set with finitely many elements. For example, the set of all permutations 

of a finite set form a finite group. The set of all operations on the Rubik cube form a finite group.

group 

A [group] (X, +, 0) is a set X with a binary operation + and a zero element 0 (also called neutral element or 

identity) with the following properties 

Examples: 

(a + b) + c = a + (b + c) associativity 

a + 0 = a zero element 

∀a∃ba + b = 0 inverse 

• the real numbers form a group under addition 5 + 2.34 = 7.34, 3 − 3 = 0. 

• the set GL(n, R) of real matrices with nonzero determinant form a group under matrix multiplication 

• the nonzero integers form a group under multiplication 4 ∗ 7 = 28. 

• all the invertible linear transformations of the plane plane form a group under composition. The ”zero 

element” is the identity transformation T (x) = x. 

• all the continuous functions on the unit interval form a group with addition (f + g)(x) = f(x) + g(x). 

• all the permutations on a finite set form a group under composition. 

• the set of subsets Y of a set X with the operation A∆B = (A ∪ B) \ (A ∩ B) form a group. The inverse 

of A is A itself because A∆A = ∅, the zero element is ∅. 

normal subgroup 

a [normal subgroup] of a group (G, ∗) is a subgroup (H, ∗) of (G, ∗) which has the property that for all g in H 

and all g in G one has g −1 hg is in H. For an abelian group all subgroups are normal. The subgroup Sl(n, R) 

of Gl(n, R) is a normal subgroup. 

ring 

A [ring] (X, +, ∗, 0) is a set X with a binary operation + and a binary operation ∗ such that (X, +, 0) is a 

commutative group and (X, ∗) is a semigroup and such that the distributivity laws a ∗ (b + c) = a ∗ b + a ∗ c, 

(a + b) ∗ c − a ∗ c + b ∗ c hold. Examples: 

• the integers Z form a ring with addition and multiplication 

• the set of rational numbers Q, the set of real numbers R or the complext numbers C form a ring with 

addition and multiplication. 

• the set of 3x3 matrices with real entries form a ring with addition and matrix multiplication. 

• the set P of polynomials with real coefficients form a ring with addition and multiplication. 

• the set of subsets Y of a set X with addition ∆ and multiplication ∩ forms a ring. 

• the set of continuous functions on an interval [0, 1] with addition (f +g)(x) = f(x)+g(x) and multiplication 

f ∗ g(x) = f(x)g(x).

commutative group 

A [commutative group] is a group (X, +, 0) which is commutative: a + b = b + a. 

• the set of real numbers R forms a commutative group under addition. 

• the set of permutations S of a set X form a noncommutative group under composition. 

commutative ring 

A [commutative ring] is a ring (X, +, ∗, 0) for which the multiplicative semigroup (X, ∗) is commutative: a ∗ b = 

b ∗ a. Examples: 

• the integers form a commutative ring. 

• the set of 2 × 2 matrices form a noncommutative ring 

• the set of polyomials with real coefficients (x 2 + πx + 2) ∗ (x + 5x) = 6x 3 + 6πx 2 + 12x. 

function field 

A [function field] is a finite extension of the field C(z) of rational functions in the variable z. 

homomorphism 

An [homomorphism] φ between two groups G, H is a map f : G → H which has the property φ(g∗h) = φ(g)∗φ(h) 

and φ(0) = 0 for all elements g, h ∈ G. Examples: 

• if G is the multiplicative group (R + , ∗) of positive real numbers and H is the additive group (R, +) of all 

positive real numbers then φ(x) = log(x) is a homomorphism: 

• if G is the group of matrices with nonzero determinant and H is the group of nonzero real numbers and 

φ(A) = det(A), we have φ(x ∗ y) = φ(x)φ(y). 

isomorphism 

An [isomorphism] φ between two groups G, H is a homomorphism between groups which is also invertible.

number field 

A [number field] is a finite extension of Q, the field of rational numbers. It is a field extension of Q which is also 

a vector space of finite dimension over Q. Since the elements of a number field are algebraic numbers, roots of 

a fixed polyonomial a0 + a1z + ... + z n with integer coefficitients, one calls them also algebraic number fields. 

The study of algebraic number fields is part of algebraic number theory. 

Examples: 

• quadratic fields: Q( √ d), where d is a rational number. It is in general a field extension of degree 2 over 

the field of rational number. 

• cyclotimic fields: Q(ξ), where ξ is a n’th root of 1. It is a field extension of degree φ(n), where φ(n) is the 

Euler function. 

octonions 

The [octonions] can be written as linear combinations of elements e0, e1, e2, ..., e7. The multiplication is determined 

by the multiplication table 

* 1 e1 e2 e3 e4 e5 e6 e7 

1 1 e1 e2 e3 e4 e5 e6 e7 

e1 e1 −1 e4 e7 -e2 e6 -e5 -e3 

e2 e2 -e4 −1 e5 e1 -e3 e7 -e6 

e3 e3 -e7 -e5 −1 e6 e2 -e4 e1 

e4 e4 e2 -e1 -e6 −1 e7 e3 -e5 

e5 e5 -e6 e3 -e2 -e7 −1 e1 e4 

e6 e6 e5 -e7 e4 -e3 -e1 −1 e2 

e7 e7 e3 e6 -e1 e5 -e4 -e2 −1 

Octonions are also called Cayley numbers. The multiplication of octonions is not associative. Octonions have 

been discovered by John T. Graves in 1843 and independently by Arthur Cayley. 

order 

The [order] of a finite group is the set of elements in the group. 

p-group 

A [p-group] is a finite group with order p n , where p is a prime integer and n > 0.

quaterions 

The [quaterions] can be written as linear combinations of elements 1, i, j, k. The multiplication is determined 

by the multiplication table 

* 1 i j k 

1 1 i j k 

i i −1 k −j 

j j −k −1 i 

k k j −i −1 

Quaternions are useful to compute rotations in three dimensions. 

semigroup 

A [semigroup] (X, +) is a set X with a binary operation + which satisfies the associativity law (a + b) + c = 

a + (b + c). Examples: 

• a group is a semigroup. 

• the set of finite words in an alphabet with composition form a semigroup word1 + word2 = word1word2 

• the natural numbers form a semigroup under addition. 

commutative semigroup 

A [commutative semigroup] is a semigroup (X, +) which is commutative. a + b = b + a. 

• the natural numbers form a commutative semigroup under addition. 

• composition of words over a finite alphabet form a noncommutative semigroup 

kernel 

The [kernel] of a homomorphism between two groups G, H is the set of all elements in G which are maped to 

the zero element of H. For example, SL(n, R) is the kernel of the homomorphism from GL(n, R) to R \ {0} 

defined by φ(A) = det(A). 

subgroup 

A [subgroup] of a group G is a subset of G which is also a group. Examples: 

• the set of n × n matrices with determinant 1 is a subgroup of the set of n × n matrices with nonzero 

determinant. 

• the trivial subgroup {0} is always a subgroup of a group (G, ∗, 0).

Theorem of Cauchy 

The [Theorem of Cauchy] in group theory states that every finite group whose order is divisible by a prime 

number p contains a subgroup of order p. 

sedenions 

[sedenions] form a zero Divisor Algebra. By a theorem of Frobenius (1877), there are three and only three 

associative finite division algebras: the real numbers R, the complex numbers C and the quaternions Q. Similar 

algebras in higher dimensions have zero divisors: sedenions are examples. 

field 

A [field] is a commutative ring (R, +, ∗, 0, 1) such that (R, +, 0) and (R \ 0, ∗, 1) are both commutative groups. 

theorem of Zorn 

By a [theorem of Zorn] (1933), every alternative, quadratic, real non-associative algebra without zero divisors 

is isomorphic to the 8-dimensional octonions O. 

Theorem of Hurwitz 

[Theorem of Hurwitz]: the normed composition algebras with unit are: real numbers, complex numbers, quaternions; 

and octonions. 

Theorem of Kervaire and Milnor 

[Theorem of Kervaire and Milnor] In 1958, Kervaire and Milnor proved independently of each other that the 

finite-dimensional real division algebras have dimensions 1, 2, 4, or 8. 

Theorem of Adams 

[Theorem of Adams] In 1960, Adams proved that a continuous multiplication in R n+1 with two-sided unit and 

with norm product exists only for n + 1 = 1, 2, 4, or 8.

Theorem of Hurwitz 

[Theorem of Hurwitz]: the normed composition algebras with unit are: 

• real numbers 

• complex numbers 

• quaternions 

• octonions 

Theorems of Sylov 

[Theorems of Sylov] If G is a finite group of order |G| = p n q, where p is a prime number, then G has a subgroup 

of order p n . Such groups are called Sylov groups and all of them are isomorphic. Furthermore, the number N 

of different p-Sylov groups in G satisfies N = 1 mod(p). 



Index 

additive, 1 

algebra, 1 

alternating group, 1 

An algebraic number field, 1 

Artinian module, 1 

Artinian ring, 1 

Cayley’s theorem, 2 

center, 2 

commutative group, 4 

commutative ring, 4 

commutative semigroup, 6 

commutator, 2 

commutator subgroup, 2 

factor group, 2 

field, 7 

finite group, 2 

function field, 4 

group, 3 

homomorphism, 4 

isomorphism, 4 

kernel, 6 

normal subgroup, 3 

number field, 5 

octonions, 5 

order, 5 

p-group, 5 

quaterions, 6 

ring, 3 

sedenions, 7 

semigroup, 6 

subgroup, 6 

Theorem of Adams, 7 

Theorem of Cauchy, 7 

Theorem of Hurwitz, 7, 8 

Theorem of Kervaire and Milnor, 7 

theorem of Zorn, 7 

Theorems of Sylov, 8 

9

AMS FIELDS 

[AMS FIELDS] Authors: Oliver Knill: September 2003 Literature: AMS Website

[AMS CLASSIFICATION] 

AMS CLASSIFICATION 

00-xx General 

01-xx History and biography 

03-xx Mathematical logic and foundations 

05-xx Combinatorics 

06-xx Order, lattices, ordered algebraic structures 

08-xx General algebraic systems 

11-xx Number theory 

12-xx Field theory and polynomials 

13-xx Commutative rings and algebras 

14-xx Algebraic geometry 

15-xx Linear and multilinear algebra; matrix theory 

16-xx Associative rings and algebras 

17-xx Nonassociative rings and algebras 

18-xx Category theory; homological algebra 

19-xx K-theory 

20-xx Group theory and generalizations 

22-xx Topological groups, Lie groups 

26-xx Real functions 

28-xx Measure and integration 

30-xx Functions of a complex variable 

31-xx Potential theory 

32-xx Several complex variables and analytic spaces 

33-xx Special functions 

34-xx Ordinary differential equations 

35-xx Partial differential equations 

37-xx Dynamical systems and ergodic theory 

39-xx Difference and functional equations 

40-xx Sequences, series, summability 

41-xx Approximations and expansions 

42-xx Fourier analysis 

43-xx Abstract harmonic analysis 

44-xx Integral transforms, operational calculus 

45-xx Integral equations 

46-xx Functional analysis 

47-xx Operator theory 

49-xx Calculus of variations and optimal control; optimization 

51-xx Geometry 

52-xx Convex and discrete geometry 

53-xx Differential geometry 

54-xx General topology 

55-xx Algebraic topology 

57-xx Manifolds and cell complexes 

58-xx Global analysis, analysis on manifolds 

60-xx Probability theory and stochastic processes 

70-xx Mechanics of particles and systems 

74-xx Mechanics of deformable solids 

76-xx Fluid mechanics 

78-xx Optics, electromagnetic theory 

80-xx Classical thermodynamics, heat transfer 

81-xx Quantum theory 

82-xx Statistical mechanics, structure of matter 

83-xx Relativity and gravitational theory 

85-xx Astronomy and astrophysics 

86-xx Geophysics 

90-xx Operations research, mathematical programming 

91-xx Game theory, economics, social and behavioral sciences 

92-xx Biology and other natural sciences 

93-xx Systems theory; control 

94-xx Information and communication, circuits



Index 

AMS CLASSIFICATION, 2 

4

ENTRY MATH CITATIONS 

[ENTRY MATH CITATIONS] Collected by Oliver Knill: 2000-2002 

solution 

[solution] Every problem in the calculus of variations has a solution, provided the word solution is suitably 

understood. – David Hilbert 

enhusiast 

[enhusiast] The real mathematician is an enthusiast per se. Without enthusiasm no mathematics. – Novalis 

[royal] There is no royal road to geometry. – Euclid 

royal 

computer 

[computer] One may be a mathematician of the first rank without being able to compute. It is possible to be a 

great computer without having the slightest idea of mathematics – Novalis 

analysis 

[analysis] Geometry may sometimes appear to take the lead over analysis, but in fact precedes it only as a 

servant goes before his master to clear the path and light him on the way. – James Joseph Sylvester 

freedom 

[freedom] The essence of mathematics lies in its freedom. – Georg Cantor 

fantasy 

[fantasy] Fantasy, energy, self-confidence and self-criticism are the characteristic endowments of the mathematician. 

– Sophus Lie 

magacian 

[magacian] Pure mathematics is the magician’s real wand. – Novalis

axiomatics 

[axiomatics] When a mathematician has no more ideas, he pursues axiomatics. – Felix Klein 

turbulence 

[turbulence] The paper ”On the nature of turbulence” with F. Takens was eventually published in a scientific 

journal. (Actually, I was an editor of the journal, and I accepted the paper by myself for publication. This is 

not a recommended procedure in general, but I felt that it was justified in this particular case). – D. Ruelle, in 

Chance and Chaos 

hairy-ball 

[hairy-ball] A good topological theorem to mention any time is the theorem which, in essence, states that 

however you try to comb the hair on a hairy ball, you can never do it smoothly - the so-called ’hairy-ball’ 

theorem. You can make snide comments about the grooming of the hosts’ dog or cat in the meantime as you 

pick hairs off your trouser leg. – R. Ainsley in Bluff your way in Maths, 1988 

large 

[large] LARGE NUMBERS: (10 n means that 10 is raised to the n’th power) 

104 One ”myriad”. The largest numbers, the Greeks were considering. 

105 The largest number considered by the Romans. 

10 10 The age of our universe in years. 

1022 Distance to our neighbor galaxy Andromeda in meters. 

1023 Number of atoms in two gram Carbon (Avogadro). 

1026 Size of universe in meters. 

1041 Mass of our home galaxy ”milky way” in kg. 

1051 Archimedes’s estimate of number of sand grains in universe. 

1052 Mass of our universe in kg. 

1080 The number of atoms in our universe. 

10100 One ”googol”. (Name coined by 9 year old nephew of E. Kasner). 

10153 Number mentioned in a myth about Buddha. 

10155 Size of ninth Fermat number (factored in 1990). 

10 (106 ) Size of large prime number (Mersenne number, Nov 1996). 

10 (107 ) Years, ape needs to write ”hound of Baskerville” (random typing). 

10 (10( 33)) Inverse is chance that a can of beer tips by quantum fluctuation. 

10 (10( 42)) Inverse is probability that a mouse survives on sun for a week. 

10 (1051)) Inverse is chance to find yourself on Mars (quantum fluctuations) 

10 (10100) One ”Gogoolplex”, Decimal expansion can not exist in universe. 

– from R.E. Crandall, Scient. Amer., Feb. 1997 

analytic 

[analytic] The statement sometimes made, that there exist only analytic functions in nature, is in my opinion 

absurd. – F. Klein, Lectures on Mathematics, 1893

violence 

[violence] The introduction of numbers as coordinates ... is an act of violence... – H. Weyl, Philosophy of 

Mathematics and Natural Science 1949 

beauty 

[beauty] Mathematics possesses not only truth but supreme beauty - a beauty cold and austere, like that of a 

sculpture – Bertrand Russell 

geometry 

[geometry] Geometry is magic that works... – R. Thom. Stability Structurelle et Morphogenese, 1972 

Zermelo 

[Zermelo] Ernst Zermelo, who created a system of axioms for set theory, was a Privatdozent at Goettingen when 

Herr Geheimrat Felix Klein held sway over the fabled mathematics department. As Pauli told it, ”Zermelo taught 

a course on mathematical logic and stunned his students by posing the following question: All mathematicians 

in Goettingen belong to one of two classes. In the first class belong those mathematicians who do what Felix 

Klein likes, but what they dislike. In the second class are those mathematicians who do what Felix Klein likes, 

but what they dislike. To what class does Felix Klein belong?” Jordan, having listened intently, broke into 

roaring laughter. Pauli paused, took a sip of wine and said disapprovingly, ”Herr Jordan, you have laughed 

too soon”. He continued: ”None of the awed students could solve this blasphemous problem. Zermelo then 

crowed in his high-pitched voice, ’But, meine Herren, it’s very simple. Felix Klein isn’t a mathematician.’” 

Jordan laughed again. Pauli drained his wine glass approvingly and concluded with ”Zermelo was not offered a 

professorship at Goettingen”. – E.L. Schucking, in ’Jordan, Pauli,Politics, Brecht and a variable gravitational 

constant’ Physics Today, Oct. 1999 

Conway 

[Conway] In the beginning, everything was void, and J.H.W.H.Conway began to create numbers. Conway said, 

”Let there be two rules which bring forth all numbers large and small. This shall be the first rule: Every 

number corresponds to two sets of previously created numbers, such that no member of the left set is greater 

than or equal to any member of the right set. And the second rule shall be this: One number is less than or 

equal to another number if and only if no member of the first number’s left set is greater than or equal to the 

second number, and no member of the second number ’s right set is less than or equal to the first number.” 

And Conway examined these two rules he had made, and behold! they were very good. And the first number 

was created from the void left set and the void right set. Conway called this number ”zero”, and said that it 

shall be a sign to separate positive numbers from negative numbers. Conway proved that zero was less than or 

equal to zero, and he saw that it was good. And the evening and the morning were the day of zero. On the 

next day, two more numbers were created, one with zero as its left set and one with zero as its right set. And 

Conway called the former number ”one”, and the latter he called ”minus one”. And he proved that minus one 

is less than but not equal to zero and zero is less than but not equal to one. And the evening... – D. Knuth, 

Surreal numbers, 1979

[obvious] Mathematics consists essentially of : 

a) proving the obvious 

b) proving the not so obvious 

c) proving the obviously untrue 

obvious 

For example, it took mathematicians until the 1800’ies to prove that 1+1=2 and not before the late 1970 were 

they confident of proving that any map requires no more than four colors to make it look nice, a fact known by 

cartographers for centuries. There are many not-so-obvious things which can be proved true too. Like the fact 

that for any group of 23 people, there is an even chance two or more of them share birthdays. (With groups 

of twins this becomes almost certain. Not quite certain as you will of course point out: they might all have 

been born either side of midnight). Mathematicians are also fond of proving things which are obviously false, 

like all straight lines being curved, and an engaged telephone being just as likely to be free if you ring again 

immediately after, as if you wait twenty minutes. – R. Ainsley in Bluff your way in Maths, 1988 

infimum 

[infimum] There exists a subset of the real line such that the infimum of the set is greater then the supremum 

of the set. – Gary L. Wise and Eric B. Hall, Counter examples in probability and real analysis, 1993, First 

Example in book 

transcendental 

[transcendental] Transcendental number : A number which is not the root of any polynomial equation, like pi 

and e, and which can only be understood after several hours meditation in the lotus position. – R. Ainsley in 

Bluff your way in Maths, 1988 

illiteracy 

[illiteracy] There are great advantages to being a mathematician: a) you do not have to be able to spell b) you 

do not have to be able to add up The illiteracy of mathematicians is taken for granted. There still persists a 

myth that mathematics somehow involves numbers. Many fondly believe that university students spend their 

time long dividing by 173 and learning their 39 times table; in fact, the reverse is true. Mathematicians are 

renowned for their inability to add up or take away, in much the same way as geographers are always getting 

lost, and economists are always borrowing money off you. – R. Ainsley in Bluff your way in Maths, 1988 

prime 

[prime] In this note we would like to offer an elementary ’topological’ proof of the infinitude of the prime 

numbers. We introduce a topology into the space of integers S, by using the arithmetic progressions (from 

-infinity to +infinity) as a basis. It is not difficult to verify that this actually yields a topological space. In 

fact, under this topology, S may be shown to be normal and hence metrisable. Each arithmetic progression is 

closed as well as open, since its complement is the union of the other arithmetic progressions (having the same 

difference). As a result, the union of any finite number of arithmetic progressions is closed. Consider now the 

set A which is the union of A(p), where A(p) consists of primes greater or equal to p. The only numbers not 

belonging to A are -1 and 1, and since the set -1,1 is clearly not an open set, A cannot be closed. Hence A is 

not a finite union of closed sets, which proves that there is an infinity of primes. – H. Fuerstenberg, On the 

infinitude of primes, American Mathematical Montly, 62, 1955, p. 353

arber 

[barber] The barber in a certain town shaves all the people who don’t shave themselves. Who shaves the barber? 

This is meant to be a clever little paradox with no solution but you can annoy the asker intensely by saying it’s 

easy and that the barber is a women. You can then ask the following (a version of Russell’s Paradox, - point 

this out too): in a library there are some books for the catalogue section which is a list of all books which don’t 

list themselves. Shold he or she include this book in its own list? If so, then it becomes a book which lists 

itself, so it shouldn’t be in the list of books which don’t and vice versa. This should keep the most determined 

assailant at bay while you attack the wine. – R. Ainsley in Bluff your way in Maths, 1988 

Hadamard 

[Hadamard] Hadamard, trying to find a job in a US university, came to a small university and was received 

by the chairman of the department of mathematics. He explained who he was and gave his curriculum vitae. 

The chairman said: ’our means are very limited and I can not promise that we shal take you’. Then Hadamard 

noticed that among the portraits on he wall was his own. ’That’s me!’ he said. ’Well, come again in a week, 

we shal think about this’. On his next visit, the answer was negative and his portrait had been removed. – 

Vladimir Mazya and Tatyana Shaposhnikova, in Jacques Hadamard, a universal Mathematician, AMS History 

of Mathematics Volume 14 

Cantor 

[Cantor] The appropriate object is known as the Cantor set, because it was discovered by Henry Smith in 1875. 

(The founder of set theory, Georg Cantor, used Smith’s invention in 1883. Let’s fact it, ’Smith set’ isn’t very 

impressive, is it?) – Ian Stewart, in Does God Play Dice, 1989 p. 121 

jouissance 

[jouissance] ... Thus the erectile organ comes to symbolize the place of jouissance, not in itself, or even in the 

form of an image, but as a part lacking in the desired image: that is why it is equivalent to the (−1) ( 1/2) of the 

signification produced above, of the Jouissance that it restores by the coefficient of its statement to the function 

of lack of signifier (−1). 

– Lacan, Ecrits, Paris 1966 (cited in ’Fashionable nonsense’ by Alan Sokal and Jean Bricmont) 

Mandelbrot 

[Mandelbrot] Mandelbrot made quite good computer pictures, which seemed to show a number of isolated 

”islands” (in the Mandelbrot set M). Therefore, he conjectured that the set M has many distinct connected 

components. (The editors of the journal thought that his islands were specks of dirt, and carefully removed 

them from the pictures). – John Milnor, in Dynamics in one complex variable, 1991 

sin 

[sin] sin, cos, tan, cot, sec, cosec - Formulae derived from the sides of triangles but which crop up in completely 

unexpected places. Sins are extremely common, but rarely do you encounter secs in mathematics. – R. Ainsley 

in Bluff your way in Maths, 1988

Moser 

[Moser] This reminds me of the Hilbert story, which I learned from my teacher Franz Rellich in Goettingen: 

When Hilbert - who was old and retired - was asked at a party by the newly appointed Nazi-minister of 

education: ”Herr Geheimrat, how is mathematics in Goettingen, now that it has been freed of the Jewish 

influences” he replied: ”Mathematics in Goettingen? That does not EXIST anymore”. – Jurgen Moser, in 

Dynamical Systems-Past and Present, Doc. Math. J. DMV I p. 381-402, 1998 

wine 

[wine] There are two glasses of wine, one white and one red. A teaspoonful of wine is taken from the red and 

mixed in with the white. Then a teaspoonful of this mixture is taken and mixed in with the red. Which is 

bigger, the amount of red in the white or the amount of white in the red? The answer is that the’re both the 

same, because there’s the same volume in each glass, so whatever quantity of red is in the white must be equal 

to the quantity of white in the red. However in practice it is impossible to do this because the white always 

runs out first at parties and the red always gets spilt on someone’s white trousers. – R. Ainsley in Bluff your 

way in Maths, 1988 

Monty-Hall 

[Monty-Hall] ”Suppose you’re on a game show and you are given a choice of three doors. Behind one door is a 

car and behind the others are goats. You pick a door-say No. 1 - and the host, who knows what’s behind the 

doors, opens another door-say, No. 3-which has a goat. (In all games, the host opens a door to reveal a goat). 

He then says to you, ”Do you want to pick door No. 2?” (In all games he always offers an option to switch). Is 

it to your advantage to switch your choice?” – The three doors problem, also known as Monty-Hall Problem 

sex 

[sex] Pure mathematician - Anyone who prefers set theory to sex. – R. Ainsley in Bluff your way in Maths, 

1988 

mad 

[mad] There was a mad scientist ( a mad ...social... scientist ) who kidnaped three colleagues, an engineer, a 

physicist, and a mathematician, and locked each of them in separate cells with plenty of canned food and water 

but no can opener. A month later, returning, the mad scientist went to the engineer’s cell and found it long 

empty. The engineer had constructed a can opener from pocket trash, used aluminum shavings and dried sugar 

to make an explosive, and escaped. The physicist had worked out the angle necessary to knock the lids off 

the tin cans by throwing them against the wall. She was developing a good pitching arm and a new quantum 

theory. The mathematician had stacked the unopened cans into a surprising solution to the kissing problem; 

his dessicated corpse was propped calmly against a wall, and this was inscribed on the floor in blood: 

Theorem: If I can’t open these cans, I’ll die. Proof: assume the opposite...

induction 

[induction] Proof by induction - A very important and powerful mathematical tool, because it works by assuming 

something is true and then goes on to prove that therefore it is true. Not surprisingly, you can prove almost 

everything by induction. So long as the proof includes the following phrases: 

a) Assume true for n; then also true for n+1 because.. (followed by some plausible but messy working out in 

which n, n+1 appear prominently). 

b) But is true for n=0 (a little more messy working out with lots of zeros sprayed at random through the proof). 

c) So is true for all n. Q.E.D. 

– R. Ainsley in Bluff your way in Maths, 1988 

horse 

[horse] LEMMA: All horses are the same color. Proof (by induction): Case n=1: In a set with only one horse, 

it is obvious that all horses in that set are the same color. Case n=k: Suppose you have a set of k+1 horses. 

Pull one of these horses out of the set, so that you have k horses. Suppose that all of these horses are the same 

color. Now put back the horse that you took out, and pull out a different one. Suppose that all of the k horses 

now in the set are the same color. Then the set of k+1 horses are all the same color. We have k true =¿ k+1 

true; therefore all horses are the same color. 

THEOREM: All horses have an infinite number of legs. Proof (by intimidation): Everyone would agree that 

all horses have an even number of legs. It is also well-known that horses have fore-legs in front and two legs in 

back. But 4 + 2 = 6 legs is certainly an odd number of legs for a horse to have! Now the only number that is 

both even and odd is infinity; therefore all horses have an infinite number of legs. However, suppose that there 

is a horse somewhere that does not have an infinite number of legs. Well, that would be a horse of a different 

color; and by the Lemma, it doesn’t exist. QED 

dean 

[dean] Dean, to the physics department. ”Why do I always have to give you guys so much money, for laboratories 

and expensive equipment and stuff. Why couldn’t you be like the maths department - all they need is money 

for pencils, paper and waste-paper baskets. Or even better, like the philosophy department. All they need are 

pencils and paper.” 

astronomer 

[astronomer] An astronomer, a physicist and a mathematician were holidaying in Scotland. Glancing from a 

train window, they observed a black sheep in the middle of a field. ”How interesting,” observed the astronomer, 

”all Scottish sheep are black!” To which the physicist responded, ”No, no! Some Scottish sheep are black!” 

The mathematician gazed heavenward in supplication, and then intoned, ”In Scotland there exists at least one 

field, containing at least one sheep, at least one side of which is black.” – J. Steward in ’Concepts of Modern 

Mathematics’

coffee 

[coffee] An engineer, a chemist and a mathematician are staying in three adjoining cabins at an old motel. First 

the engineer’s coffee maker catches fire. He smells the smoke, wakes up, unplugs the coffee maker, throws it out 

the window, and goes back to sleep. Later that night the chemist smells smoke too. He wakes up and sees that 

a cigarette butt has set the trash can on fire. He says to himself, ”Hmm. How does one put out a fire? One 

can reduce the temperature of the fuel below the flash point, isolate the burning material from oxygen, or both. 

This could be accomplished by applying water.” So he picks up the trash can, puts it in the shower stall, turns 

on the water, and, when the fire is out, goes back to sleep. The mathematician, of course, has been watching 

all this out the window. So later, when he finds that his pipe ashes have set the bed-sheet on fire, he is not in 

the least taken aback. He says: ”Aha! A solution exists!” and goes back to sleep. 

logs 

[logs] Taking logs - Broadly speaking, any equation which looks difficult will look much easier when logs are 

taken on both sides. Taking logs on one side only is tempting for many equations, but may be noticed. – R. 

Ainsley in Bluff your way in Maths, 1988 

cat 

[cat] Theorem: A cat has nine tails. Proof: No cat has eight tails. A cat has one tail more than no cat. 

Therefore, a cat has nine tails. 

chocolate 

[chocolate] Prime number - A number with no divisors. Boxes of chocolates always contain a prime number so 

that, whatever the number of people present, somebody has to have that one left over. – R. Ainsley in Bluff 

your way in Maths, 1988 

aleph 

[aleph] Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, You take one down, and pass it around, 

Aleph-null bottles of beer on the wall. 

qed 

[qed] At the end of a proof you write Q.E.D, which stands not for Quod Erat Demonstrandum as the books 

would have you believe, but for Quite Easily Done. – R. Ainsley in Bluff your way in Maths, 1988 

[1+1] 1+1 = 3, for large values of 1 

1+1

painting 

[painting] Group theory - An exceedingly beautiful branch of pure mathematics used for showing in how many 

ways blocks of wood can be painted. – R. Ainsley in Bluff your way in Maths, 1988 

engeneer 

[engeneer] 

Mathematician: 3 is prime,5 is prime,7 is prime, by induction - every odd integer higher than 2 is prime. 

Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error, 11 is prime,... 

Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime,... 

Programmer: 3’s prime, 5’s prime, 7’s prime, 7’s prime, 7’s prime,... 

Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 – we’ll do for you the best we can,... 

Software seller: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release,... 

Biologist: 3 is prime, 5 is prime, 7 is prime, 9 – results have not arrived yet,... 

Advertiser: 3 is prime, 5 is prime, 7 is prime, 11 is prime,... 

Lawyer: 3 is prime, 5 is prime, 7 is prime, 9 – there is not enough evidence to prove that it is not prime,... 

Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime, deducing 10 percent tax and 5 percent other obligations. 

Statistician: Let’s try several randomly chosen numbers: 17 is prime, 23 is prime, 11 is prime... 

Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to suppress it,... 

[pi] PI= 3.14159265358979323846264338327950288419716939937510582097494459230781640628 

[e] Euler E= 2.71828182845904523536028747135266249775724709369995957496696762772407663035 

pi 

e 

cancel 

[cancel] THEOREM: The limit as n goes to infinity of sin x/n is 6. PROOF: cancel the n in the numerator and 

denominator. 

coffee 

[coffee] A mathematician is a device for turning coffee into theorems. – P. Erdos 

stupider 

[stupider] Finally I am becoming stupider no more. – Epitaph, P. Erdos wrote for himself

Erdoes 

[Erdoes] 

epsilon child 

bosses women 

slaves men 

captured married 

liberated divorced 

recaptured remarried 

trivial beings nonmathematicians 

noise music 

poison alcohol 

preaching giving a lecture 

supreme fascist god 

died stopped doing mathematics 

preach lecture 

Joedom UDSSR 

Samland USA 

on the long wave length communists on the short wave length fashists – from the vocabulary of P. Erdos ’the 

man who loved only numbers’ 

Chebyshev 

[Chebyshev] Chebyshev said it, and I say it again There is always a prime between n and 2n – P. Erdos 

Outrage 

[Outrage] Outrage, disgust, the characterization of group theory as a plague or as a dragon to be slain - this is 

not an atypical physist’s reaction in the 1930s-50s to the use of group theory in physics. – S. Sternberg 

digits 

[digits] Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. 

– J. von Neumann 

poet 

[poet] The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors 

or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in 

the world for ugly mathematics... It may be very hard to define mathematical beauty, but that is just as true 

of beauty of any kind - we may not know quite what we mean by a beautiful poem, but that does not prevent 

us from recognizing one when we read it. – G.H. Hardy 

melancholy 

[melancholy] It is a melancholy experience for a professional mathematician to find himself writing about 

mathematics. – G.H. Hardy

Hilbert 

[Hilbert] There is a much quoted story about David Hilbert, who one day noticed that a certain student had 

stopped attending class. When told that the student had decided to drop mathematics to become a poet, 

Hilbert replied, ”Good- he did not have enough imagination to become a mathematician”. – R. Osserman 

refreree 

[refreree] Referee’s report: This paper contains much that is new and much that is true. Unfortunately, that 

which is true is not new and that which is new is not true. – H. Eves ’Return to Mathematical Circles’, 1988. 

weapons 

[weapons] Structures are the weapons of the mathematician. – N. Bourbaki 

undogmatic 

[undogmatic] Mathematics is the only instructional material that can be presented in an entirely undogmatic 

way. – M. Dehn 

solve 

[solve] Each problem that I solved became a rule which served afterwards to solve other problems – R. Decartes 

tool 

[tool] For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the 

main source of concepts and principles by means of which new theories can be created. – F. Dyson 

sheet 

[sheet] If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary 

we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only 

mathematicians have microscopes fine enough to actually observe them. – J. Eving

ecommendation 

[recommendation] Sample letter of recommendation: 

Dear Search Committee Chair, I am writing this letter for Mr. Still Student who has applied for a position 

in your department. I should start by saying that I cannot recommend him too highly. In fact, there is no 

other student with whom I can adequately compare him, and I am sure that the amount of mathematics he 

knows will surprise you. His dissertation is the sort of work you don’t expect to see these days. It definitely 

demonstrates his complete capabilities. In closing, let me say that you will be fortunate if you can get him to 

work for you. Sincerely, A. D. Advisor (Prof.) – from MAA Focus Newsletter 

cube 

[cube] To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers 

of the same denomination above the second is impossible, and I have assuredly found an admirable proof of 

this, but the margin is too narrow to contain it. – P. de Fermat 

reality 

[reality] Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, 

obviously. And matter is made of particles. It’s made of electrons and neutrons and protons. So the entire 

universe is made out of particles. Now what are the particles made out of? They’re not made out of anything. 

The only thing you can say about the reality of an electron is to cite its mathematical properties. So there’s a 

sense in which matter has completely dissolved and what is left is just a mathematical structure. – M. Gardner 

[arithmetic] God does arithmetic. – K.F. Gauss 

arithmetic 

hypothesis 

[hypothesis] Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own 

proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What 

about the degenerate cases? Where does the proof use the hypothesis? – P.R. Halmos 

dice 

[dice] God not only plays dice. He also sometimes throws the dice where they cannot be seen. – S.W. Hawking 

wissen 

[wissen] ’Wir muessen wissen. Wir werden wissen.’ (We have to know. We will know.) – D. Hilbert (engraved 

in tombstone)

physics 

[physics] Physics is much too hard for physicists. – D. Hilbert 

Hofstadter 

[Hofstadter] Hofstadter’s Law: It always takes longer than you expect, even when you take into account Hofstadter’s 

Law. – D.R. Hofstadter, Goedel-Escher-Bach 

experience 

[experience] The science of mathematics presents the most brilliant example of how pure reason may successfully 

enlarge its domain without the aid of experience. – E. Kant 

doughnut 

[doughnut] A topologist is one who doesn’t know the difference between a doughnut and a coffee cup. – J. 

Kelley 

Kovalevsky 

[Kovalevsky] Say what you know, do what you must, come what may. – S. Kovalevsky 

god 

[god] God made the integers, all else is the work of man. – L. Kronecker 

abstract 

[abstract] There is no branch of mathematics, however abstract, which may not some day be applied to phenomena 

of the real world. – N. Lobatchevsky 

medicine 

[medicine] Medicine makes people ill, mathematics make them sad and theology makes them sinful. – M. Luther

intelligence 

[intelligence] The mathematician who pursues his studies without clear views of this matter, must often have 

the uncomfortable feeling that his paper and pencil surpass him in intelligence. – E. Mach 

flesh 

[flesh] I tell them that if they will occupy themselves with the study of mathematics they will find in it the best 

remedy against the lusts of the flesh. – T. Mann 

philosophers 

[philosophers] Today, it is not only that our kings do not know mathematics, but our philosophers do not know 

mathematics and - to go a step further - our mathematicians do not know mathematics. – J.R. Oppenheimer 

obvious 

[obvious] Mathematics consists of proving the most obvious thing in the least obvious way. – G. Polya 

whispers 

[whispers] However successful the theory of a four dimensional world may be, it is difficult to ignore a voice 

inside us which whispers: ”At the back of your mind, you know a fourth dimension is all nonsense”. I fancy 

that voice must have had a busy time in the past history of physics. What nonsense to say that this solid table 

on which I am writing is a collection of electrons moving with prodigious speed in empty spaces, which relative 

to electronic dimensions are as wide as the spaces between the planets in the solar system! What nonsense to 

say that the thin air is trying to cursh my body with a load of 14 lbs. to the square inch! What nonsense that 

the star cluster which I see through the telescope, obviously there NOW, is a glimpse into a past age 50’000 

years ago! Let us not be beguiled by this voice. It is discredited... – Sir Arthur Eddington 

decimal 

[decimal] The first million decimal places of pi are comprised of: 

99959 0’s 

99758 1’s 

100026 2’s 

100229 3’s 

100230 

100359 

4’s 

5’s 

–David Blatner, the joy of pi 

99548 6’s 

99800 7’s 

99985 8’s 

100106 9’s

historians 

[historians] Math historians often state that the Egyptians thought pi = 256/81. In fact, there is no direct 

evidence that the Egyptians conceived of a constant number pi, much less tried to calculate it. Rather, they 

were simply interested in finding the relationship between the circle and the square, probably to accomplish the 

task of precisely measuring land and buildings. –David Blatner, the joy of pi 

[pi] 

2000 BC Babilonians use pi=25/8, Egyptians use pi=256/81 

1100 BC Chinese use pi=3 

200 AC Ptolemy uses pi=377/120 

450 Tsu Ch’ung-chih uses pi=255/113 

530 Aryabhata uses pi=62832/20000 

650 Brahmagupta uses pi=sqrt(10) 

1593 Romanus finds pi to 15 decimal places 

1596 Van Ceulen calculates pi to 32 places 

1699 Sharp calculates pi to 72 places 

1719 Tantet de Lagny calculates pi to 127 places 

1794 Vega calculates pi to 140 decimal places 

1855 Richter calculates pi to 500 decimal places 

1873 Shanks finds 527 decimal places 

1947 Ferguson calculates 808 places 

1949 ENIAC computer finds 2037 places 

1955 NORC computer computes 3089 places 

1959 IBM 704 computer finds 16167 places 

1961 Shanks-Wrench (IBM7090) find 100200 places 

1966 IBM 7030 computes 250000 places 

1967 CDC6600 computes 500000 places 

1973 Guilloud-Bouyer (CDC7600) find 1 Mio places 

1983 Tamura-Kanada (HITACM-280H) compute 16 Mio places 

1988 Kanada (HITAC M-280H) computes 16 Mio digits 

1989 Chudnovsky finds 1000 Mio digits 

1995 Kanada computes pi to 6000 Mio digits 

1996 Chudnovsky computes pi to 8000 Mio digits 

1997 Kanada determines pi to 51000 Mio digits 

–David Blatner, the joy of pi 

pi 

FBI 

[FBI] The following is a transcript of an interchange between defence attorney Robert Blasier and FBI Special 

Agent Roger Martz on July 26, 1995, in the courtroom of the O.J. Simpson trial: 

Q: Can you calculate the area of a circle with a five-millimeter diameter? A: I mean I could. I don’t...math 

I don’t ... I don’t know right now what it is. Q: Well, what is the formula for the area of a circle? A: Pi R 

Squared Q: What is pi? A: Boy, you ar really testing me. 2.12... 2.17... Judge Ito: How about 3.1214? Q: 

Isn’t pi kind of essential to being a scientist knowing what it is? A: I haven’t used pi since I guess I was in 

high school. Q: Let’s try 3.12. A: Is that what it is? There is an easier way to do... Q: Let’s try 3.14. And 

what is the radius? A: It would be half the diameter: 2.5 Q: 2.5 squared, right? A: Right. Q: Your honor, may 

we borrow a calculator? [pause] Q: Can you use a calculator? A: Yes, I think. Q: Tell me what pi times 2.5 

squared is. A: 19 Q: Do you want to write down 19? Square millimeters, right? The area. What is one tenth 

of that? A: 1.9 Q: You miscalculated by a factor of two, the size, the minimum size of a swatch you needed to 

detect EDTA didn’t you? A: I don’t know that I did or not. I calculated a little differently. I didn’t use this. 

Q: Well, does the area change by the different method of calculation? A: Well, this is all estimations based on 

my eyeball. I didn’t use any scientific math to determine it. –David Blatner, the joy of pi

eauty 

[beauty] To those who do not know Mathematics it is difficult to get across a real feeling as to the beauty, 

the deepest beauty of nature. ... If you want to learn about nature, to appreciate nature, it is necessary to 

understand the language that she speaks in. – Richard Feynman in ”The Character of Physical Law” 

Bacon 

[Bacon] All science requires Mathematics. The knowledge of mathematical things is almost innate in us... This 

is the easiest of sciences, a fact which is obvious in that no one?s brain rejects it; for laymen and people who 

are utterly illiterate know how to count and reckon. – Roger Bacon 

deductions 

[deductions] Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is 

true of anything, then such and such another proposition is true of that thing... It’s essential not to discuss 

whether the proposition is really true, and not to mention what the anything is of which it is supposed to 

be true... If our hypothesis is about anything and not about some one or more particular things, then our 

deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know 

what we are talking about, nor whether what we are saying is true. – Bertrand Russell 

ambitious 

[ambitious] The more ambitious plan may have more chances of success – G. Polya, How To Solve It 

fourteen 

[fourteen] THEOREM: Every natural number can be completely and unambiguously identified in fourteen words 

or less. PROOF: 1. Suppose there is some natural number which cannot be unambiguously described in fourteen 

words or less. 2. Then there must be a smallest such number. Let’s call it n. 3. But now n is ”the smallest 

natural number that cannot be unambiguously described in fourteen words or less”. 4. This is a complete 

and unambiguous description of n in fourteen words, contradicting the fact that n was supposed not to have 

such a description! 5. Since the assumption (step 1) of the existence of a natural number that cannot be 

unambiguously described in fourteen words or less led to a contradiction, it must be an incorrect assumption. 

6.Therefore, all natural numbers can be unambiguously described in fourteen words or less! 

1=2 

[1=2] THEOREM: 1=2 PROOF: 

1. Let a = b. 

2. Then a 2 = ab, 

3. a 2 + a 2 = a 2 + ab, 

4. 2a 2 = a 2 + ab, 

5. 2a 2 − 2ab = a 2 + ab − 2ab, 

6. and 2a 2 − 2ab = a 2 − ab 

7. Writing this as 2(a 2 − ab) = 1(a 2 − ab), 

8. and cancelling the (a 2 − ab) from both sides gives 1 = 2.

primes 

[primes] II III V VII XI XIII XVII XIX XXIII XXIX ... 

Queen 

[Queen] ”Can you do addition?” the White Queen asked. ”What’s one and one and one and one and one and 

one and one and one and one and one?” ”I don’t know,” said Alice, ”I lost count.”. – Lewis Carrol alias Charles 

Lutwidge Dodgson, Alice’s Adventures in Wonderland 

subtraction 

[subtraction] ”She can’t do Subtraction”, said the White Queen. ”Can you do Division? Divide a loaf by a knife 

– what’s the answer to that?” ”I suppose –” Alice was beginning, but the Red Queen answerd for her. ”Bread 

and butter, of course ...” – Lewis Carrol alias Charles Lutwidge Dodgson, Alice’s Adventures in Wonderland 

subtraction 

Theorem: the square root x of 2 is irrational. Proof: x=n/m with gcd(n, m) = 1 implies 2 = n 2 /m 2 which is 

2m 2 = n 2 so that n must be even and n 2 a multiple of 4. Therefore m is even. This contradicts gcd(n,m)=1. 

blackboard 

[blackboard] It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a 

sheet of paper could change the course of human affairs. – Stanislaw Ulam. 

ephermeral 

[ephermeral] Of all escapes from reality, mathematics is the most successful ever. It is a fantasy that becomes 

all the more addictive because it works back to improve the same reality we are trying to evade. All other 

escapes- sex, drugs, hobbies, whatever - are ephemeral by comparison. The mathematician’s feeling of triumph, 

as he forces the world to obey the laws his imagimation has created, feeds on its own success. The world is 

premanently changed by the workings of his mind, and the certainty that his creations will endure renews his 

confidence as no other pursuit. – Gian-Carlo Rota 

[joke] A good mathematical joke is better, and better mathematics than a dozen mediocre papers. 

– John Edensor Littlewood 

joke

[Leibniz] pi/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9.... 

– Wilhelm von Leibniz 

Leibniz 

war 

[war] It has been said that the First World War was the chemists’ war because mustard gas and chlorine were 

empolyed for the first time, and that the Second World War was the physicists war, because the atom bomb 

was detonated. Similarly, it has been argued that the Third World War would be the mathematicians’ war, 

because mathematics will have control over the next great weapon of war - information. – Simon Singh, in ’The 

code book’ 

clearly 

[clearly] Never speak more clearly than you think. – Jeremy Bernstein 

Piaget 

[Piaget] What, in effect are the conditions for the construction of formal thought? The child must not only apply 

operations to objects - in other words, mentally execute possible actions on them - he must also ’reflect’ those 

operations in the absence of the objects which are replaced by pure propositions. Thus ’reflection’ is thought 

raised to the second power. Concrete thinking is the representation of a possible action, and formal thinking 

is the representation of a representation of possible action... It is not surprising, therefore, that the system of 

concrete operations must be completed during the last years of childhood before it can be ’reflected’ by formal 

operations. In terms of their function, formal operations do not differ from concrete operations except that they 

are applied to hypotheses or propositions whose logic is an abstract translation of the system of ’inference’ that 

governs concrete operations. – Jean Piaget 

Mersenne 

[Mersenne] An integer 2 n − 1 is called a Mersenne number. If it is prime, it is called a Mersenne prime. In 

that case, n must be prime. Known examples are n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 

1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 

216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377. It is not known whether there are infinitely many 

Mersenne primes. 

Mersenne 

A positive integer n is called a perfect number if it is equal to the sum of all of its positive divisors, excluding n 

itself. Examples are 6=1+2+3, 28=1+2+4+7+14. An integer k is an even perfect number if and only if it has 

the form 2 ( n − 1)(2 n − 1) and 2 n − 1 is prime. In that case 2 n − 1 is called a Mersenne prime and n must be 

prime. It is unknown whether there exists an odd perfect number.

Wilson 

[Wilson] WILSON’S THEOREM: p prime if and only if (p − 1)! == −1 (modp) PROOF. 1, 2, ..., p − 1 are 

roots of x ( p − 1) == 0 (modp). A congruence has not more roots then its degree, hence x ( p − 1) − 1 == 

(x − 1)(x − 2)...(x − (p − 1)) mod p. For x=0, this gives −1 == (−1) ( p − 1)(p − 1)! == (p − 1)! which is also 

true for p=2. 

– from P. Ribenboim, ’The new book of prime number records’ 

twin 

[twin] There is keen competition to produce the largest pair of twin primes. On October 9, 1995, Dubner 

discovered the largest known pair of twin primes p, p + 2, where p = 570918348 ∗ 10 5120 − 1. It took only one 

day with 2 crunchers. The expected time would be 150 times longer! What luck! – from P. Ribenboim, ’The 

new book of prime number records’ 

[lion] How to catch a lion: 

lion 

• THE HILBERT METHOD. Place a locked cage in the desert. Set up the following axiomatic system. (i) 

The set of lions is non-empty 

(ii) If there is a lion in the desert, then there is a lion in the cage. 

Theorem. There is a lion in the cage 

• THE PEANO METHOD. There is a space-filling curve passing through every point of the desert. Such 

a curve may be traversed in as short a time as we please. Armed with a spear, traverse the curve faster 

than the lion can move his own length. 

• THE TOPOLOGICAL METHOD. The lion has a least the connectivity of a torus. Transport the desert 

into 4-space. It can now be deformed in such a way as to knot the lion. He is now helples. 

• THE SURGERGY METHOD. The lion is an orientable 3-manifold with boundary and so may be rendered 

contractible by surgery. 

• THE UNIVERSAL COVERING METHOD. Cover the lion by his simply-connected covering space. Since 

this has no holes, he is trapped. 

• THE GAME THEORY METHOD. The lion is a big game, hence certainly a game. There exists an 

optimal strategy. Follow it. 

• THE SCHROEDINGER METHOD. At any instant there is a non-zero probability that the lion is in the 

cage. Wait. 

• THE ERASTOSHENIAN METHOD. Enumerate all objects in the desert: examine them one by one; 

discard all those that are not lions. A refinement will capture only prime lions. 

• THE PROJECTIVE GEOMETRY METHOD. The desert is a plane. Project this to a line, then project 

the line to a point inside the cage. The lion goes to the same point. 

• THE INVERSION METHOD. Take a cylindrical cage. First case: the lion is in the cage: Trivial. Second 

case: the lion is outside the cage. Go inside the cage. Invert at the boundary of the cage. The lion is 

caught. Caution: Don’t stand in the middle of the cage during the inversion!

Euler 

[Euler] Euler’s formula: A connected plane graph with n vertices, e edges and f faces satisfies n - e + f = 2 

Proof. Let T be the edge set of a spanning tree for G. It is a subset of the set E of edges. A spanning tree is 

a minimal subgraph that connects all the vertices of G. It contains so no cycle. The dual graph G* of G has 

a vertex in the interior of each face. Two vertices of G* are connected by an edge if the correponding faces 

have a common boundary edge. G* can have double edges even if the original graph was simple. Consider 

the collection T* of edges E* in G* that correspond to edges in the complement of T in E. The edges of T* 

connect all the faces because T does not have a cycle. Also T* does not contain a cycle, since otherwise, it 

would seperate some vertices of G contradicting that T was a spanning subgraph and edges of T and T* don’t 

intersect. Thus T* is a spanning tree for G*. Clearly e(T)+e(T*)=2. For every tree, the number of vertices is 

one larger than the number of vertices. Applied to the tree T, this yields n = e(T)+1, while for the tree T* it 

yields f=e(T*)+1. Adding both equations gives n+f=(e(T)+1)+(e(T*)+1)=e+2. – from M.Aigner, G. Ziegler 

”Proofs from THE BOOK” 

irrational 

[irrational] Theorem: e = sum(k) 1/k! is irrational. Proof. e=a/b with integers a,b would imply N = n! (e 

- sum(k¡n+1) 1/k!) is an integer for n¿b because n! e and n!/k! were both integers. However, 0 < N = 

� 

k>n n!/k! = 1/(n + 1) + 1/(n + 1)(n + 2) + ... < 1/(n + 1) + 1/(n + 1)2 + ... = 1/n (second sum is a geometric 

series) for every n is not possible. 

– from M.Aigner, G. Ziegler ”Proofs from THE BOOK” 

Wiener 

[Wiener] After a few years at MIT, the Mathematician Norbert Wiener moved to a larger house. His wife, 

knowing his nature, figured that he would forget his new address and be unable to find his way home after 

work. So she wrote the address of the new home on a piece of paper that she made him put in his shirt pocket. 

At lunchtime that day, the professor had an inspiring idea. He pulled the paper out of his pocket and used it 

to scribble down some calculations. Finding a flaw, he threw the paper away in disgust. At the end of the day 

he realized he had thrown away his address, he now had no idea where he lived. Putting his mind to work, he 

came up with a plan. He would go to his old house and await rescue. His wife would surely realize that he was 

lost and go to his old house to pick him up. Unfortunately, when he arrived at his old house, there was no sign 

of his wife, only a small girl standing in front of the house. ”Excuse me, little girl” he said ”but do you happen 

to know where the people who used to live here moved to?” ”It’s okay, Daddy,” said the little girl, ”Mommy 

sent me to get you”. Moral 1. Don’t be surprised if the professor doesn’t know your name by the end of the 

semester. Moral 2. Be glad your parents aren’t mathematicians. if your parents are mathematicians, introduce 

yourself and get them to help you through the course. - From the introduction of ”How to ace calculus” by C. 

Adams, A. Thompson and J. Hass 

funeral 

[funeral] David Hilbert was one of the great European mathematicians at the turn of the century. One of his 

students purchased an early automobile and died in one of the first car accidents. Hilbert was asked to speak 

at the funeral. ”Young Klaus” he said, ”was one of my finest students. He had an unusual gift for doing 

mathematics. He was insterested in a great variety of problems, such as...” There was a short pause, follwed by 

”Consider the set of differentiable functions on the unit interval and take their closure in the ...” Moral 1. Sit 

near the door. Moral 2. Some mathematicians can be a little out of touch with reality. If your professor falls 

in this category, look at the bright side. You will have lots of funny stories by the end of the semester. - From 

the introduction of ”How to ace calculus” by C. Adams, A. Thompson and J. Hass

abbit 

[rabbit] In a forest a fox bumps into a little rabbit, and says, ”Hi, junior, what are you up to?” ”I’m writing a 

dissertation on how rabbits eat foxes,” said the rabbit. ”Come now, friend rabbit, you know that’s impossible!” 

”Well, follow me and I’ll show you.” They both go into the rabbit’s dwelling and after a while the rabbit emerges 

with a satisfied expression on his face. Along comes a wolf. ”Hello, what are we doing these days?” ”I’m writing 

the second chapter of my thesis, on how rabbits devour wolves.” ”Are you crazy? Where is your academic 

honesty?” ”Come with me and I’ll show you.” ...... As before, the rabbit comes out with a satisfied look on 

his face and this time he has a diploma in his paw. The camera pans back and into the rabbit’s cave and, as 

everybody should have guessed by now, we see an enormous mean-looking lion sitting next to the bloody and 

furry remains of the wolf and the fox. The moral of this story is: It’s not the contents of your thesis that are 

important – it’s your PhD advisor that counts. - Unknown Usenet Source 

poet 

[poet] It is true that a mathematician who is not also something of a poet will never be a perfect mathematician. 

- K. Weierstrass, Quoted in D MacHale, Comic Sections (Dublin 1993) 

equilateral 

[equilateral] THEOREM: All triangles are equilateral. PROOF: 1) Given an arbitrary triangle ABC. Construct 

the middle orthogonal on AB in D and cut it with the line dividing the angle at C. Call the intersection E. 

Form the normal from E to AC in F and from E to BC in G. Draw the lines AE und BE. C * / / *F *G / E* 

/ — / — / —D A*———*————*B 

2. The angles ECF and ECG are gleich. The angles EFC and EGC are both right angles. Because the triangles 

ECF and ECG have also EC common, they must be congruent. Therefore CF=CG and EF=EG. 

3. The sides DA and DB are equal. The angle EDA and EDB are both right angles. Because the triangles EDA 

and EDB have also ED in common, they have to be congruent and EA=EB. 

4. The angle EGB and EFA are both right angle. Also, EF=EG and EA=EB. Therefore both triangles EGB 

and EFA are congruent. Therefore FA=GB. 

5. Since CF=CG and FA=GB, addition of the sides gives also CA=CB. 

6. Having proved that two arbitrary sides are equal, all are equal. 

widow 

[widow] I married a widow, who had an adult stepdaughter. My father, a widow and who often visited us, fell in 

love with my stepdaughter and married her. So, my father became my son-in-law and my stepdaughter became 

my stepmother. But my wife became the mother-in-law of her father-in-law. My stepmother, stepdaughter 

of my wife had a son and I therefore a brother, because he is the son of my father and my stepmother. But 

since he was in the same time the son of our stepdaughter, my wife became his grandmother and I became the 

grandfather of my stepbrother. My wife gave me also a son. My stepmother, the stepsister of my son, is in 

the same time his grandmother, because he is the son of her stepson and my father is the brother-in-law of my 

child, because his sister is his wife. My wife, who is the mother of my stepmother, is therefore my grandmother. 

My son, who is the child of my grandmother, is the grandchild of my father. But I’m the husband of my wife 

and in the same time the grandson of my wife. This means: I’m my own grandfather. 

dots 

[dots] I never could make out what those damned dots meant. – Lord Randolph Churchill (1849-1895) Brittish 

conservative politician, referring to decimal points.

ladder 

[ladder] The mathematician has reached the highest rung on the ladder of human thought. – Havelock Ellis 

ignorant 

[ignorant] Let no one ignorant of mathematics enter here. – Plato, Inscription written over the entrance to the 

academy 

god 

[god] I knew a mathematician, who said ’I do not know as much as God. But I know as much as God knew at 

my age’. – Milton Shulman, Candian writer 

english 

[english] English professor: In English, a double negative makes a positive. In other languages such as Russian, 

a double negative is still a negative. There are, however, no languages in which a double positive makes a 

negative. Student in back of class: ”Yea, right” 



Index 

1+1, 8 

1=2, 16 

abstract, 13 

aleph, 8 

ambitious, 16 

analysis, 1 

analytic, 2 

arithmetic, 12 

astronomer, 7 

axiomatics, 2 

Bacon, 16 

barber, 5 

beauty, 3, 16 

blackboard, 17 

cancel, 9 

Cantor, 5 

cat, 8 

Chebyshev, 10 

chocolate, 8 

clearly, 18 

coffee, 8, 9 

computer, 1 

Conway, 3 

cube, 12 

dean, 7 

decimal, 14 

deductions, 16 

dice, 12 

digits, 10 

dots, 21 

doughnut, 13 

e, 9 

engeneer, 9 

english, 22 

enhusiast, 1 

ephermeral, 17 

equilateral, 21 

Erdoes, 10 

Euler, 20 

experience, 13 

fantasy, 1 

FBI, 15 

flesh, 14 

fourteen, 16 

freedom, 1 

funeral, 20 

geometry, 3 

god, 13, 22 

Hadamard, 5 

hairy-ball, 2 

Hilbert, 11 

historians, 15 

Hofstadter, 13 

23 

horse, 7 

hypothesis, 12 

ignorant, 22 

illiteracy, 4 

induction, 7 

infimum, 4 

intelligence, 14 

irrational, 20 

joke, 17 

jouissance, 5 

Kovalevsky, 13 

ladder, 22 

large, 2 

Leibniz, 18 

lion, 19 

logs, 8 

mad, 6 

magacian, 1 

Mandelbrot, 5 

medicine, 13 

melancholy, 10 

Mersenne, 18 

Monty-Hall, 6 

Moser, 6 

obvious, 4, 14 

Outrage, 10 

painting, 9 

philosophers, 14 

physics, 13 

pi, 9, 15 

Piaget, 18 

poet, 10, 21 

prime, 4 

primes, 17 

qed, 8 

Queen, 17 

rabbit, 21 

reality, 12 

recommendation, 12 

refreree, 11 

royal, 1 

sex, 6 

sheet, 11 

sin, 5 

solution, 1 

solve, 11 

stupider, 9 

subtraction, 17 

tool, 11 

transcendental, 4 

turbulence, 2

twin, 19 

undogmatic, 11 

violence, 3 

war, 18 

weapons, 11 

whispers, 14 

widow, 21 

Wiener, 20 

Wilson, 19 

wine, 6 

wissen, 12 

Zermelo, 3

ENTRY COMPUTABILITY 

[ENTRY COMPUTABILITY] Authors: Oliver Knill: nothing real yet Literature: not yet, some lectures of 

E.Engeler on computation theory 

Church’s theses 

The generally accepted [Church’s theses] tells that everything which is computable can be computed using a 

Turing machine. In that case, the problem to determine, whether a Turing machine will halt, is not computable. 

cipher 

A [cipher] is a secret mode of writing, often the result of subsituting numbers of letters and then carrying out 

arithmetic operations on the numbers. 

Coding theory 

[Coding theory] is the theory of encryption of messages employed for security during the transmission of data 

or the recovery of information from corrupted data. 

Cooks hypothesis 

[Cooks hypothesis] P = NP . A proof or disproof is one of the millenium problems. 

Graph isomorphism problem 

[Graph isomorphism problem] It is not known whether graph isomorphism can be decided in deterministic 

polynomial time. It is an open problem in computational complexity theory. 

Inductive structure 

[Inductive structure] A set U with a subset A and operations g1, ..., gn define an inductive structure 

(U, A, g1, ..., gn) If all elements of U can be generated by repeated applications of the operations gi on elements 

of A. Examples: 

• (N, A = {0, 1}, g1(a, b) = a + b, g2(a, b) = a ∗ b defines an inductive structure. 

• If U = N is the set of natural numbers, A = {1, 2, 3} g1(x, y) = 3x − 4, g2(x, y, z) = 7x + 5y − z, then 

(U, A, g1, g2) define an inductive structure.

syntactic structure 

An inductive structure (U, A, g1, ..., gn) is called a [syntactic structure] if it is uniquely readable that is if 

g1(u1, ..., uk) = g2(v1, ..., vl), then g1 = g2, k = l and u1 = v1, ..., uk = vk. Example: if X is the set of finite 

words in the alphabet {p, q, r, K, N} and A = {p, q, r}. Define g1(x, y) = Kxy and g2(x, y) = Nx and U the 

set of words generated from A. The structure is the language of elementary logic in polnic notation. It is a 

syntactic structure. Syntactic structures are in general described by grammers. 

grammar 

A [grammar] (N, T, G) is given by two sets of symbols N, T and a finite set G of pairs (ni, ti) which define 

transitions ni → ti. For example: N = {S}, T = {K, N, p, q, r}, G = {S → p, S → q, S → r, S → KSS, S → 

NS}. Acoording to Chomsky, one classifies grammers with additional conditions like context sensitivity or 

regularity. 

context sensitive 

A grammer (N, T, G) is called [context sensitive] if (n, t) ∈ G then |t| ≥ |n|. 

context sensitive 



Index 

Church’s theses, 1 

cipher, 1 

Coding theory, 1 

context sensitive, 2 

Cooks hypothesis, 1 

grammar, 2 

Graph isomorphism problem, 1 

Inductive structure, 1 

syntactic structure, 2 

3

ENTRY COMPUTER 

[ENTRY COMPUTER] Authors: Oliver Knill: May 2001 Literature: for video stuff: http://www.doom9.org, 

foldoc 

AAC 

[AAC] Advanced Audio Coding will be the successor of AC3 audio. It is based on AC3 while adding a number 

of improvements in various areas. Currently player and hardware support for this upcoming audio format is 

still very limited. 

acrobat 

[acrobat] A product from Adobe for manipulating documents stored in the PDF (Portable Document Format). 

[amd] Daemon which enables the NFS automount. 

[AMD] Advanced Micro Devices, Chip company. 

amd 

AMD 

arpwatch 

[arpwatch] Daemon to log and buil a database of Ethernet address/IP address pairings it sees on a LAN interface. 

ASCII 

[ASCII] American Standard Code for Information Interexchange, an industry standard, which assigns letters, 

numbers and other characters within the 256 slots available in the 8-bit code. 

AC3 

[AC3] Initially known as Audio Coding 3 AC3 is a synonym for Dolby Digital these days. Dolby Digital is an 

advanced audio compression technology allowing to encode up to 6 separate channels at bitrates up to 448kbit/s. 

For more information please check out the Dolby website.

ASF 

[ASF] Advanced Streaming Format. Microsoft’s answer to Real Media and streaming media in general. 

[AT] keyboard The standard keyboard used with the IBM compatible computer. 

AT 

backdoor 

A [backdoor] is a ”mechanism surreptitiously introduced into a computer system to facilitate unauthorized 

access to the system”. An example of a backdoor is ”bindshell”. 

AVI 

[AVI] Audio Video Interleave. The video format most commonly used on Windows PC’s. It defines how video 

and audio are attached to each other, without specifying a codec. 

Bandwidth 

[Bandwidth] Bandwidth measures how much information can be carried in a given time period over a 

wired or wireless communications link. A typical broadband speed is 1270 Kbps (kilo bit per sec- 

Technology Speed mbit/s 

56k modem 0.056 

DSL varies 

cable varies 

T1 1.544 

Ethernet 10.000 

ond) which is 155.6 KBytes/sec T3 44.736 see http://home.cfl.rr.com/cm3/speedtest7.htm 

OC-3 155.520 

OC-12 622.080 

OC-48 2,488.320 

OC-96 4,976.640 

OC-192 9,953.280 

OC-255 13,219.200 

http://jetstreamgames.co.nz/speed/ADSLdownload1MB.html http://home.cfl.rr.com/eaa/Bandwidth.htm 

BUP file 

[BUP file] A bup file is a Back UP file of an IFO file. These files are commonly found on DVDs. 

Byte 

One [Byte] is an information unit of a sequence of 8 bits.

[CASE]: Computer Aided Software Engineering. 

CASE 

Cell (ID) 

[Cell (ID)] A cell is the smallest video unit on a DVD. Normally used to contain a chapter it can also be used 

to contain a smaller unit in case of multiangles or seamless branching titles. 

certificate 

A [certificate] is a digital identifcation of a physical or abstract object, a person, business, computer, program 

or document. A digital certificate is much like a passport. It is issued by a certificate authority, which vouches 

for its authenticity. 

Codec 

[Codec] COder/DECoder. A codec is a piece of software that allows to encode something - usually audio or 

video - to a specific format and can decode media encoded in this specific format again. Popular Codecs are 

MPEG1, MPEG2, MPEG-4 (=divx=xvid), realvideo, wmv, dv Indeo, etc. MPEG, AVI, ASF, Quicktime is 

not a codec but a container format - that can be encoded using different codecs. In avi container files, there’s 

mostly mpeg4 video content and mp3 audio content but this is not obligatory. For DVD, the video should be 

in mpg2, the audio in mp2 and both of these will be in a mpeg-ps (program stream aka ”vob”) container. 

Container 

[Container] A container is, like the name says, a construct to contain data - in this case video and audio date 

and possibly subtitles and navigational information. For instance, you would like to put a soundless video 

stream and the audio track together in one file. To do that you need a container format. Examples of container 

formats are: AVI, ASF, OGM, Quicktime, VOB and MPG. In avi container files, there’s mostly mpeg4 video 

content and mp3 audio content but this is not obligatory. For DVD, the video should be in mpg2, the audio 

in mp2 and both of these will be in a mpeg-ps (program stream aka ”vob”) container. A [cookie] is a block of 

information recorded and stored within the client’s browser. 

CSS 

[CSS] Cascading Style Sheets is a simple mechanism for adding style (e.g. fonts, colors, spacing) to Web 

documents. For example: 

body, table font-family: verdana, arial, geneva, sans-serif;

CSS 

[CSS] Content Scrambling System. Prioprietary scrambling system for video DVDs. Designed to stop people 

from making copies of DVDs, most commercial DVDs are encrypted using CSS. During playback, DVDs are 

then decrypted on the fly. Only parts of the DVD are encrypted (for instance all IFO and BUP files are not 

encrypted, and VIDEOTS.VOB often isn’t encrypted either) and the encryption scheme is rather weak and was 

quickly defeated. If you want to know what CSS does, insert a DVD video disc into your PC, start playing the 

disc using a software DVD player, then close the player. Now copy a 0.99GB VOB file from the disc to your 

harddisk and try to play back that VOB file in your software DVD player. You’ll see a lot of funny colored 

blocks all over the picture making the movie unwatchable. But you’ll also see parts of the movie (the parts that 

are not encrypted). 

DAR 

[DAR] Display Aspect Ratio. Indicates the dimension of a screen. Most PC screens have a DAR of 4:3, meaning 

that the horizontal size is 4/3 as large as the vertical size. For TVs we have a lot of old 4:3 displays and more 

and more 16:9 displays. As you can guess from the numbers 16:9 displays are broader than 4:3 displays having 

the same diagonal size. 16:9 screens are more suited to display Hollywood movies which are usually shot with 

an aspect ratio of 1:2.35 or 1:1.85 (meaning that the horizontal size of the picture is 1.85 times as wide as the 

vertical size). 

Deinterlace 

[Deinterlace] The process of restoring a progressive video stream out of an interlaced one is called deinterlacing. 

Demultiplexing 

[Demultiplexing] The opposite of multiplexing. In this process a combined audio/video stream will be separated 

into the number of streams it consists of (a video stream, at least one audio stream and a navigational stream). 

Every VOB encoder demultiplexes the VOB files before encoding (FlaskMpeg, mpeg2avi, dvd2mpg, ReMpeg2) 

and every DVD player does the same (audio and video are being treated by different circuits, or decoded by 

different filters on a PC). 

Descrambling 

[Descrambling] DVDs are usually CSS scrambled - imagine you decide to give a number to each letter, starting 

with 1 for a, etc. A sentence would become a couple of digits - that’s what we call scrambled. Of course CSS is 

much better than that but it’s still quite easy to crack. Descrambling means reversing the scrambling process, 

rendering our digits to a sentence again, or making our movie playable again - you can try to copy a movie to 

your hard disk when you’ve authenticated your DVD drive and play it, you’ll get a garbled picture because it’s 

still scrambled. Common CSS descramblers either use a pool of known descrambling keys (DeCSS or DODSrip 

- they contain a large number of keys but not all of them) or try to derive the key by a cryptographic attack 

(VobDec - that’s why it works on most disc since it’s not dependent on a pool of discs).

Digital Video 

[Digital Video] Digital video is usually compressed. Since standard loss less compression is insufficient for video, 

the video codecs have to get rid of unimportant information - stuff the human eye won’t see or is unlikely to 

see. Since that is still not enough modern compression algorithms use keyframes, I and P frames in order to 

save space. 

DivX 

[DivX] There are 2 flavors of DivX today: DivX is the name of the hacked Microsoft MPEG4 codecs (Windows 

Media Video V3). Those codecs were developed by Microsoft for use in its proprietary Windows Media architecture 

and initially supported encoding AVIs and ASFs but all non-beta versions included an AVI lock, making it 

impossible to use them to encode to the AVI format - and only a few tools support ASF today. What the makers 

of DivX did is remove that AVI lock making it possible to encode to AVI again, and changed the name to DivX 

video in order to prevent confusion of codecs, since it’s possible to have both the unhacked and hacked codecs 

on the same computer if you use the Windows Media Encoder. The latest releases of DivX also include a hacked 

Windows Media Audio Codec called DivX audio - the hack of that codec is not perfect yet and its use is limited 

for higher bitrates. This codec is also known as DivX3. The other DivX is a brand-new MPEG-4 video codec 

developed by DivXNetworks. It offers much advanced encoding controls and 2 pass encoding. Furthermore the 

codec can play the old DivX3 movies. The codec is commonly called DivX4. 

DHCPD 

[DHCPD] Daemon to service which can dynamically assign IP addresses to its client hosts. 

DOM 

[DOM] The Document Object Model is a platform- and language-neutral interface that will allow programs and 

scripts to dynamically access and update the content, structure and style of documents. 

DOS 

[DOS] is a Disk operating system, based on a command line user interface. MS-DOS 1.0 was released in 1981 

for IBM computers. While MS-DOS is not much used by itself today, it still can be accessed from Windows 95, 

Windows 98 or Windows ME by clicking Start/Run and typing command or CMD in Windows NT, 2000 or 

XP. 

DRC 

[DRC] Dynamic Range Compression. AC3 Tracks contain a much larger dynamic range that most audio 

equipment can handle, therefore most standalone and software DVD player will compress the dynamic range 

somewhat, according to the actual dynamic range. In layman terms the volume will be augmented dynamically, 

e.g. explosions won’t become louder or only a bit louder, whereas in normal dialogues the volume will be 

augmented quite a bit. Since your player will do the same this is the way to go to have augmented volume.

[DTML] document template markup language. 

[DTP] Desktop publishing. 

DTML 

DTP 

Dynamic HTML 

[Dynamic HTML] is a term used by some vendors to describe the combination of HTML, style sheets and scripts 

that allows documents to be animated. 

Elementary Stream (ES) 

[Elementary Stream (ES)] An elementary stream is a single (video or audio) stream without container. For 

instance a basic MPEG-2 video stream (.m2v or .mpv) is an MPEG-2 ES, and on the audio side we have AC3, 

MP2, etc files that are ES. Most DVD authoring program require ES as input. 

[EULA] End user licence agreement. 

EULA 

FAT 

[FAT] File allocation table. Filesystem used by Windows. Example: Windows 95 users rely on the FAT 16, In 

1996 Microsoft introduced the FAT 32 file system, which is still very widely used today besides NTFS on the 

windows platform. 

FUD 

[FUD] stands for Fear, Uncertainty, Doubt. It is a marketing technique used when a competitor launches a 

product that is both better than yours and costs less, i.e. your product is no longer competitive. Unable 

to respond with hard facts, scare-mongering is used via ’gossip channels’ to cast a shadow of doubt over the 

competitors offerings and make people think twice before using it. 

GUI 

[GUI] - Graphical User Interface; A desktop-like interface usually containing icons, menus and windows. Invented 

by Xerox, later ”borrowed” by Microsoft and Apple.

HTML 

[HTML] Hypertext markup language. Will be replaced by XHTML, and XHTML 2.0 in particular. 

[HTTP] Hypertext Transfer Protocol. 

[HTTPD] Daemon to Apache webserver. 

HTTP 

HTTPD 

Hypertext 

[Hypertext] - shortcuts or links between different parts of a document, article, website or world wide web. 

While early hypertext formats were already Apples Hypercard, it is now common in HTML (Hypertext markup 

language). 

inetd 

[inetd] Daemon which is at the heart of providing network services like telnet or ftp. 

IFO file 

[IFO file] InFOrmation file commonly found on DVDs. Such files contain navigational information for DVD 

players. 

Interlaced 

[Interlaced] Interlaced is a video storage mode. An interlaced video stream doesn’t contain frames but fields 

with each field containing either even or odd lines of one frame. 

IP 

[IP] Internet Protocol. Standard which defines the structure of a message sent between two computers over the 

network. 

[IPFW] IP firewall. 

IPFW

ICMP 

[ICMP] Internet Control Message Protocol. ICMP messages contain information about communication between 

two computers. 

Java 

[Java] is a true compiler-based, low level programming language. [Javascript] is a scripting programming 

language. It was developed by Netscape and used to create interactive Web sites. JavaScript is a popular 

client-side scripting language because it is supported by virtually all browsers. 

KISS 

[KISS] - Keep It Simple Stupid. Rule of thumb for software designers. Keep design small to minimize confusion. 

LDAP 

[LDAP] Lightweight Directory Access Protocol. A network directory which can substitute DNS and much more. 

Not to be confused with a database. A directory is mostly looked up and not written often into. 

LDIF 

[LDIF] LDAP interchange Format is a standard text file for storing LDAP configuration information and 

directory contents. 

MathML 

[MathML] is a low-level specification for describing mathematics as a basis for machine to machine communication. 

It provides a foundation for the inclusion of mathematical expressions in Web pages. 

miniDVD 

[miniDVD] Basically a DVD on a CD. A miniDVD can contain bitrates up to 10mbit/s (audio and video 

combined). Video is MPEG2, preferably VBR and audio can be MPEG1 audio layer 2, raw uncompressed PCM 

or AC3. Video quality can be up to an actual DVD level if a limited playtime is accpted. 

MPEG 

[MPEG] MPEG means Motion Picture Expert Group and it’s the resource for video formats in general. This 

group defines standards in digital video, among it the MPEG1 standard (used in Video CDs), the MPEG2 

standard (used on DVDs and SVCDs), the MPEG4 standard and several audio standards - among them MP3 

and AAC. Files containing MPEG-1 or MPEG-2 video often use either the .mpg or .mpeg extension.

MPEG4 

[MPEG4] Is pretty much a collection of standards defined by the MPEG Group, and it should become the next 

standard in digital video. MPEG4 allows the use of different encoding methods, for instance a keyframe can be 

encoded using ICT or Wavelets resulting in different output qualities. 

MPG 

[MPG] MPG can be either an abbreviation for MPEG or is used as a file extension for MPEG-1 and MPEG-2 

video data. It is a container to contain MPEG-1/2 video stream and MPEG1 layer 2 audio (aka mp2 files). 

MPG containers are also refered to as program streams (PS). 

MM4 

[MM4] Multiple MPEG 4: A combination of different bitrate encoded files. For instance you could take a 

2000kbit/s encode, a 910kbit/s encode and combine the files together, use the lower bitrate file and replace 

scenes where the quality gets too bad due to a lot of action with the parts taken from the 2000kbit/s one. 

NAT 

[NAT] Network Address translation. A typical home user with broadband access and router performs Network 

Address Translation, or NAT allowing multiple computers to share a single fast Internet connection. 

.Net 

[.Net] A collectionof technologies pushed by Microsoft. It contains C# programming language (an alternative 

to Java). Part of the .Net initiative. It builds on standards like XML and SOAP. 

Network layers 

[Network layers] Application layer: Client and server programs. Transport layer: TCP and UDP protocols, 

service ports Network layer: IP packets, IP addresses, ICMP messsages Data link layer: Ethernet frames and 

MAC addresses Physical layer: Copper wire, fiberoptic cable, radio 

Newbie 

[Newbie] - (Also n00b and newb) a newcomer to a certain computer topic or program asking help from experienced 

user. 

[NFS] Network file system. 

NFS

OGM 

[OGM] OGM stands for OGg Media which is the name of the Ogg container implementation by Tobias Waldvogel. 

OGM can be used as an alternative to the AVI container and it can contain Ogg Vorbis, MP3 and AC3 

audio, all kinds of video formats, chapter information and subtitles. 

Perl 

[Perl] Perl is a high-level programming language. It derives from the C programming language and to a lesser 

extent from sed, awk, the Unix shell, and at least a dozen other tools and languages. Perl’s process, file, 

and text manipulation facilities make it particularly well-suited for tasks involving quick prototyping, system 

utilities, software tools, system management tasks, database access, graphical programming, networking, and 

world wide web programming. These strengths make it especially popular with system administrators and CGI 

script authors, but mathematicians, geneticists, journalists, and even managers also use Perl. 

PHP 

[PHP] PHP is a widely-used general-purpose scripting language that is especially suited for Web development 

and can be embedded into HTML. 

Pocket PC 

[Pocket PC] Operating system for handhelds. Usually running Microsoft CE or the Palm OS. 

PNG 

[PNG] is graphics file format for the lossless, portable, well-compressed storage of raster images. Indexed-color, 

grayscale, and truecolor images are supported, plus an optional alpha channel for transparency. Sample depths 

range from 1 to 16 bits per component (up to 48bit images for RGB, or 64bit for RGBA). 

Python 

[Python] is an interpreted, high-level, object-oriented programming language. 

QNX 

[QNX] is a realtime, microkernel, preemptive, prioritized, message passing, network distributed, multitasking, 

multiuser, fault tolerant operating system.

Qt 

[Qt] (”kjut”) Multi platform toolkit and graphics library. Developed by Trolltech. Runs on Windows systems 

including XP, all unix derivates with X windows as well as Mac OS X. 

PCMCIA 

[PCMCIA] Personal Computer Memory Card International Association. 

[RDBM] relational data base manager. 

RDBM 

rff/tff 

[rff/tff] RFF means repeat first frame, it’s a technique used to make the necessary 29.97 frames per second 

out of a 24 frames per second source - the movie like it was recorded with a traditional movie camera used by 

Hollywood. The rff flag tells the player to repeat one field of the video stream. Tff means top field first and is 

also used to perform a telecine to make a 24fps movie into 29.97fps. 

Real time operating systems 

[Real time operating systems] Operating systems which are used in handhelds, robots, telphone switches. Examples: 

QNX, VxWorks (as used in Mars rovers), Windows CE (as used in handhelds), Nucleus RTX. Realtime 

systems must function reliably in event of failures. It is said that the three most important things in Realtime 

system design are timing, timing and timing. 

Ripping 

[Ripping] Ripping means copying a DVD movie to the hard disk of the computer. This includes the authentication 

process for the DVD Drive and the actual CSS Descrambling. CSS (Content Scrambling System) is a copy 

protection scheme designed to prevent unauthorized copying of DVD movies, although many argue that it was 

also designed to control where DVD movies can be played since without a CSS license you essentially have to 

crack the encryption to play a DVD movie. The term ”ripping” is also often used to describe the whole process 

of descrambling a DVD, then convert the audio and video into another format. 

RSS 

[RSS] is a method of distributing links to content in a web site so that others can use use it. It’s a mechanism 

to ”syndicate” the content. The original RSS, version 0.90, was designed by Netscape as a format for building 

portals of headlines to mainstream news sites. RSS is an acronym for Really Simple Syndication.

RTFM 

[RTFM] - Read the fucking manual. Common answer to basic and often repeated questions, that could be 

avoided in the first place just by looking at the manual. 

RSS 

Really Simple Syndication [RSS] is an XML-based format for content distribution. For example, News.com 

offers several RSS feeds with headlines, descriptions and links back to News.com for the full story. [ROUTER] 

a machine designed to direct packets from their source host to their destination. 

RDBMS 

[RDBMS] A Relational Database Management System. Stores data related tables. A single database can be 

spread across several tables unlike flat-file databases where each database is self-contained in a single table. 

SBC 

[SBC] Smart Bitrate Control. A new kind of DivX encoder called Nandub can modify many internal codec 

parameters on the fly during compression, giving you better quality and a lot more control over the encoding 

session. More information can be found in the SBC guide in the DivX guides section. 

SGML 

[SGML] The standard Generalized Markup Language (SGML) is a meta Markup Languages like XML. They are 

used for defining markup languages. A markup language defines using SGML or XML has a specific vocabulary. 

SMIL 

[SMIL] Synchronized Multimedia Integration Language. It enables simple authoring of interactive audiovisual 

presentations. SMIL is typically used for ”rich media”/multimedia presentations which integrate streaming 

audio and video with images, text or any other media type. SMIL is an easy-to-learn HTML-like language, and 

many SMIL presentations are written using a simple text-editor. 

[SOAP] Simple object access protocol. 

[SQL] structured query language. 

SOAP 

SQL

Sun ONE 

[Sun ONE] Sun Open net initiative. Answer to .Net initiative of Microsoft. Has Java, XML and SOAP as 

foundation. 

SVG 

[SVG] The Scalable Vector Graphics is a language for describing two-dimensional graphics in XML. SVG 

allows for three types of graphic objects: vector graphic shapes, images and text. Graphical objects can be 

grouped, styled, transformed and composited into previously rendered objects. The feature set includes nested 

transformations, clipping paths, alpha masks, filter effects and template objects. 

TCP 

[TCP] Transmission control protocol. An IP message type. Most network services run over TCP. A typical 

TCP connection is visiting a remote web site. 

TCPA 

[TCPA] (Trusted Computing Platform Architecture) belongs to DRM (Digital rights management). TCPA 

aims at integrity of kernel and system components - to assure you that your system can be trusted. Palladium, 

on the other hand, uses similar technology to make sure that the user does not do anything else than what is 

allowed by content owners. 

TLD 

[TLD] Top level domain. The last entry in a webaddress. The TLD of www.w3c.org is ”org”. In the 1980s, 

seven TLDs (.com, .edu, .gov, .int, .mil, .net, and .org) were created. Later four of the new TLDs (.biz, .info, 

.name, and .pro) as well as sponsered TLD’s .aero, .coop, and .museum) were created. TLDs with two letters 

(such as .de, .mx, and .jp) have been established for over 240 countries and external territories and are referred 

to as ”country-code” TLD. 

UDP 

[UDP] User Datagramm Protocol. Sendes transport-level data between two network-based programs. For 

example, internet-time servers are assigned UDP services. 

UML 

[UML] Unified modelling language is a language for specifying, visualizing, constructing, and documenting 

software systems, as well as for business modeling and modeling of other non-software systems.

VCD 

[VCD] Video CD, works on many DVD players, there are software players on almost every operating systems, 

doesn’t need a fast computer but the image is VHS-like. Video is MPEG1 at 1150kbit/s and audio MPEG1 

audio layer 2 at 224kbit/s. 

[VLDB] Very large data base. 

[VML] Vector Markup Language 

VLDB 

VML 

W3C 

[W3C] The World Wide Web Consortium (W3C) develops interoperable technologies (specifications, guidelines, 

software, and tools) to lead the Web to its full potential. W3C is a forum for information, commerce, 

communication, and collective understanding. 

Wavelets 

[Wavelets] Wavelets are an alternative basis space. There are infinitely many wavelet bases (Daubechies, Haar, 

Mexican Hat, ”Spline”, Zebra, etc), but their primary feature is that they are localized. Fourier basis functions 

span all space (from negative to positive infinity). Wavelets are basically individual pulses of waves (at various 

positions and scales). Their value in compression stems from factors like the grouping which generally shows 

that a good 90filters, with the high-pass filters generally showing very small values that are mostly details. (of 

course, this is not true if the source is noisy in the first place). For images, the greatest value comes from 

localization of the basis, which means that we can model discontinuities (e.g. edges) VERY well with wavelets. 

You will NOT get those weird JPEG halos if you use wavelets. 

WebDAV 

[WebDAV] Web-based Distributed Authoring and Versioning. A set of extensions to the HTTP protocol which 

allows users to collaboratively edit and managa files on remote web servers. 

Widget 

[Widget]- objects that make up interfaces, i.e. mouse, menus, textbox, buttons; basic tools and objects.

Windows Media 

[Windows Media] Microsoft’s proprietary architecture for audio and video on the PC. It’s based on a collection 

of codecs which can be used by the WindowsMedia Player to play files encoded in any supported format. 

WindowsMedia 7.0 offers a new set of codecs, among them a fully ISO compliant MPEG4 codec (called MS 

Windows Video V1), an improved MPEG-4 codec called MS Video V7 (although I did not notice any improvement 

compared with MS Windows Video V3 on which DivX is based), an encoder that supports Deinterlacing 

and Inverse Telecine. 

Win FS 

[Win FS] Windows Future Storage file system, planned in Windows Longhorn, the successor of Windows XP. 

WYSIWYG 

[WYSIWYG]- What You See Is What You Get. Usually to distinguish document authoring tools. Writing 

a Latex file as a text document is not WYSIWYG, while authoring a word document is. Writing a HTML 

document with a text editor is not WYSIWYG, while writing it with an authoring tool is. 

WORM 

[WORM] a program that connects to other machines and replicates itself. Worms have the potential to both 

damage infected machines and to interfere with networks and services due to congestion caused by the spread 

of the worm. 

WORM 

XHTML 2 is a general purpose markup language designed for representing documents for a wide range of 

purposes across the World Wide Web. To this end it does not attempt to be all things to all people, supplying 

every possible markup idiom, but to supply a generally useful set of elements. Here is an element of a XHTML 2.0 

document: ¡?xml version=”1.0” encoding=”UTF-8”?¿ ¡!DOCTYPE html PUBLIC ”-//W3C//DTD XHTML 

2.0//EN” ”TBD”¿ ¡html xmlns=”http://www.w3.org/2002/06/xhtml2” xml:lang=”en”¿ ¡head¿ ¡title¿Virtual 

Library¡/title¿ ¡/head¿ ¡body¿ ¡p¿Moved to ¡a href=”http://vlib.org/”¿vlib.org¡/a¿.¡/p¿ ¡/body¿ ¡/html¿ 

XML 

[XML] The Extensible textbased Markup Language is a format for structured documents and data on the Web. 

It is derived from SGML (ISO 8879). XML is also playing an increasingly important role in the exchange of a 

wide variety of data on the Web. 

XviD 

[XviD] XviD is a word play, read it the reverse way and you might find a familiar term. XviD is an open source 

MPEG-4 codec which depending on whom you’re asking yields even better quality than the best DivX codec.

zombie 

[zombie] A unix process that has died but has not yet relinquished its process table slot. The parent process 

hasn’t executed a ”wait” for it yet). 



Index 

.Net, 9 

AAC, 1 

AC3, 1 

acrobat, 1 

AMD, 1 

amd, 1 

arpwatch, 1 

ASCII, 1 

ASF, 2 

AT, 2 

AVI, 2 

backdoor, 2 

Bandwidth, 2 

BUP file, 2 

Byte, 2 

CASE, 3 

Cell (ID), 3 

certificate, 3 

Codec, 3 

Container, 3 

CSS, 3, 4 

DAR, 4 

Deinterlace, 4 

Demultiplexing, 4 

Descrambling, 4 

DHCPD, 5 

Digital Video, 5 

DivX, 5 

DOM, 5 

DOS, 5 

DRC, 5 

DTML, 6 

DTP, 6 

Dynamic HTML, 6 

Elementary Stream (ES), 6 

EULA, 6 

FAT, 6 

FUD, 6 

GUI, 6 

HTML, 7 

HTTP, 7 

HTTPD, 7 

Hypertext, 7 

ICMP, 8 

IFO file, 7 

inetd, 7 

Interlaced, 7 

IP, 7 

IPFW, 7 

Java, 8 

KISS, 8 

17 

LDAP, 8 

LDIF, 8 

MathML, 8 

miniDVD, 8 

MM4, 9 

MPEG, 8 

MPEG4, 9 

MPG, 9 

NAT, 9 

Network layers, 9 

Newbie, 9 

NFS, 9 

OGM, 10 

PCMCIA, 11 

Perl, 10 

PHP, 10 

PNG, 10 

Pocket PC, 10 

Python, 10 

QNX, 10 

Qt, 11 

RDBM, 11 

RDBMS, 12 

Real time operating systems, 11 

rff/tff, 11 

Ripping, 11 

RSS, 11, 12 

RTFM, 12 

SBC, 12 

SGML, 12 

SMIL, 12 

SOAP, 12 

SQL, 12 

Sun ONE, 13 

SVG, 13 

TCP, 13 

TCPA, 13 

TLD, 13 

UDP, 13 

UML, 13 

VCD, 14 

VLDB, 14 

VML, 14 

W3C, 14 

Wavelets, 14 

WebDAV, 14 

Widget, 14 

Win FS, 15 

Windows Media, 15 

WORM, 15 

WYSIWYG, 15

XML, 15 

XviD, 15 

zombie, 16

N[FromContinuedFraction[Table[k,k,0,100]]] 

FromContinuedFractionTablek,k,0,100 



ENTRY CONSTANTS 

[ENTRY CONSTANTS] Authors: Oliver Knill: March 2000 - March 2004 Literature: Some from Mario Livio 

”The golden ratio”, www.mathworld.com David Wells: ”The Penguin Dictionary of Curious and Interesting 

Numbers”. 

Archimedes Constant, pi 

The [Archimedes Constant, pi] π = 3.14159 is the length of a half circle with radius 1. It is the area of a disc 

of radius 1. 

Bruns constant 

[Bruns constant] is the sum of the reciprocals of all twin primes. Brun has proven that this sum converges 

evenso it is unknown whether there are infinitely many twin primes. 

Catalan constant 

The [Catalan constant] is defined as the sum (−1) n /(2n + 1) 2 = 0.91596. 

Champernown’s number 

[Champernown’s number] is 0.12345678910111213... whose digits are those of all natural numbers in succession. 

Continued fraction constant 

[Continued fraction constant] is the number with continued fraction (0, 1, 2, 3, 4, 5, 6, ...) it is about 0.697774658. 

Euler Mascheroni constant 

[Euler Mascheroni constant] is defined as the limit of (1 + 1/2 + 1/3 + ... + 1/n) − log(n) as n goes to infinity. 

Aperi constant 

[Aperi constant] It is an irrational number ζ(3) = 1.20206, the value of the zeta function at 3. [Feigenbaum 

constant] When iterating maps f(x) = ax(1 − x) on the unit interval the stable periodic orbits bifurcate when 

varing a. If an are the bifurcation values, then δ = lim(an − an−1)/(an+1 − an) is a Feigenbaum constant.

golden ratio 

The [golden ratio] is τ = (1 + √ 5)/2 = 0.618... If 1, 1, 2, 3, 5, 8, 13, 21... are the Fibonnachi numbers (the next 

number is always the sum of the two previous once), then the ratio of neighboring entries approaches the 

golden mean. 13/21 = 0.61904 is already quite close to the golden mean. The Golden ratio has the continued 

fraction expansion [1, 1, 1, 1...] which means that the number can be written as τ = 1 + 1/(1 + 1/(1 + ...)). The 

golden mean is an example of a Diophantine number, a number which can not be approximated well by rational 

numbers. Especially, it is irrational. The golden ratio is also called ”golden mean” or ”divine constant”. 

The [golden mean] see golden ratio. 

golden mean 

Khinchin constant 

The [Khinchin constant] is defined as the limit (a1a2...an) 1/n where [a1, a2, ...] is the continued fraction of a 

random number in the sense that the limit is known to exist for almost all real numbers. It is not known for 

example, if π is a typical number in the sense that it produces the Khinchin constant. 

natural logarithmic base 

The [natural logarithmic base] e = 2.7182818... can be defined as exp(1) = 1 + 1/1! + 1/2! + 1/3! + ... or 

limn→∞(1 + 1/n) n . 

number of the beast 

The [number of the beast] is the integer 666. The ”beast” is associated with the ”antichrist”. The origin of 

the association is the bible: the book of revelations (13:18) reads: ”this calls for wisdom: let anyone with 

understanding calculate the number of the beast, for it is the number of a man. Its number is six hundred and 

sixty six.” 

Pythagoras constant 

The [Pythagoras constant] is the square root of 2 x = √ 2 = 1.41421.... It is the length of the diagonal of the 

unit square. It is irrational because x = p/q would imply 2q 2 = x 2 q 2 = p 2 , which is impossible because the 

prime factorization on the left contains an odd number of 2’s, while it contains an even number of 2’s on the 

right. 

Smith number 

[Smith number] Smith numbers are integers n such that the sum of its digits in the decimal exapansion of n is qual 

to the sum of the digits of its prime factorization, excluding 1. Smith numbers were defined by A. Wilansky. He 

called it Smith numbers after his brother in law H. Swmith, whose telephone number 4937775 = 3 ∗ 5 ∗ 5 ∗ 65 ′ 837 

is a Smith number. Here are the first Smith numbers: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265....

Wallis constant 

The [Wallis constant] is the real solution to the polynomial x 3 −2x+5 which is 2.0945514815.... This equation was 

solved by the English mathematicaian John Wallis [1616-1703] to illustrate Newton’s method for the numerical 

solution of equations. It has since served as a test for many subsequent methods of approximation. 

Zero 

[Zero] The integer zero 0 is the neutral element in the additive group of integers n + 0 = n. 



Index 

Aperi constant, 1 

Archimedes Constant, pi, 1 

Bruns constant, 1 

Catalan constant, 1 

Champernown’s number, 1 

Continued fraction constant, 1 

Euler Mascheroni constant, 1 

golden mean, 2 

golden ratio, 2 

Khinchin constant, 2 

natural logarithmic base, 2 

number of the beast, 2 

Pythagoras constant, 2 

Smith number, 2 

Wallis constant, 3 

Zero, 3 

4

ENTRY CURVES 

[ENTRY CURVES] Authors: Oliver Knill, Andrew Chi, 2003 Literature: www.mathworld.com, 

www.2dcurves.com 

astroid 

An [astroid] is the curve t ↦→ (cos 3 (t), a sin 3 (t)) with a > 0. An asteroid is a 4-cusped hypocycloid. It is 

sometimes also called a tetracuspid, cubocycloid, or paracycle. 

Archimedes spiral 

An [Archimedes spiral] is a curve described as the polar graph r(t) = at where a > 0 is a constant. In words: 

the distance r(t) to the origin grows linearly with the angle. 

bowditch curve 

The [bowditch curve] is a special Lissajous curve r(t) = (asin(nt + c), bsin(t)). 

brachistochone 

A [brachistochone] is a curve along which a particle will slide in the shortest time from one point to an other. 

It is a cycloid. 

Cassini ovals 

[Cassini ovals] are curves described by ((x + a) + y 2 )((x − a) 2 + y 2 ) = k 4 , where k 2 < a 2 are constants. They 

are named after the Italian astronomer Goivanni Domenico Cassini (1625-1712). Geometrically Cassini ovals 

are the set of points whose product to two fixed points P = (−a, 0), Q = (0, 0) in the plane is the constant k 2 . 

For k 2 = a 2 , the curve is called a Lemniscate. 

cardioid 

The [cardioid] is a plane curve belonging to the class of epicycloids. The fact that it has the shape of a heart 

gave it the name. The cardioid is the locus of a fixed point P on a circle roling on a fixed circle. In polar 

coordinates, the curve given by r(φ) = a(1 + cos(φ)). 

catenary 

The [catenary] is the plane curve which is the graph y = c cosh(x/c). It was discovered by Jacques Bernoulli. 

It has the shape of a uniform flexible chain hung from two points.

catenoid 

The [catenoid] is the surface obatained by rotating the catenary about the x-axis. The minimal surface bounded 

by two coaxial rings can be a catenoid. 

circle 

A [circle] in the plane is the curve r(t) = (r cos(t), r sin(t)) where the radius r is a constant. It is the set of 

points which have a fixed distance r from the origin. More generally, a circle is the set of points in a metric 

space which have a fixed distance from a given point. 

cissoid 

A [cissoid] is a plane curve given in Euclidean coordinates by y 2 (2a − x) = x 3 . In polar coordinates, it satisfies 

r(t) = 2a tan(t) sin(t) or in Euclidean coordinates r(t) = (2a sin 2 (t), 2a sin 3 (t)/ cos(t)). The curve has a cusp at 

the origin. It was first mentioned by Diocles in 180 B.C. 

conic section 

A [conic section] is a nondegenerate curve generated by intersecting a plane with one or two nappes of a cone. 

The three congruence classes of conic sections are the ellipse, the parabola, and the hyperbola. 

ellipse 

An [ellipse] is the locus of all points in the plane the sum of whose distances from two fixed points is a positive 

constant. It is also the conic section which results from a plane which intersects only one nappe of the cone. 

The general formula for an ellipse up to rotation and translation is x2 

a 2 + y2 

b 2 = 1. 

hyperbola 

A [hyperbola] is the set of points in the plane for which the difference of the distances from two fixed points is 

a constant. It is also the conic section which results from a plane intersecting a cone. The general formula for 

a hyperbola up to rotation and translation is x2 

a 2 − y2 

b 2 = 1. 

curve 

A [curve] is a continuous map from the real line to an the plane or to space. The word ”curve” is often used to 

mean the image of this map. Curves can be represented parametrically by r(t) = (x(t), y(t)) or implicitely as 

f(x, y) = 0.

Airy function 

The [Airy function] is commonly found as a solution to boundary value problems in quantum mechanics and 

electromagnetism. It is the solution to the differential equation: y ′′ = xy. The two independent solutions are 

(without constants): 

Bi(x) = 

Ai(x) = 

� ∞ 

0 

� ∞ 

0 

� � 3 t 

cos + xt dt 

3 

� 

t3 − 

e 3 +xt + sin 

� t 3 

algebraic curve 

3 

�� 

+ xt dt 

A plane curve is an [algebraic curve] if it is given by g(x, y) = 0 where g is algebraic a polynomial in x and y. 

An algebraic curve with degree greater than 2 is called a higher plane curve. The circle g(x, y) = x 2 +y 2 −1 = 0 

is an example of an algebraic curve, the catenary g(x, y) = y − c cosh(x/c) = 0 is an example of a nonalgebraic 

curve. 

cubic curve 

A [cubic curve] is an algebraic curve of order three. Newton showed that all cubics can be generated as 

projections of the five divergent cubic parabolas. Examples include the cissoid of Diocles and ellptic curves. 

ampersand curve 

The [ampersand curve] is a quartic curve with implicit equation (y 2 − x 2 )(x − 1)(2x − 3) = 4(x 2 + y 2 − 2x) 2 . 

It looks like an ampersand. 

bean curve 

The [bean curve] is a quartic curve given by the implicit equation: x 4 + x 2 y 2 + y 4 = x(x 2 + y 2 ). It looks like a 

bean. 

bicorn 

The [bicorn] is the name of a collection of quartic curves studied by Sylvester in 1864 and Cayley in 1867. It is 

given by y 2 (a 2 − x 2 ) = (x 2 + 2ay − a 2 ) 2 . 

bicuspid 

The [bicuspid] is the quartic curve given by the implicit equation: (x 2 − a 2 )(x − a) 2 + (y 2 − a 2 ) 2 = 0.

ow 

The [bow] is a quartic curve with the implicit equation: x 4 = x 2 y − y 3 . 

cartesian oval 

A [cartesian oval] is a quartic curve consisting of two ovals. It is the locus of a point P whose distances from 

two foci F1 and F2 in two-center bipolar coordinates satisfy 

mr ± nr ′ = k 

where m and n are positive integers, k is a positive real, and r and r ′ are the distances from F1 and F2. If 

m = n, then the oval becomes an ellipse. 

Cassini oval 

A [Cassini oval] is one of a family of quartic curves, also called Cassini ellipses, described by a point such that 

the product of its distances from two fixed points a distance 2a apart is constant b 2 . The shape of the curve 

depends on b/a. The Cassini ovals are defined in two-center bipolar coordinates by the equation r1r2 = b 2 

where b is a positive constant. 

cruciform 

A [cruciform] is a plane quartic curve also called the cross curve or policeman on point duty curve. It is given 

by the implicit equation: x 2 y 2 − b 2 x 2 − a 2 y 2 = 0. 

lemniscate 

The [lemniscate], also known as the lemniscate of Bernoulli, is a polar curve whose most common form is the 

locus of points the product of whose distances from two fixed points a distance 2a away is the constant a 2 . The 

usual polar coordinate form is as follows: r 2 = a 2 cos(2θ). 

natural equation 

A [natural equation] is an equation which specifies a curve independent of any choice of coordinates or 

parametrization. This arose in the solution to the following problem: given two functions of one parameter, 

find the space curve for which the functions are the curvature and torsion. Often, the natural equation will 

be in terms of integrals. 

polynomial curve 

A [polynomial curve] is a curve obtained by fitting polynomials to a sequence of points. To fit curves better, 

splines like Bezier curve are more suited.

quadrifolium 

A [quadrifolium] is a rose curve with n = 2. It has polar equation 

r = a sin(2θ). 

sextic curve 

A [sextic curve] is an algebraic curve of degree 6. Examples include the atriphtaloid and the butterfly curve 

y 6 = x 2 − x 6 . 

atriphtaloid 

The [atriphtaloid] is a sextic curve also known as atriphtothlassic curve and given by the equation: x 4 (x 2 + 

y 2 ) − (ax 2 − b) 2 = 0. 

butterfly curve 

The [butterfly curve] is a sextic plane curve given by the implicit equation y 6 = x 2 − x 6 . 

trifolium 

A [trifolium] is the 3-petalled rose given in polar form as r(t) = a| cos(3t)|. 

spiral 

A [spiral], in general, is a curve with τ(s)/κ(s) constant for 



Index 

Airy function, 3 

algebraic curve, 3 

ampersand curve, 3 

Archimedes spiral, 1 

astroid, 1 

atriphtaloid, 5 

bean curve, 3 

bicorn, 3 

bicuspid, 3 

bow, 4 

bowditch curve, 1 

brachistochone, 1 

butterfly curve, 5 

cardioid, 1 

cartesian oval, 4 

Cassini oval, 4 

Cassini ovals, 1 

catenary, 1 

catenoid, 2 

circle, 2 

cissoid, 2 

conic section, 2 

cruciform, 4 

cubic curve, 3 

curve, 2 

ellipse, 2 

hyperbola, 2 

lemniscate, 4 

natural equation, 4 

polynomial curve, 4 

quadrifolium, 5 

sextic curve, 5 

spiral, 5 

trifolium, 5 

6

ENTRY FUNCTIONAL ANALYSIS 

[ENTRY FUNCTIONAL ANALYSIS] Authors: Oliver Knill: 2002 Literature: various notes 

adjoint 

The [adjoint] of a bounded linear operator A on a Hilbert space is the unique operator B which satisfies 

(Ax, y) = (x, By) for all x, y ∈ H. One calls the adjoint A ∗ . An bounded linear operator is selfadjoint, if 

A = A ∗ . 

Alaoglu’s theorem 

[Alaoglu’s theorem] (=Banach-Alaoglu theorem): the closed unit ball in a Banach space is weak-* compact. 

angle 

The [angle] φ between two vectors v and w of a Hilbert space is a solution φ of the equation cos(φ)||v||||w|| = 

(v, w), usually the smaller of the two solutions. 

balanced 

A subset Y of a vector space X is called [balanced] if tx is in Y whenever x is in Y and t < 1. 

B*-algebra 

A [B*-algebra] is a Banach algebra with a conjugate-linear anti-automorphic involution ∗ satisfying ||xx ∗ || = 

||x|| 2 . 

Banach algebra 

A [Banach algebra] is an algebra X over the real numbers or complex numbers which is also a Banach space 

such that ||xy|| ≤ ||x||||y|| for all x, y ∈ X. 

Banach limit 

A [Banach limit] is a translation-invariant functional f on the Banach space of all bounded sequence such that 

f(c) = c1 for constant sequences.

A [Banach space] is a complete normed space. 

Banach space 

barrel 

A [barrel] is a closed, convex, absorbing, balanced subset of a topological vector space. 

basis 

A [basis] (= Schauder basis) of a separable normed space is a sequence of vectors xj such that every vector x 

can uniquely be written as y = � 

j yjxj. 

basis 

A [basis] in a vector space is a linearly independent subset that generates the space. 

baralled space 

A [baralled space] is a topological vector space in which every barrel contains a neighborhood of the origin. 

biorthogonal 

Two sequences an and bn in a Hilbert space are called [biorthogonal] if Anm = (an, bm) is an unitary operator. 

Bergman space 

The [Bergman � �space] 

for an open subset G of the complex plane C is the collection of all anlytic function f on 

G for which G |f(x + iy)|2 dxdy is finite. It is an example of a Hilbert space. 

Buniakovsky inequality 

The [Buniakovsky inequality] (=Cauchy-Schwarz inequality) in a Hilbert space tells that |(a, b)| ≤ ||a|| ||b||. 

Cauchy-Schwartz inequality 

The [Cauchy-Schwartz inequality] in a Hilbert space H states that |(f, g)| ≤ ||f|| ||g||. It is also called Buniakovsky 

inequality or CBS inequality.

compact operator 

A [compact operator] is a bounded linear operator A on a Hilbert space, which has the property that the image 

A(B) of the unit ball B has compact closure in H. 

compact operator 

A bounded operator A on a separable Hilbert space is called [diagonalizable] if there exists a basis in H such 

that Hvi = λivi for every basis vector vi. Compact normal operators are diagonalizable. 

dimension 

The [dimension] of a Hilbert space H is the cardinality of a basis of H. A Hilbert space is called seperable, if 

the cardinality of the basis is the cardinality of the integers. 

Egorov’s theorem 

[Egorov’s theorem] Let (X, S, m) be a measure space, where m(S) has finite measure. If a sequence fn of 

measurable functions converges to f almost everywhere, then for every d¿0, there is a set Ed ⊂ X such that 

fn → f uniformly on E \ Ed and m(Ed) < d. 

finite rank operator 

A bounded linear operator A on a Hilbert space H is called a [finite rank operator] if the rank of A is finite 

dimensional. Finite rank operators are examples of compact operators. 

Hilbert space 

A [Hilbert space] H is a vector space equiped with an inner product (x, y) for which the corresponding metric 

d(x, y) = ||x − y|| = (x − y, x − y) makes (H, d) into a complete metric space. Examples: 

• l2 (N) is the collection of sequences an such that � 

n |an| < ∞ is a Hilbert space with inner product 

(a, b) = � 

n anbn. 

• L 2 (G) the space of all analytic functions on an open subset of the complex plane which are also in L 2 (G, µ), 

where µ is the Lebesgue measure on G. 

• All vectors of a finite dimensional vector space, where the inner product is the usual dot product. 

• All square integrable functions L 2 (X, µ) on a measure space (X, S, µ). 

idempotent 

A bounded linear operator A on a Hilbert space is called [idempotent] if A 2 = A. Projections P are examples 

of idempotent operators.

Lusin’s theorem 

[Lusin’s theorem] If (X, S, m) is a measure space and f is a measureable function on S. For every d > 0, there 

is a set Ed with m(Ed) < d and a measurable function g such that g is continuous on Ed. 

linear operator 

A [linear operator] is a linear map between two Hilbert spaces or two Banach spaces. Important examples are 

bounded linear operators, linear operators which also continuous maps. Linear operators are also called linear 

transformations. 

norm 

The [norm] of a bounded linear operator A on a Hilbert space H is defined as ||A|| = sup ||x||≤1,x∈H ||Ax||. 

normal 

A bounded linear operator A is called [normal] if AA ∗ = A ∗ A, where A ∗ is the adjoint of A. Examples: 

• selfadjoint operators are normal. 

• unitary operators are normal 

Open mapping theorem 

[Open mapping theorem] If a map A from X to Y is a surjective continuous linear operator between two Banach 

spaces X and Y , and U is an open set in X, then A(U) is open in Y . 

The proof of the theorem which is also called the Banach-Schauder theorem uses the Baire category theorem. 

implications: 

• A bijective continuous linear operator between the Banach spaces X and Y has a continuous inverse. 

• Closed graph theorem: if for every sequence xn ∈ X with xn → 0 and Axn → y follows y = 0, then A is 

continuous. 

Riesz representation theorem 

[Riesz representation theorem] If f is a bounded linear functional on a Hilbert space H, then there exists a 

vector y ∈ H such that f(x) = (x, y) for all x ∈ H. 

Sturm Liouville operator 

A [Sturm Liouville operator] L is an unbounded operator on the Hilbert space L 2 [a, b] defined by L(f) = 

−f ′′ + gf, where g is a continuous function on [a, b].

unitary 

A bounded linear operator A on a Hilbert space is called [unitary] if AA ∗ = A ∗ A = 1 if 1 is the identity operator 

1(x) = x. 

unit ball 

The [unit ball] B in a Hilbert space H is the set of all points x ∈ H satisfiying ||x| ≤ 1. 



Index 

adjoint, 1 

Alaoglu’s theorem, 1 

angle, 1 

B*-algebra, 1 

balanced, 1 

Banach algebra, 1 

Banach limit, 1 

Banach space, 2 

baralled space, 2 

barrel, 2 

basis, 2 

Bergman space, 2 

biorthogonal, 2 

Buniakovsky inequality, 2 

Cauchy-Schwartz inequality, 2 

compact operator, 3 

dimension, 3 

Egorov’s theorem, 3 

finite rank operator, 3 

Hilbert space, 3 

idempotent, 3 

linear operator, 4 

Lusin’s theorem, 4 

norm, 4 

normal, 4 

Open mapping theorem, 4 

Riesz representation theorem, 4 

Sturm Liouville operator, 4 

unit ball, 5 

unitary, 5 

6

ENTRY FUNCTIONS 

[ENTRY FUNCTIONS] Authors: Oliver Knill: 2003, Literature: no 

real-valued function 

A [real-valued function] is usually assumed to be map to the reals. 

abscissa 

[abscissa] The x-coordinate in an (x,y) graph of a function. The y-coordinates is called ordinate. 

ordinate 

[ordinate] The y-coordinate in an (x,y) graph of a function. The x-coordinates is called abscissa. 

Airy function 

The [Airy function] is defined as the solution of the differential equation y ′′ − xy = 0. 

Briggsian logarithm 

The [Briggsian logarithm] also called common logarithm is the logarithm to the base 10. 

Bessel function 

THe [Bessel function] is a special function. Bessel function of the first kind of order zero is defined as J0 = 

� ∞ 

k=0 (−1)k (x/2) 2k /(k!) 2 . 

Sin 

The [Sin] is a trigonometric function. It can be defined by its series sin(x) = x − x 3 /3! + x 5 /5! − ..., where 

5! = 54321 is the factorial of 5. The sine function can also be defined as the imaginary part of exp(ix) = 

cos(x) + i sin(x), where i = (−1) ( 1/2) is the imaginary unit. Examples of values sin(0) = 0, sin(π/2) = 

1, sin(π) = 0, sin(3π/2) = −1. 

[Csc] The cosecant is defined as csc(x) = 1/ sin(x). 

Csc

Arcsin 

[Arcsin] The arcsin is the inverse of sin. It is also denoted by sin ( −1)(x) or asin(x). One has the identities 

arcsin(sin(x)) = x, or sin(arcsin(x)) = x. 

Sinh 

[Sinh] The hyperbolic sine can be defined as sinh(x) = (exp(x) − exp(−x))/2. Examples: sinh(0) = 0. 

[ArcSinh] The inverse of sinh is called arcsinh. 

ArcSinh 

Cos 

The trigonometric function [Cos] can be defined by its series cos(x) = 1 − x 2 /2! + x 4 /4! − x 6 /6! − ..., where 

4! = 4321 is the factorial of 4. It can also be defined as the real part of exp(ix), where i = (−1) 1 /2 is the 

imaginary unit, the square root of -1. Examples: cos(0) = 1, cos(pi/2) = 0 sin(π) = −1, sin(3π/2) = 0. 

Arccos 

[Arccos] The inverse of the function cos is written arccos(x), also denoted by cos ( −1)(x) or acos(x). We have 

the identities arccos(cos(x)) = x, or cos(arccos(x)) = x. 

[Sec] The secant is defined as sec(x) = 1/ cos(x). 

Sec 

Cosh 

[Cosh] The hyperbolic cosine can be defined as sinh(x) = (exp(x) + exp(−x))/2. Examples: cosh(0) = 1. 

[ArcCosh] The inverse of cosh is called arccosh. 

ArcCosh 

Tan 

The [Tan] is a trigonometric function. It can be defined as tan(x) = sin(x)/ cos(x). Examples: tan(0) = 

0, tan(π/4) = 1.

Arctan 

[Arctan] The inverse of tan is the function arctan(x). It is also called tan ( − 1)(x). One has arctan(tan(x)) = x 

and tan(arctan(x)) = x. Examples: arctan(1) = π/2. 

Cot 

[Cot] is a trigonometric function. It can be defined as cot(x) = cos(x)/ sin(x). It can also be defined 

geometrically as the relation of two sides in a right angle triangle if x is one of the angles. Examples: 

cot(π/2) = 0, cot(π/4) = 1. 

Exp 

[Exp] is the exponential function. It can be defined by its series exp(x) = 1 + x + x 2 /2! + x 3 /3! + x 4 /4! + ... 

where 4! = 4321 is the factorial of 4. Examples: exp(0) = 1, exp(1) = e = 2.712.... 

Sqr 

[Sqr] The square of a number is the product of the number by itself. For example, the square of 4 is 16. The 

square of a function sin(x) is denoted by sin 2 (x). 

Zeta 

[Zeta] ζ(s) is the Riemann zeta function. It is defined for complex numbers s which have Re(s) > 1 as 

ζ(s) = 1+1/2 s +1/3 s +.... The function can be continued to the entire complex plane except at s = 1, where the 

function has a singularity. The zeta function has zeros at −2, −4, −6 and also zeros on the real line Re(s) = 1/2. 

The famous Riemann hypothesis claims that all the nontrivial zeros are on this line. This conjecture remains 

unproven until today and is considered one of the most important open problems in mathematics. 

Log 

[Log] The logarithm is the inverse to the exponential function: log(exp(x)) = x and exp(log(x)) = x. For 

example: log(1) = 0, log(e) = 1. The logarithm function satisfies for example the laws log(xy) = log(x)+log(y), 

log(x/y) = log(x) − log(y), log(x y ) = y log(x). 

Sqrt 

[Sqrt] The square root of a number x is the number which 



Index 

abscissa, 1 

Airy function, 1 

Arccos, 2 

ArcCosh, 2 

Arcsin, 2 

ArcSinh, 2 

Arctan, 3 

Bessel function, 1 

Briggsian logarithm, 1 

Cos, 2 

Cosh, 2 

Cot, 3 

Csc, 1 

Exp, 3 

Log, 3 

ordinate, 1 

real-valued function, 1 

Sec, 2 

Sin, 1 

Sinh, 2 

Sqr, 3 

Sqrt, 3 

Tan, 2 

Zeta, 3 

4

ENTRY GROUP THEORY 

[ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: ”Algebra” by Michael 

Artin, Mathworld 

Group theory 

[Group theory] is studies algebraic objects called groups. The German mathematician Karl Friedrich Gauss 

(1777-1855) developed but did not publish some of the mathematics of group theory. The French mathematician 

Evariste Galois (1811-1832) is generally credited with being the first to develop the theory, which he did by 

developing new techniques to study the solubility of equations. Group theory is a powerful method for analyzing 

abstract and physical systems in which symmetry –the intrinsic property of an object to remain invariant under 

certain classes of transformations– is present because the mathematical study of symmetry is systematized and 

formalized in group theory. Consequently, group theory is an important tool in physics particularly in quantum 

mechanics. 

group 

A [group] is an object consisting of a set G and a law of composition (or binary operation) L on G satisfying: 

• L is associative. 

• L has an identity in G. 

• Every element of G has an inverse. 

The study of groups is known as group theory. If a group G has n elements where n is a positive integer, then 

G is a finite group with order n. If a group is not finite it is infinite. Examples: 

• Z + , the integers under addition; 

• R + = (R, +), the real numbers under addition; 

• R × = (R − {0}, ·), the real numbers without zero under multiplication; 

• GLn(C), the n × n general linear group under matrix multiplication; 

• Sn, the symmetric group on n objects under composition. 

law of composition 

A [law of composition] or, binary operation, on a set S is a function from S × S into S. That is, a law of 

composition on S prescribes a rule for combining pairs of elements in S to get an element in S. For convenience, 

functional notation is not used; that is, if a law of composition f sends (a, b) to c, one does not usually write 

f(a, b) = c. It is customary to instead use notation that resembles that used for multiplication or addition of 

real numbers, such as ab = c, a · b = c, a ◦ b = c, a + b = c, and so on. 

An example of a law of composition is multiplication on the real numbers, R. If m: R × R → R defines 

multiplication on R then m(x, y) = x · y. For example m(2, 5) = 2 · 5 = 10.

inary operation 

A [binary operation], or law of composition, on a set S is a function from S × S into S. That is, a binary 

operation on S prescribes a rule for combining pairs of elements in S to get an element in S. For convenience, 

functional notation is not used; that is, if a binary operation f sends (a, b) to c, one does not usually write 

f(a, b) = c. It is customary to instead use notation that resembles that used for multiplication or addition of 

real numbers, such as ab = c, a · b = c, a ◦ b = c, a + b = c, and so on. 

An example of a binary operation is multiplication on the real numbers, R. If m: R × R → R defines multiplication 

on R then m(x, y) = x · y. For example m(2, 5) = 2 · 5 = 10. 

associative 

A law of composition on a set S is [associative] if for all a, b, c ∈ S, (ab)c = a(bc). The informal intuition 

behind associativity (the property of being associative) is that if one has an expression in which there are 

many parentheses and the only operation performed in this expression is that defined by an associative law of 

composition, then one may ignore the parentheses. For example, if · is an associative law of composition on S 

and a, b, c, d ∈ S, then ((a · (b · c)) · d = ((a · b) · c) · d = (a · b) · (c · d) and so on; thus one may write a · b · c · d 

without being ambiguous. 

An example of an associative law of composition is addition on the integers, Z. That is, for all a, b, c ∈ Z, 

(a + b) + c = a + (b + c). 

identity 

An [identity] for a law of composition on a set S is an element e such that, for all a ∈ S, ea = a and ae = a. 

Note that a law of composition has at most one identity. The symbols e, 0 and 1 are commonly used to denote 

the identity element of a group. The number 0 is an identity for addition on the real numbers. 

identity 

Suppose a set S has a law of composition with identity 1. For every element a ∈ S, if there exists an element 

b ∈ S such that ab = 1 and ba = 1 then b is the [inverse] of a. When using multiplicative notation for the law 

of composition, the inverse of a can be written as a −1 . As an example, the inverse of any integer n is −n where 

the law of composition is addition and the identity is 0. As another example, the inverse of any nonzero real 

, where the law of composition is multiplication and the identity is 1. 

number x is 1 

x 

general linear group 

The n × n [general linear group] GLn(F ) is the set of n × n matrices with entries in the field F and nonzero 

determinant, under the law of composition of matrix multiplication. Thus GLn(F ) is the group of n × n 

invertible matrices with entries in F . If F is a finite field of field order q then sometimes the general linear 

group GLn(F ) is denoted by GLn(q). The general linear group often appears with respect to the real numbers, 

R, or the complex numbers, C; that is, the general linear group often appears as GLn(R) or GLn(C). 

The special linear group SLn(F ) is the subgroup of GLn(F ) whose elements have determinant equal to 1.

special linear group 

The n × n [special linear group] SLn(F ) is the set of n × n matrices with entries in the field F and determinant 

equal to 1, under the law of composition of matrix multiplication. 

If F is a finite field of field order q then sometimes the special linear group SLn(F ) is denoted by SLn(q). 

SLn(F ) is a subgroup of the general linear group GLn(F ). 

trivial 

A group is [trivial] if it contains exactly one element. The one element in the group is the identity element. As 

all trivial groups are isomorphic, one usually refers to a trivial group as the trivial group. A group that is not 

trivial is nontrivial. 

trivial 

The group containing exactly one element (the identity) is unique up to isomorphism and is therefore called the 

[trivial group]. The trivial group is a normal subgroup of every group. 

A group is [nontrivial] if it is not trivial. 

nontrivial 

abelian 

A group is [abelian] if its law of composition is commutative. Examples of abelian groups include the following: 

• R + = (R, +), the real numbers under addition; 

• R × = (R − {0}, ·), the real numbers without zero under multiplication; 

• any cyclic group. 

Examples of nonabelian groups, i.e. groups that are not abelian: 

• GLn(C), the general linear group; 

• The symmetric group on n objects, where n is a positive integer greater than 2. 

commutative 

A law of composition on a set S is [commutative] if for all a, b ∈ S ab = ba. An example of a commutative law 

of composition is addition on the real numbers: for example, 3.2 + 4 = 7.2 = 4 + 3.2.

cancellation Law 

The [cancellation Law] states that if a, b, and c are elements of a group and if ab = ac then b = c. Similarly, if 

ba = ca then b = c. The Cancellation Law follows from the fact that every element of a group has an inverse. 

permutation 

If S is a set, then a [permutation] of S is a bijective map from S into S. The intuition underlying the definition 

of a permutation is that a permutation determines a reordering of the elements in a list or the rearrangement 

of objects. For example, the permutation σ : {1, 2, 3} → {1, 2, 3} defined by σ(1) = 2, σ(2) = 1, and σ(3) = 3 

can be thought to represent the reordering of the list 1,2,3 that results in the list 2,1,3. There is an important 

kind of permutation called a transposition. A transposition of a set S is a permutation σ: S → S satisfying the 

following: there exist s1, s2 ∈ S such that 

• σ(s1) = s2 

• σ(s2) = s1 

• and for all s ∈ S, if s �= s1 and s �= s2, then σ(s) = s. 

Every permutation of a finite set can be written as the composition of a finite number of transpositions of that 

set. For example, the permutation σ : {1, 2, 3} → {1, 2, 3} defined by σ(1) = 2, σ(2) = 3, and σ(3) = 1 is 

equivalent to the composition of two transpositions. Define σ1 : {1, 2, 3} → {1, 2, 3} by σ1(1) = 2, σ1(2) = 1, 

and σ(3) = 3, and define σ2 : {1, 2, 3} → {1, 2, 3} by σ2(1) = 3, σ1(2) = 2, and σ(3) = 1. Then σ1 and σ2 are 

transpositions and σ = σ2 ◦ σ1. 

The sign of a permutation σ is (−1) n where n is a finite positive integer such that there exist n transpositions 

whose composition equals σ. If a permutation has sign 1, then it is called an even permutation. If a permutation 

has sign -1 then it is called an odd permutation. Thus the identity permutation is an even permutation (since 

it is equal to the composition of any transposition with itself), and any transposition is an odd permutation 

(since it is equal to one transposition). The sign map encapsulates the notion of the sign of a permutation of a 

finite set. 

The set of permutations on a set forms a group where the law of composition is composition of functions. 

One example of a group of permutations that appears frequently in group theory is the symmetric group on n 

objects, i.e. the group of permutations of the set {1, 2, . . . , n}. 

transposition 

A [transposition] of a set S is a permutation σ: S → S satisfying the following: there exist s1, s2 ∈ S such that 

• σ(s1) = s2 

• σ(s2) = s1 

• and for all s ∈ S, if s �= s1 and s �= s2, then σ(s) = s. 

Every permutation of a finite set can be written as the composition of a finite number of transpositions of that 

set. For example, the permutation σ : {1, 2, 3} → {1, 2, 3} defined by σ(1) = 2, σ(2) = 3, and σ(3) = 1 is 

equivalent to the composition of two transpositions. Define σ1 : {1, 2, 3} → {1, 2, 3} by σ1(1) = 2, σ1(2) = 1, 

and σ(3) = 3, and define σ2 : {1, 2, 3} → {1, 2, 3} by σ2(1) = 3, σ1(2) = 2, and σ(3) = 1. Then σ1 and σ2 are 

transpositions and σ = σ2 ◦ σ1. Every transposition is an odd permutation.

sign 

The [sign] of a permutation σ is (−1) n where n is a finite positive integer such that there exist n transpositions 

whose composition equals σ. If a permutation has sign 1, then it is called an even permutation. If a permutation 

has sign -1 then it is called an odd permutation. Thus the identity permutation is an even permutation (since 

it is equal to the composition of any transposition with itself), and any transposition is an odd permutation 

(since it is equal to one transposition). 

even permutation 

An [even permutation] is a permutation that has sign 1. That is, an even permutation is the composition of an 

even number of transpositions. Thus the identity permutation is an even permutation. 

odd permutation 

An [odd permutation] is a permutation that has sign -1. That is, an odd permutation is the composition of an 

odd number of transpositions. Thus every tranposition is an odd permutation. 

symmetric group 

The [symmetric group] on n objects, denoted Sn, is the group of permutations of the set {1, 2, . . . , n}; the law 

of composition is composition of functions. The order of Sn is n! for all positive integers n. For example, 

S2 = {e, σ}, where e is the identity permutation, and σ is a transposition. That is, e is the identity element of 

S2 and is defined by e(1) = 1 and e(2) = 2; σ is defined by σ(1) = 2 and σ(2) = 1. 

sign map 

The [sign map], denoted sign, is a group homomorphism from the symmetric group, Sn, into the group {1, −1} 

(under multiplication). The sign map is defined by sign(σ) = (−1) k where σ is equal to the composition of k 

transpositions. The sign map is well-defined because it is a standard result that if σ is any permutation (of 

any set), and if σ is equal to the composition of k transpositions and is also equal to the composition of m 

transpositions, then (−1) k = (−1) m . The kernel of the sign map is the alternating group, An; that is, An is the 

group of even permutations on n objects. 

alternating group 

The [alternating group] is the kernel of the sign map. In other words, the alternating group on n objects, usually 

denoted An, is a normal subgroup of the symmetric group on n objects: An = {σ ∈ Sn | sign(σ) = 1}. Thus 

An is the group of even permutations in Sn. For n ≥ 5, An is a simple group, i.e. a group that has no proper 

normal subgroup.

simple group 

A group G is a [simple group] if every normal subgroup N of G is not a proper subgroup. That is, G is simple 

if its only normal subgroups are G and the trivial group. The alternating group An is simple for n ≥ 5. Any 

cyclic group of prime order is simple. Any cyclic group of prime order is simple. In fact, any simple abelian 

group is a cyclic group of prime order. 

subgroup 

A subset H of a group G is a [subgroup] of G if it satisfies the following properties: 

• If a ∈ H and b ∈ H, then ab ∈ H. 

• 1 ∈ H, where 1 is the identity element of G. 

• If a ∈ H then the inverse of a, a −1 , is also in H. 

When it is clear that G is a group, sometimes H ⊆ G is used to denote that H is a subgroup of G (as opposed 

to merely being a subset of G). 

Every nontrivial group G has at least two subgroups: the whole group G and the subgroup {1} consisting 

exactly of the identity element of G. If G is trivial then these two subgroups are the same and G has exactly 

one subgroup. A subgroup is a proper subgroup if it is neither the whole group nor the trivial group. 

As an example, the integers under addition are a subgroup of the real numbers under addition. 

By Lagrange’s Group Theorem, if H is a subgroup of a finite group G, the order of H divides the order of G. 

proper subgroup 

A [proper subgroup] of a group G is a nontrivial subgroup of G that is not equal to G. 

order 

A finite group G is said to have [order] n if G has n elements. More generally, the order of a group G is the 

cardinality of the set G, both of which are often denoted |G|. 

For any given element x of a given group, if there exists a positive integer k such that x k = 1, then x is said to 

have order m, where m is the least positive integer satisfying x m = 1. If x k �= 1 for all positive integers k, then 

x is said to have infinite order.

cyclic group 

If G is a group and if x is an element of G, the [cyclic group] 〈x〉 generated by x is the set of all powers of x: 

〈x〉 = {. . . , x −2 , x −1 , 1, x, x 2 , . . .}. 

Note that 〈x〉 is the smallest subgroup of G which contains x. Further note that any cyclic group is abelian. If 

x has infinite order then 〈x〉 is said to be infinite cyclic. Note that if 〈x〉 is infinite cyclic then 〈x〉 is isomorphic 

to Z + , the integers under addition. As a result, one sometimes refers to any infinite cyclic group as the infinite 

cyclic group, denoted Z + . 

If x has order n, then 〈x〉 has order n and is called a cyclic group of order n: 

〈x〉 = {1, x, x 2 , . . . , x n }. 

If 〈x〉 is a cyclic group of order n, then 〈x〉 is isomorphic to Z/n, where Z/n is the group satisfying the following 

properties (we use additive notation as opposed to multiplicative notation for the law of composition of Z/n): 

1. Z/n = {0, 1, 2, . . . , n − 1}. 2. for any x, y ∈ Z/n, x +1 y is the unique element in Z/n which is congruent 

modulo n to x + y, where +1 denotes the law of composition on Z/n and + denotes conventional integer 

addition. Normally + is used to denote the law of composition on Z/n, but +1 is used here to distinguish 

it from conventional addition. Two integers a and b are congruent modulo n, written a ≡ b(modulo n), if n 

divides b − a. 

As a result, one sometimes refers to any cyclic group of order n as the cyclic group of order n, often denoted 

Z/n. 

cyclic group 

Let G1 and G2 be groups. A map ϕ: G1 → G2 is a group [homomorphism] if ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ G1. 

Here we use the same multiplicative notation for the laws of composition of G1 and G2, even though there is 

no requirement that their laws of composition be the same. 

Note that ϕ(1G1 ) = 1G2 and ϕ(a−1 ) = ϕ(a) −1 for all a ∈ G, where 1Gi is the identity of Gi. 

The kernel of ϕ, sometimes denoted ker ϕ, is the set {x ∈ G1 | ϕ(x) = 1G2 }. Note that ker ϕ is a normal 

subgroup of G2. 

Another subgroup of G2 determined by ϕ is the image of ϕ, sometimes denoted im ϕ. The image of ϕ is im 

ϕ = {x ∈ G2 | x = ϕ(a) for some a ∈ G1}. Sometimes the image of ϕ is denoted ϕ(G1). The following are 

examples of homomorphisms: 

• The inclusion map i: H → G defined by i(a) = a, where H is a subgroup of G. ker i = {1G} and im 

i = H. 

• For a fixed a ∈ G, the map ϕ: Z + → G defined by ϕ(n) = a n , where Z + denotes the integers with addition. 

ker ϕ = {n | a n = 1G} and im ϕ = 〈a〉 (the cyclic subgroup generated by a). 

• The determinant map det: GLn(R) → R × , where GLn(R) denotes the general linear group and R × 

denotes the real numbers without zero under multiplication. kerdet = SLn(R), the special linear group, 

and imdet = R × . 

• The sign map on permutations sign: Sn → {1, −1}, where Sn denotes the symmetric group on n objects. 

ker sign = An, the alternating group, and im sign = {1, −1}. 

Let R1 and R2 be rings. A map ϕ: R1 → R2 is a ring homomorphism if 

• ϕ(a + b) = ϕ(a) + ϕ(b), 

• ϕ(ab) = ϕ(a)ϕ(b), and 

• ϕ(1R1 ) = 1R2 , for all a, b ∈ R1. 

Here we use the same additive and multiplicative notation for the laws of composition of R1 and R2, even 

though there is no requirement that their laws of composition be the same.

kernel 

The [kernel] of a group homomorphism ϕ: G1 → G2 is the set ker ϕ = {x ∈ G1 | ϕ(x) = 1G2 }, where 1G2 denotes 

the identity of G2. The kernel of a homomorphism is an important example of a normal subgroup. There are 

many results involving the kernel of a homomorphism. 

image 

The [image] of a map ϕ: G1 → G2 is the set {x ∈ G2 | x = ϕ(a) for some a ∈ G1}. In general, the image of ϕ is 

often denoted ϕ(G1). If ϕ is a group homomorphism, then the image of ϕ is a subgroup of G2 and is sometimes 

denoted im ϕ. 

isomorphism 

A group [isomorphism] is a bijective group homomorphism. 

A ring isomorphism is a bijective ring homomorphism. 

isomorphic 

Two groups G1 and G2 are [isomorphic] if there exists a group isomorphism from G1 into G2. Sometimes 

G1 ∼ = G2 is used to denote ‘G1 and G2 are isomorphic.’ Note that ∼ = is an equivalence relation on the set of 

all groups. When one speaks of classifying groups, hat is usually referred to is the classification of isomorphism 

classes. Thus one might say that there are two groups of order 6 up to isomorphism, meaning that there are 

two isomorphism classes of groups of order 6. 

One sometimes says that ‘G1 is isomorphic to G2’ instead of saying ‘G1 and G2 are isomorphic.’ 

automorphism 

An [automorphism] of a group G is an isomorphism from G into G. The identity map is a simple example of an 

automorphism. Conjugation by an element of the group is an important example of an automorphism. That 

is, for a fixed element b ∈ G, conjugation by b is the map ϕ: G → G defined by ϕ(a) = bab −1 . Here we use 

multiplicative notation for the group law of composition. Note that if G is abelian, then conjugation by any 

element is the identity map. However, if G is not abelian, then there exists a nontrivial conjugation (i.e. a 

conjugation not equal to the identity map) of G. 

automorphism 

Let G be a group and let b ∈ G. The map ϕ: G → G defined by ϕ(a) = bab −1 is [conjugation] by b. Note that 

conjugation is an automorphism of G. Further note that if G is abelian, then conjugation by any element is the 

identity map. However, if G is not abelian, then there exists a nontrivial conjugation (i.e. a conjugation not 

equal to the identity map) of G.

conjugation 

Let G be a group and let H be a subgroup of G. H is a [normal subgroup] of G (sometimes written H ⊳ G) 

if for all a ∈ H and for all x ∈ G, xax −1 ∈ H. Note that it follows that any subgroup of an abelian group is 

normal. 

Normal subgroups appear often in group theory. Every group G has at least one normal subgroup, called the 

center of G, denoted by Z or Z(G). The center of G is the set of elements that commute with every element of 

G: Z(G) = {z ∈ G | zx = xz for all x ∈ G}. Another important example of a normal subgroup is the kernel of 

a group homomorphism. 

center 

The [center] of a group G, denoted by Z or Z(G) is the set of elements that commute with every element of G: 

Z(G) = {z ∈ G | zx = xz for all x ∈ G}. Note that if G is abelian then Z(G) = G. 

coset 

Given a subgroup H of a group G, a [coset] of H is a subset H ′ of G such that there exists an a ∈ G such that 

(1) H ′ = aH = {ah | h ∈ H}, in which case H ′ is said to be a left coset; or (2) H ′ = Ha = {ha | h ∈ H}, in 

which case H ′ is said to be a right coset. 

Given a ∈ G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal 

subgroup if and only if aH = Ha for every a ∈ G. 

In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. 

The left cosets of H are the equivalence classes of the equivalence relation ∼ defined by a ∼ b if there exists 

h ∈ H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. 

The cardinality of the set of left cosets of H is called the index of H in G and is denoted by [G : H]. Given 

a ∈ G, h ↦→ ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|[G : H], where 

|G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order 

of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is 

finite then the order of an element of G divides the order of G. These results follow from a special case of what 

is known as Lagrange’s Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then 

[G : K] = [G : H][H : K], where the products are taken as products of cardinals. 

An important result that follows from Lagrange’s Theorem is that if the order of G is a prime number then 

G = 〈a〉 for any a ∈ G such that a is not the identity, where 〈a〉 denotes the cyclic group generated by a. 

Note that if ϕ: G → G ′ is a group [homomorphism], then [G : kerϕ] = |imϕ|. Thus another result of Langrange’s 

Theorem is that |G| = |kerϕ| · |imϕ|.

left coset 

Given a subgroup H of a group G, a [left coset] of H is a subset H ′ of G such that there exists an a ∈ G such 

that H ′ = aH = {ah | h ∈ H}. 















Note that if ϕ: G → G ′ is a group [homomorphism], then [G : ker(ϕ)] = |im(ϕ)|. Thus another result of 

Langrange’s Theorem is that |G| = |ker(ϕ)| · |im(ϕ)|. 

right coset 

Given a subgroup H of a group G, a [right coset] of H is a subset H ′ of G such that there exists an a ∈ G such 

that H ′ = Ha = {ha | h ∈ H}. 















Note that if ϕ: G → G ′ is a group homomorphism, then [G : ke(ϕ)] = |im(ϕ)|. Thus another result of 

Langrange’s Theorem is that |G| = |ker(ϕ)| · |im(ϕ)|. 

index 

The [index] of subgroup H of a group G is the cardinality of the set of left cosets of H in G. The index of H 

in G is denoted [G : H].

quotient group 

Given a group G and a normal subgroup N of G, the [quotient group] of N in G, written G/N and read 

“G mod(ulo) N”, is the set of cosets of N in G, under the law of composition that is defined as follows: 

(aN)(bN) = abN, where xN = {xn | n ∈ N}. Note that since N is normal, aN = Na for all a ∈ G, so it is not 

necessary to define this law of composition in terms of left cosets instead of right cosets. 

The order of G/N is the index [G : N] of N in G. 

Quotient groups can be identified by the First Isomorphism Theorem: if ϕ: G → G ′ is a surjective group 

homomorphism and if N = ke(ϕ) then ψ: G/N → G ′ is an isomorphism, where ψ is defined by ψ(aN) = ϕ(a). 

First Isomorphism Theorem 

The [First Isomorphism Theorem]. Suppose ϕ: G → G ′ is a surjective group homomorphism, and let N denote 

the kernel of ϕ. Then the quotient group G/N is isomorphic to G ′ by the map ψ defined by ψ(aN) = ϕ(a). 

The First Isomorphism Theorem is the principle method of identifying quotient groups. As an example, consider 

the group homomorphism ϕ from C × , the nonzero complex numbers under multiplication, into R × , the nonzero 

real numbers under multiplication, defined by ϕ(z) = |z|, where |z| denotes the absolute value of z. The kernel 

of ϕ is the unit circle, U, and the image of ϕ is the group of positive real numbers. So C × /U is isomorphic to 

the multiplicative group of positive real numbers. 

operation 

Given a group G and a set S, an [operation] of a G on S is a map from G × S into S - often written using 

multiplicative notation: (g, s) ↦→ gs - satisfying: 

• 1s = s for all s ∈ S, where 1 is the identity of G; and 

• (gg ′ )s = g(g ′ s), for all g, g ′ ∈ G and for all s ∈ S. 

There are some terms that are sometimes associated with a group operation: S is often called a G-set; G is 

sometimes called a transformation group; and the group operation is often also called a group action. 

Mathworld: ”Historically, the first group action studied was the action of the Galois group on the roots of 

a polynomial. However, there are numerous examples and applications of group actions in many branches 

of mathematics, including algebra, topology, geometry, number theory, and analysis, as well as the sciences, 

including chemistry and physics.” 



Index 

abelian, 3 

alternating group, 5 

associative, 2 

automorphism, 8 

binary operation, 2 

cancellation Law, 4 

center, 9 

commutative, 3 

conjugation, 9 

coset, 9 

cyclic group, 7 

even permutation, 5 

First Isomorphism Theorem, 11 

general linear group, 2 

group, 1 

Group theory, 1 

identity, 2 

image, 8 

index, 10 

isomorphic, 8 

isomorphism, 8 

kernel, 8 

law of composition, 1 

left coset, 10 

nontrivial, 3 

odd permutation, 5 

operation, 11 

order, 6 

permutation, 4 

proper subgroup, 6 

quotient group, 11 

right coset, 10 

sign, 5 

sign map, 5 

simple group, 6 

special linear group, 3 

subgroup, 6 

symmetric group, 5 

transposition, 4 

trivial, 3 

12

ENTRY HARVARD 

[ENTRY HARVARD] Authors: Oliver Knill: 2000, Literature: no 

[Harvard] 

Harvard 

Science center 

[Science center] The science center is the Polaroid camera shape building near Harvard square. The actual 

building is close to Memorial Hall. I actually live and think in the science center. You can virtually walk into 

the science center 

president of Harvard 

[president of Harvard] The President of Harvard is currently Lawrence H. Summers. To find out more about 

him, visit the website 

chairman of math department 

[chairman of math department] The chairman of the Mathematics department is currently Joe Harris. 

number people math department 

[number people math department] Currently, there are over 190 people at the Math department. 

preceptor 

[preceptor] Preceptors work alongside other faculty on teaching, developing and supporting sections of entry 

level courses at the Harvard Mathematics department. 

ca 

[ca] A course assistant (CA) is an undergraduate student who assists the teaching fellow (TF) with grading, 

running problem sessions and tutoring in the question center.

tf 

[tf] TF stands for teaching fellow. Everybody who is teaching is called a TF. It can be a senior or junior faculty, 

a visiting fellow or a graduate student. 

concentrator 

[concentrator] A (math) concentrator is a sophomore or senior undergraduate student. A math concentrator 

would also be called a math major. There are about one hundred math concentrators. Each year, there are 

about 30 new concentrators. 

head tutor 

[head tutor] The head tutor is a professor who is the chief undergraduate advisor. 

question center 

[question center] The question center QC is a place to work on homework problems or exam preparation. Tutors 

(both course assistants or teaching fellows) are available for questions. 

[qc] see question center 

qc 

math table 

[math table] The math table is a ”dinner seminar” which takes place in one of the student houses. A faculty or 

student presents a half an hour talk just after a dinner. 

grade 

[grade] Grades are an important issue for most students. Unfortunately, I have no access to your grades. 

where harvard 

[where harvard] Harvard University is located in Cambridge, Massachusetts USA. We are on the east cost. One 

can see from here Boston. The Campus is quite close to the Charles River. To look up something specific, it is 

best to start online with the Harvard search page.

get into Harvard 

[get into Harvard] Work hard, have lots of interests. You need of course some luck. 

life at Harvard 

[life at Harvard] Fun. Beside a great academic environment, there are a lot of things to see. You can spend 

weeks at Harvard square and still find new things. 

grade inflation 

[grade inflation] Grade inflation is when most students get A’s. One of the problems with grade inflation is that 

grades start losing their purpose. 

computer type 

[computer type] At the Mathematics department, people use all kind of operating systems 

• Sun work stations running Solaris 

• Macintoshs running OSX 

• PC’s running Linux 

• PC’s running Windows. 

software 

[software] Have a look at http://www.math.harvard.edu/computing. 

mathematics 

[mathematics] Mathematics is both a science and an art. It is also a language for other sciences. Quite many 

topics in Mathematics have turned out to be useful. Examples is the theory of operators which provided the 

framework for Quantum mechanics. An other example is number theory which provides the foundation for 

many encryption algorithms. 

afread of math 

[afread of math] You probably had some bad experiences in the past. You should chat more with me!

learn math 

[learn math] Just do it! There are hundreds of nice books about Math, many resources on the internet. Take a 

math class with a good teacher. 

physics and math 

[physics and math] There are not many differences. Indeed, there are branches of Mathematics like computational 

number theory where people do a lot of experiments and there are branches of physics, where people do 

very abstract theory which presumably never can be tested in laboratories. 

[charles river] A great place to row, run bike and relax. 

[harvard yard] the center of the Harvard campus 

charles river 

harvard yard 

math professors 

[math professors] They are all excellent Mathematicians. 

[pi day] 

pi day 

student number 

[student number] There are more than 18’000 degree candidates at Harvard. 

Harvard founded 

[Harvard founded] Harvard College was established in 1636, already 16 years after the arrival of the pilgrims at 

Plymouth.

people at harvard 

[people at harvard] There are over 14’000 people at Harvard including more than 2’000 faculty. There are also 

7,000 faculty appointments in affiliated teaching hospitals. 

Nobel Laureates 

[Nobel Laureates] Harvard produced nearly 40 Nobel Laureates. 

US presidents 

[US presidents] Harvard produced seven presidents of the United States: John Adams, John Quincy Adams, 

Theodore and Franklin Delano Roosevelt, Rutherford B. Hayes, John Fitzgerald Kennedy and George W. Bush. 

[Millenium prize problems] 

• P versus NP 

• Hodge Conjecture 

• Poincare Conjecture 

• Riemann Hypothesis 

• Yang-Mills Existence and Mass Gap 

• Navier-Stokes Existence and Smoothness 

• Birch-Swinnerton-Dyer Conjecture 

Millenium prize problems 

Core course 

[Core course] In 1978, Harvard adopted a ’core’ of courses in fields of inquiry that spanned domains, including 

historical study, moral reasoning, social analysis, science, music and art, literature and so on. These courses are 

designed to introduce ’approaches to knowledge’ rather than specific information and thus legitimized a trend 

throughout education toward ways of knowing rather than knowledge. 



Index 

afread of math, 3 

ca, 1 

chairman of math department, 1 

charles river, 4 

computer type, 3 

concentrator, 2 

Core course, 5 

get into Harvard, 3 

grade, 2 

grade inflation, 3 

Harvard, 1 

Harvard founded, 4 

harvard yard, 4 

head tutor, 2 

learn math, 4 

life at Harvard, 3 

math professors, 4 

math table, 2 

mathematics, 3 

Millenium prize problems, 5 

Nobel Laureates, 5 

number people math department, 1 

people at harvard, 5 

physics and math, 4 

pi day, 4 

preceptor, 1 

president of Harvard, 1 

qc, 2 

question center, 2 

Science center, 1 

software, 3 

student number, 4 

tf, 2 

US presidents, 5 

where harvard, 2 

6

ENTRY JOKES 

[ENTRY JOKES] Authors: Oliver Knill: 2002 Literature: not yet 

JOKE GARAGE SALE 

[JOKE GARAGE SALE] Pride is what you feel when your kids net 143 dollars from a garage sale. Panic is 

what you feel when you realize your car is missing. 

JOKE DOORBELL 

[JOKE DOORBELL] A priest was walking down the street when he saw a little boy jumping up and down to 

try to reach a doorbell. So the priest walked over and pressed the button for the youngster. ”And now what, 

my little man?” he asked. ”Now.” said the boy, ”run like hell!” 

[JOKE FAMOUS LAST WORDS] 

JOKE FAMOUS LAST WORDS 

postman: ”good doggy, nice doggy” 

butcher: ”could you throw me the big knife, please?” 

computer: ”are you sure? (yes/no)” 

stuntman: ”what? reality TV?” 

doorman: ”only over my dead body” 

detective” ”clear case: you are the murderer” 

muchroom picker: ”I never saw this one” 

boss: ”nice present, a lighter which looks like a revolver” 

submarine crew: ”I need some fresh air, open the window” 

sysadmin: ”I recently had a fresh backup” 

student: ”I’m going to eat in the mensa, anybody coming?” 

bungee jumper: ”hurrey!” 

PC: ”loading windows - please wait” 

JOKE PAINT JOB 

[JOKE PAINT JOB] There was a college student trying to earn some pocket money by going from house to 

house offering to do odd jobs. He explained this to a man who answered one door. ”How much will you charge 

to paint my porch?” asked the man. ”Forty dollars.” ”Fine” said the man, and gave the student the paint 

and brushes. Three hours later the paint-splattered lad knocked on the door again. ”All done!”, he says, and 

collects his money. ”By the way,” the student says, ”That’s not a Porsche, it’s a Ferrari.” 



Index 

JOKE DOORBELL, 1 

JOKE FAMOUS LAST WORDS, 1 

JOKE GARAGE SALE, 1 

JOKE PAINT JOB, 1 

2

ENTRY K12 

[ENTRY K12] Authors: Oliver Knill: 2000 Literature: not yet 

abacus 

An [abacus] is an acient mechanical computing device. It is made of beads arranged on a frame. 

absolute value 

The [absolute value] |n| of a real number n is the maximum of n and its negative −n. For example, the absolute 

value of −6 is | − 6| = 6. The absolute value is the distance from 0. 

adjacent angles 

Two angles that share a ray are called [adjacent angles]. 

affine cipher 

An [affine cipher] uses affine functions to scramble the letters in an alphabet of a secret message. For 

example, with an alphabet of 26 letters, f(x) = bx + a = 5x + 2mod(26) produces a new alphabet of 

the same size if b has no common multiple with 26. The simplest example is the Caesar cipher, where 

b = 2. It rotates the letters in an alphabet x ↦→ x + amod(26). For example, for a = 1, we get 

a b c d e f g h i j k l m n o p q r s t u v w x y z 

This ci- 

b c d e f g h i j k l m n o p q r s t u v w x y z a 

pher changes the word hello to the word ifmmp. A requently used Caesar cipher is ”rot13” defined by 

f(x) = x + 13mod(13). It has the property that encryption and decryption are the same. For example, applying 

rot13 on the word ”decryption” produces qrpelcgvba and applying rot13 on that word again gives back 

”decryption. More complicated versions of affine cyphers can be obtained by writing the to encoded text as a 

sequence of vectors x and then applying Ax + b on each vector. Affine cyphers are very easy to crack. They are 

only used to illustrate the concept like for educational purposes. 

algebra 

[algebra] is a branch of elementary mathematics that generalizes arithmetic by using variables. An example of 

an algebraic identity is x ∗ (y + z) = x ∗ y + x ∗ z. 

acute 

An angle is called [acute], if is smaller than 90 degrees. An angle which is 90 degrees is called a right angle.

addition 

[addition] is a basic operation for numbers. The result is called the sum of the two numbers. Examples: 5+3 = 8. 

2 3 4 5 

+ 9 2 3 5 

1 1 5 8 0 

More generally, a group operation in a commutative group is often called addition. Examples of groups are 

integers, real numbers, vectors or matrices. 

alternate exterior angles 

[alternate exterior angles] are angles located outside a set of two parallel lines and on opposite sides of the 

transversal line. They are equal. 

alternate interior angles 

[alternate interior angles] are angles located inside a set of parallel lines and on opposite sides of the transversal. 

angle bisector 

A ray that divides an angle into two equal angles is called an [angle bisector]. The bisector can be constructed 

with ruler and compass. An angle trisector on the other hand, a ray which splits an angle into three equal parts 

can not be constructed by ruler and compass. 

apex 

The [apex] is the highest vertex in a given orientation of a polygon. 

Arabic numerals 

[Arabic numerals]: symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that represent successive entries of words representing numbers 

in the decimal system. For example, 2347 is the number 2000 + 300 + 40 + 7. 

area 

The [area] of a surface is a measure for the number of square units needed to cover the surface. For example, 

the sphere of radius 1 has the surface area 4π.

arithmetic mean 

[arithmetic mean] Given two numbers a, b, the arithmetic mean is defined as (a + b)/2. It is sometimes also 

called the mean. Other means are the geometric mean √ ab or the harmonic mean 1/(1/a + 1/b). 

average 

The [average] of a few numbers is the sum of all the numbers divided by the number of numbers. For example, 

the average of 2, 4, 6 is (2 + 4 + 6)/3 = 4, the average of the numbers 1, 5, 8, 4 is (1+5+8+5)/3 = 19/3 The 

average is also called the mean. The average of two numbers is also called the arithmetic mean. 

base 

A [base] is the number of distinct single-digit numbers in a counting system. Example: the binary system 

has base 2. The decimal system has base 10. The base is also called radix. Numbers can be represented 

in any base r > 1. Because humans have 10 fingers, the decimal system is the one favoured by this species. 

Because computers work with circuits which are based on the principle ”on” or ”off”, they like the base 2. The 

hexadecimal system (base 16) or octal system (base 8) are also used a lot by computers. Modern computers 

can even work directly with numbers written in base 32 or 64. 

bell curve 

The [bell curve] is an other term for graph of the normal distribution f(x) = 1 √ e π −x2. 

It is also called the Gauss 

distribution. The bell curve is often seen in probability distributions. There is a reason for that called the 

central limit theorem which assures that if we average independent data with some distribution, we approach 

the normal distribution. 

billion 

A [billion] is one thousand millions in the American or French system, it is a million millions in the English or 

German system. In other words 

One billion in UK,Germany: 10 12 1’000’000’000’000 

One billion in US,France: 10 9 1’000’000’000 

binary number 

A [binary number] is a number expressed in place-value notation to the base 2. For example: 101101 represents 

the decimal number 1 + 0 + 4 + 8 + 0 + 32 = 45. 

cipher 

[cipher] Ciphers are codes used to encrypt ”secret” messages.

coefficient 

The word [coefficient] is used to denote numbers in the front of the variables in an algebraic formula. For 

example: 4x + 5y = 3 has coefficients 4, 5. 

combinatorics 

[combinatorics] The science of counting things. Combinatorics is an important part of probability and statistics. 

common factor 

A [common factor] of two integers n and m is a number which is a factor of both. A common factor is also 

called a common divisor. Examples: 3 is a common factor of 18 and 27. Also, 9 is the greatest common factor 

of 18 and 27: we write 9 = gcd(18, 27), where gcd stands for the greatest common divisor. 

complex numbers 

[complex numbers] can be written as a pair of real numbers z = x+iy, where i is a symbol which satisfies i 2 = −1. 

One can add and subtract complex numbers by adding their coefficients x, y. For example 4+5i+5−7i = 9−2i. 

complementary angles 

Two angles whose sum is 90 degrees form [complementary angles]. For example, the two non-right angles in a 

right triangle form complementary angles. 

concave up 

A graph of a function f is [concave up] if f has the property f((x + y)/2) ≤ (f(x) + f(y))/2. If f is concave 

up, then −f is concave down. For example, the graph of the function f(x) = x 2 is concave up, the graph of the 

function f(x) = −x 4 is concave down. 

conditional probability 

The [conditional probability] is the probability that an event A happens provided a second event B occurs. 

One writes P [A|B]. It satisfies P [A|B] = P [A ∩ B]/P [B], where P [B] is the probability of the event B and 

P [A ∩ B] probability of the intersection of A and B. For example, if we throw 2 coins and we know one of 

the coins is head H, then the probability that the there is also a coin with tail is 2/3. Proof: The probability 

space is X = {HT, T H, T T, HH}. The event that one of the coins is head is A = {HT, T H, HH}. The event 

B that one of the coins shows tail is B = {HT, T H, T T }. The intersection of B and A is {T H, HT }. We have 

P [B|A] = P [B ∩ A]/P [A] = (1/2)/(3/4) = 2/3.

congruent 

Two figures are called [congruent] if one can move one to an other by translation and rotation. 

constant 

A quantity that does not change in an equation is called a [constant]. 

constant function 

A [constant function] is a function which takes the same value whatever input we enter to it. 

coordinate 

A [coordinate] is an entry in a collection of numbers identifiing the point in coordinate space. 

continuous graph 

The graph of a continuous function is called a [continuous graph]. Roughly speaking, a graph of a function 

defined on some interval [a, b] is a continuous graph if one can draw the graph using a pencil without having to 

the lift the pencil. Examples: 

• 1/x is not continuous on [−1, 1]. 

• x 2 + 1 is continuous on [−1, 1]. 

• 1/x 2 is not continuous on [−1, 1]. 

• sin(1/x) is not continuous on [−1, 1]. 

• x sin(1/x) is continuous on [−1, 1]. 

• f ′ (x) is not continuous on [−1, 1] if f(x) = |x|. 

corresponding angles 

[corresponding angles] are two angles in the same relative position on two straight lines when those lines are 

intersected by a a transversal straight line. 

decimal number 

[decimal number] is a fraction, where the denominator is a power of 10. It can be expressed using a decimal 

point. For example: 0.872 is the decimal equivalent of 872/1000.

degrees 

An angle is often measured in [degrees]. The entire circle has 360 degrees, a half a circle is 180 degrees, a quarter 

circle is a right angle and has 90 degrees. A more natural unit is the length unit where the entire circle has 

angle 2π and the right angle is the angle π/2. 

denominator 

The [denominator] is the integer q below the fraction in a rational number p/q. The other number p is called 

the nominator. 

discontinuous graph 

A [discontinuous graph] is the graph of a function which is not continuous. Discontinuities can occur in different 

ways. The function can jump from one value to an other. The function can also be infinite at some point or 

the function can oscillate infinitely much at some point. Examples: 

• The graph of the function f(x) = 1/x on [−1, 1]. 

• The graph of the function f(x) = sin(1/x) on [−1, 1]. 

• The graph of the function f(x) = sign(x), which is 1 for x > 0 equal to 0 for x = 0 and −1 for x = −1. 

disjoint events 

Two events are called [disjoint events] if they have no common elements. 

division 

The inverse operation of multiplication is called [division]. 

domain 

The [domain] of a function f is the set of numbers x for which f(x) is defined. For example, the domain of the 

function f(x) = 1/x is the entire real line except the point 0. 

element 

An [element] of a set is is a member of that set. For example table is an element of the set {table, chair, floor}.

empty set 

The [empty set] ∅ is the set which does not contain any elements. 

equally likely 

If two events have the same probability they are called [equally likely]. For example, the event of throwing an 

even number with one dice is equally likely than throwing an odd number. 

event 

An [event] is a subset of the entire probability space. For example, if X = {HH, HT, T H, T T } is the probability 

space of all throwing of two coins, then A = {HH, HT } is the event that in the second throw one had a head. 

exponent 

The [exponent] of an expression a x is part x. One can get the exponent of y = a x by the formula x = 

log(y)/ log(a). 

Fibonacci numbers 

[Fibonacci numbers] are numbers obtained in the Fibonacci sequence defined by starting the numbers 0, 1 and 

defining the next element as the sum of the two previous ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .... The sequence is 

named after Leonardo of Pisa, who called himself Fibonacci, short for Filius Bonacci (= Son of Bonacci). The 

original problem he investigated in 1202 A.D. was the growth of rabbits. Explicit expressions for the n’th term 

of the sequence can be obtained using linear algebra. More generally, one can find explicit formulas for the n’th 

term in a linear recursion of the form an+1 = � k 

j=0 cjan−j. 

fractal 

A [fractal] is a set which has non-integer dimension. The term was coined by Benoit Mandelbrot in 1975. Many 

objects in nature appear to be fractals, like coast lines, trees, mountains. One can mathematically define fractals 

using iterative constructions. Examples are the Koch curve, the snow flake, the Menger Sponge, the Shirpinsky 

carpet, the Cantor set. 

fraction 

A [fraction] is a rational number written in the form a/b, where a is called the numerator and b is called the 

denominator.

function 

A [function] f of a variable x is a rule that assigns to each number x in the function’s domain a single number 

f(x). For example f(x) = x 2 is a function which assigns to each number its square like f(4) = 16. 

geometric sequence 

The [geometric sequence] is a sequence where each element is a multiple of the previous element. For example: 

1, 2, 4, 8, 16, 32, 64, ... is a geometric sequence. 

graph of the function 

The [graph of the function] is the set of all points (x, f(x)) in the plane, where x in the domain of f. 

greatest common factor 

The [greatest common factor] of two numbers n,m is the larest common factor of both. One denotes the greatest 

common factor with ”gcd”. Examples: 

6 is the greatest common factor of 12 and 18 6=gcd(12,18) 

8 is the greatest common factor of 8 and 80. 8=gcd(8,80) 

1 is the greatest common factor of 7 and 11 1=gcd(7,11) 

[greatest common divisor] see greatest common factor. 

greatest common divisor 

histogram 

A [histogram] is a bar graph in which area over each range of values is proportional to the relative frequency of 

the data in this interval. 

hypotenuse 

The [hypotenuse] of a right triangle is the opposite side to the right angle. 

independent events 

Two events A and B are called [independent events] if the probability that both happen is the product of the 

probabilities that each occurs alone: P [A ∩ B] = P (A)P (B). Using conditional probability one can write this 

as P [A|B] = P [A]. Knowing A under the condition B is the same as knowing A without knowing B.

infinity 

[infinity] is a ”number” which is larger than any other number. One writes ∞. One should rather treat of it as a 

symbol evenso some computations can be extended to the real numbers including ∞ like ∞ + x = ∞, ∞ + ∞ = 

∞, x ∗ ∞ = ∞ for x > 0, x ∗ ∞ = −∞ for x < 0 or ∞ ∗ ∞ = ∞, (−∞) ∗ ∞ = −∞. One can not define ∞ − ∞ 

in a consistent way nor can one do that with 0 ∗ ∞. Also the expression 1/0 = ∞ is ill defined because 1/x 

takes near x = 0 arbitrary large and arbitrary small values. 

integer 

An [integer] is a number of the form n or −n, where n is a natural number. Examples of integers are ... − 

3, 2, 1, 0, 1, 2, 3, 4.... The fraction 2/5 is not an integer. 

intersection 

The [intersection] of two or more sets is the set of elements which are in both sets. One writes A ∩ B for the 

intersection of A and B. 

isosceles triangle 

An [isosceles triangle] is a triangle which has at least two congruent sides. A special case is the isocline triangle 

in which all sides are congruent. 

least common multiple 

The [least common multiple] of two numbers n, m is the least common multiple of both. One denotes the least 

common multiple with ”lcm”. Examples: 

18 is the least common multiple of 9 and 6 18=lcm(9,6) 

77 is the least common multiple of 7 and 11 77=gcd(7,11) 

limit 

The [limit] of a sequence of numbers is the limiting value the sequence converges to. It needs not to exist. For 

example, the sequence an = 1/n converges to 0. One sais that 0 is the limit of that sequence. The sequence 

an = n has no finite limit. One could assign infinity as a limit. The sequence 1, −1, 1, −1, 1, −1, ... has no limit. 

logarithm 

The [logarithm] of b is the exponent to which one has to rize a base number to get b. For example, 2 is the 

logarithm of 100 to the base 10 or 10 is the logarithm of 1024 to the base 2.

mean 

The [mean] of a list of numbers is their sum divided by the total number of members in the list. It is also called 

arithmetic mean. 

median 

The [median] is the ”middle value” of a list. If the list has an odd number 2m + 1 elements, the median is the 

number in the list such that m scores are smaller or equal and m scores are bigger or equal. If the list has an 

even number of elements, one usually takes the algebraic average between the middle to elements Examples: 

med(1, 1, 2, 2) = 3/2, med(1, 2, 3, 4, 7) = 3, med(1, 2, 3, 4, 5, 6) = 7/2. 

multiplication table 

[multiplication table] A table of products of numbers which has to be memorized. 

1 2 3 4 5 6 7 8 9 10 

1 1 2 3 4 5 6 7 8 9 10 

2 2 4 6 8 10 12 14 16 18 20 

3 3 6 9 12 15 18 21 24 27 30 

4 4 8 12 16 20 24 28 32 36 40 

5 5 10 15 20 25 30 35 40 45 50 

6 6 12 18 24 30 36 42 48 54 60 

7 7 14 21 28 35 42 49 56 63 70 

8 8 16 24 32 40 48 56 64 72 80 

9 9 18 27 36 45 54 63 72 81 90 

10 10 20 30 40 50 60 70 80 90 100 

The diagonal contains squares. All numbers between 11 and 99 which do not appear in this table are prime 

numbers, numbers only divisible by 1 and itself. 

obtuse angle 

An angle whose measure is greater than 90 degrees is called an [obtuse angle]. 

optical illusion 

An [optical illusion] is a drawing of an object that makes certain things appear which it does not have. 

palindrome 

A [palindrome] is a word or number that is the same when read backwards. Examples: ”otto”, ”anna”, ”racecar”, 

”78777787”.

paradox 

A [paradox] is a statement that appears to contradict itself. For example, the statement ”I always lie” is a 

paradox. If I tell the truth, then I lie, if I lie, then I tell the truth. 

Two lines which do not intersect are called [parallel]. 

parallel 

parallelogram 

A [parallelogram] is a quadrilateral that contains two pairs of parallel sides. 

pattern 

A [pattern] is a characteristic observed in one item that may be repeated in other items. For example, the 

sequence 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, ... has a pattern which is also visible in a similar way in the sequence 

1, 3, 4, 3, 1, 3, 4, 3, 1, 3, 4, 3, 1, .... 

percent 

A [percent] is one hunderth. The symbol for percent is %. For example 0.1 is 10 percent. 2 is two hundred 

percent. 

perimeter 

The [perimeter] of a polygon is the sum of the lengths of all the sides of the polygon. 

permutation 

A [permutation] is a rearrangement of objects in a set. There are for example 6 permutations of the set 

A = (a, b, c). They are (a, b, c), (a, c, b), (b, a, c), (b, c, a), (c, a, b), (c, b, a). 

polygon 

A [polygon] is a closed plane figure formed by connecting a finite set of points in such a way that they do not 

cross each other.

polyhedra 

[polyhedra] A solid figure for which the outer surface is composed of polygons. 

prime number 

A [prime number] is a number which is divisible only by 1 and itself. The first prime numbers are 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.... The number 33 for example is no prime number because it is divisible 

by 3. 

quadrant 

A [quadrant] is one of the regions in the plane obtained when cutting the plane along the coordinate axes. 

• The first quadrant contains all the points with positive x and positive y coordinates. 

• The second quadrant contains all the points with negative x and positive y coordinates. 

• The third quadrant contains all the points with negative x and negative y coordinates. 

• The fourth quadrant contains all the points with positive x and negative y coordinates. 

quadratic function 

A function of the form f(x) = ax 2 + bx + c is called a [quadratic function]. For example f(x) = x 2 + 2 is a 

quadratic function. The graph of a quadratic function is a parabola if a is not zero. If a is zero it is a linear 

function which has as the graph a line. 

quotient 

The [quotient] of two numbers n and m is the largest integer smaller or equal to n/m. for example the quotient 

of 11 and 4 is 2 with areminder of 3. 

[smallest common multiple] see least common multiple. 

smallest common multiple

subtraction 

[subtraction] is a basic operation for numbers. Examples: 5 − 3 = 2. 

2 3 5 2 

- 1 4 5 6 

8 9 6 

More generally in any commutative group, the addition of the inverse -b of an element b to an element a, 

denoted by a-b is called a subtraction. 

multiplication 

[multiplication] is a basic operation for numbers. Example: 4 ∗ 5 = 20. 

1 2 4 x 1 2 3 3 4 

+ 4 9 3 3 6 

+ 2 4 6 6 8 

+ 1 2 3 3 4 

1 5 2 9 4 1 6 

More generally, in any group the group operation is called multiplication. Example: The composition of two 

transformations in the plane is a multiplication, matrix multiplication is a multiplication. 

fraction 

[fraction]: The representation of a rational number r as p/q, where p and q �= 0 are integers is called a fraction. 

Fractions can be added, subtracted, multiplied and one can divide by nonzero fractions. 

Addition of fractions: 5/3 + 7/5 = 25/15 + 21/15 = 46/15 

Subtraction of fractions: 5/3 − 7/5 = 25/15 − 21/15 = 4/15 

Multiplication of fractions: 5/3x7/5 = 35/15 

Division of fractions: 5/3 : 7/5 = 5/3x5/7 = 25/21 

Not all numbers are rational numbers. Non rational numbers are called irrational. For example, sqr(2) can not 

be written as p/q because � (2) = p/q would imply 2 = p 2 /q 2 or 2q 2 = p 2 . But by the uniqueness of prime 

factorisation of integers the left hand side has an odd number of factors 2, the right hand side an even number. 

� (2) is an example of an irrational number. 

A [hexagon] is a polygon with six sides. 

hexagon 

parallelogram 

A [parallelogram] is a quadrilateral whose opposite sides are parallel and congruent.

polygon 

A [polygon] is a closed curve in the plane formed by three or more line segments. One usually assumes that the 

segments don’t intersect. Examples: 

A [quadrilateral] is a polygon with four sides. 

3 sides: triangle 

4 sides: quadrilateral, (i.e. rectangle, rhombus, rhombus) 

5 sides: pentagon 

6 sides: hexagon 

7 sides: septagon 

8 sides: octogon 

quadrilateral 

parallel 

Two lines in the plane are called [parallel] if they do not intersect. Two parallel lines can be translated into 

each other. Two lines in space are called [parallel] if they can be translated into each other. Unlike in the plane, 

two lines in space which are not parallel do not need to intersect. 

triangle 

A [triangle] is a polygon defined by three points in the plane. The three points form the edges of the triangles, 

the three connections of the points form the sides of the triangle. 

random number generator 

A [random number generator] is a device used to produce random numbers. In daily life like for gambling, one 

often uses dice or coin tossing to find random numbers. Computers often use pseudo random number generators, 

which are deterministic sequences which look random. Computers can also access hardware internal staes of 

the computer to improve randomness. 

range of a function 

The [range of a function] is the set of all values f(x), where x is in the domain of f. 

ratio 

[ratio] A rational number of the form a/b where a is called the numerator and b is called the denominator.

ectangle 

A [rectangle] is a parallelogram with four right angles. It is a quadrilateral, a polygon with four points in the 

plane. All angles have to be right angles. In a rectangle, opposite sides are parallel. A rectangle is therefore a 

special parallelogram. 

regular polygon 

A [regular polygon] is a polygon which has sides of equal length and equal angles. Squares, equilateral triangles 

or regular hexagons are examples of regular polygons. 

remainder 

The [remainder] of a division p/q is the amount left after subtracting the maximal integer multiple of q from p. 

For example 7/3 has the reminder 1 because 7 − 2 ∗ 3 = 1. −11/5 has the reminder −1. 

rhombus 

A [rhombus] is a parallelogram with four congruent sides. A special case is the square. 

An angle of 90 degrees is called a [right angle]. 

right angle 

right triangle 

A triangle which has a right angle is also called a [right triangle]. 

sequence 

An ordered list of elements is called a [sequence] For example, (1, 3, 2, 1) is a list of elements which form a finite 

sequence. The list (1, 2, 3, 4, 5, 6, ....) forms an infinite sequence. 

set 

A [set] is a collection of things, without regard to their order.

slope 

The [slope] of a line y = mx + b is the number m. One often measures it in percentages. m = 1 means 100 

percent. If a street has a slope of 10 percent, one climbes for every 10 meters going forwards 1 meter up. 

square 

A [square] is a polygonal shape in the plane with four sides where each side has the same length and all sides 

are perpendicular on each other. A square is also a number of the form n*n like 65=8*8. A square with integer 

side length has a square number as the area. 

subset 

A [subset] of a set is a set of elements which are all contained in that set. For example the set A = {1, 2, 3, 8, 4} 

has B = {2, 3} as a subset. 

subtraction 

[subtraction] is the operation of taking the difference between two numbers. For example 7 − 2 = 5. 

tessellation 

A [tessellation] is a cover of the plane using a finite set of polygons without leaving gaps or overlaps. Examples 

are regular tessellation into triangles or squares or regular hexagons. Semiregular tesselations allow to cover 

the plane with different types of shapes. Tesselations are also called tilings and can be defined also in higher 

dimensions. 

trapezoid 

A [trapezoid] is a quadrilateral with one pair of parallel sides. 

union of sets 

The [union of sets] is the set which contains the elements of all sets. One writes A ∪ B. For example, if 

A = {1, 2, 3, 4} and B = {0, 2, 4, 6}, then A ∪ B = {0, 1, 2, 3, 4, 6}. 

Venn Diagram 

In a [Venn Diagram] sets are represented as simple geometric shapes. It visualizes intersections and unions of 

sets. For example if A is the set of all even numbers between 0 and 10 and B is the set of all numbers divisible 

by 3 between 0 and 10 one can visualize this with two circles, one of which contains 2, 5, 8, 6, the other 3, 6, 9. 

The circles intersect in a region which has the single element 6.

whole number 

A [whole number] is one of the numbers 0, 1, 2, 3, 4, ... A whole number is also called natural number or nonnegative 

integer. 



Index 

abacus, 1 

absolute value, 1 

acute, 1 

addition, 2 

adjacent angles, 1 

affine cipher, 1 

algebra, 1 

alternate exterior angles, 2 

alternate interior angles, 2 

angle bisector, 2 

apex, 2 

Arabic numerals, 2 

area, 2 

arithmetic mean, 3 

average, 3 

base, 3 

bell curve, 3 

billion, 3 

binary number, 3 

cipher, 3 

coefficient, 4 

combinatorics, 4 

common factor, 4 

complementary angles, 4 

complex numbers, 4 

concave up, 4 

conditional probability, 4 

congruent, 5 

constant, 5 

constant function, 5 

continuous graph, 5 

coordinate, 5 

corresponding angles, 5 

decimal number, 5 

degrees, 6 

denominator, 6 

discontinuous graph, 6 

disjoint events, 6 

division, 6 

domain, 6 

element, 6 

empty set, 7 

equally likely, 7 

event, 7 

exponent, 7 

Fibonacci numbers, 7 

fractal, 7 

fraction, 7, 13 

function, 8 

geometric sequence, 8 

graph of the function, 8 

greatest common divisor, 8 

greatest common factor, 8 

hexagon, 13 

18 

histogram, 8 

hypotenuse, 8 

independent events, 8 

infinity, 9 

integer, 9 

intersection, 9 

isosceles triangle, 9 

least common multiple, 9 

limit, 9 

logarithm, 9 

mean, 10 

median, 10 

multiplication, 13 

multiplication table, 10 

obtuse angle, 10 

optical illusion, 10 

palindrome, 10 

paradox, 11 

parallel, 11, 14 

parallelogram, 11, 13 

pattern, 11 

percent, 11 

perimeter, 11 

permutation, 11 

polygon, 11, 14 

polyhedra, 12 

prime number, 12 

quadrant, 12 

quadratic function, 12 

quadrilateral, 14 

quotient, 12 

random number generator, 14 

range of a function, 14 

ratio, 14 

rectangle, 15 

regular polygon, 15 

remainder, 15 

rhombus, 15 

right angle, 15 

right triangle, 15 

sequence, 15 

set, 15 

slope, 16 

smallest common multiple, 12 

square, 16 

subset, 16 

subtraction, 13, 16 

tessellation, 16 

trapezoid, 16 

triangle, 14 

union of sets, 16

Venn Diagram, 16 

whole number, 17

[you sound tired] I worked all night 

[This is great] this is fantastic 

[ok] not ok 

[switzerland] country center europe 

[Germany] country in Europe 

[Austria] country in Europe 

you sound tired 

This is great 

ok 

switzerland 

Germany 

Austria 



Index 

Austria, 1 

Germany, 1 

ok, 1 

switzerland, 1 

This is great, 1 

2

Famous Literature 

[Famous Literature] List from http://www.literaturepage.com 

[A Christmas Carol] Charles Dickens 

A Christmas Carol 

A Connecticut Yankee in King Arthur’s Court 

[A Connecticut Yankee in King Arthur’s Court] Mark Twain 

[A Dream Within a Dream] Edgar Allan Poe 

[A Midsummer Night’s Dream] William Shakespeare 

[A Princess of Mars] Edgar Rice Burroughs 

[A Room With a View] E. M. Forster 

[A Study in Scarlet] Sir Arthur Conan Doyle 

[A Tale of Two Cities] Charles Dickens 

A Dream Within a Dream 

A Midsummer Night’s Dream 

A Princess of Mars 

A Room With a View 

A Study in Scarlet 

A Tale of Two Cities

[A Thief in the Night] E. W. Hornung 

[A Treatise on Government] Aristotle 

[A Woman of No Importance] Oscar Wilde 

[A Woman of Thirty] Honore de Balzac 

A Thief in the Night 

A Treatise on Government 

A Woman of No Importance 

A Woman of Thirty 

Adventures of Sherlock Holmes 

[Adventures of Sherlock Holmes] Sir Arthur Conan Doyle 

[Aesop’s Fables] Aesop 

[Agnes Grey] Anne Bronte 

[Alice’s Adventures in Wonderland] Lewis Carroll 

[All’s Well That Ends Well] William Shakespeare 

Aesop’s Fables 

Agnes Grey 

Alice’s Adventures in Wonderland 

All’s Well That Ends Well

[Allan Quatermain] H. Rider Haggard 

[Allan’s Wife] H. Rider Haggard 

[An Ideal Husband] Oscar Wilde 

[An Occurrence at Owl Creek Bridge] Ambrose Bierce 

[Andersen’s Fairy Tales] Hans Christian Andersen 

[Anna Karenina] Leo Tolstoy 

[Annabel Lee] Edgar Allan Poe 

[Anne of Green Gables] L. M. Montgomery 

[Antony and Cleopatra] William Shakespeare 

Allan Quatermain 

Allan’s Wife 

An Ideal Husband 

An Occurrence at Owl Creek Bridge 

Andersen’s Fairy Tales 

Anna Karenina 

Annabel Lee 

Anne of Green Gables 

Antony and Cleopatra

[Apology] Plato 

[Around the World in Eighty Days] Jules Verne 

[As You Like It] William Shakespeare 

[At the Earth’s Core] Edgar Rice Burroughs 

[Barnaby Rudge] Charles Dickens 

[Berenice] Edgar Allan Poe 

[Black Beauty] Anna Sewell 

[Bleak House] Charles Dickens 

[Buttered Side Down] Edna Ferber 

Apology 

Around the World in Eighty Days 

As You Like It 

At the Earth’s Core 

Barnaby Rudge 

Berenice 

Black Beauty 

Bleak House 

Buttered Side Down

[Cousin Betty] Honore de Balzac 

[Crime and Punishment] Fyodor Dostoevsky 

[Daisy Miller] Henry James 

[David Copperfield] Charles Dickens 

[Dead Men Tell No Tales] E. W. Hornung 

[Discourse on the Method] Rene Descartes 

[Dorothy and the Wizard in Oz] L. Frank Baum 

[Dracula] Bram Stoker 

[Eight Cousins] Louisa May Alcott 

Cousin Betty 

Crime and Punishment 

Daisy Miller 

David Copperfield 

Dead Men Tell No Tales 

Discourse on the Method 

Dorothy and the Wizard in Oz 

Dracula 

Eight Cousins

[Emma] Jane Austen 

[Essays and Lectures] Oscar Wilde 

[Essays of Francis Bacon] Sir Francis Bacon 

[Essays, First Series] Ralph Waldo Emerson 

[Essays, Second Series] Ralph Waldo Emerson 

[Ethan Frome] Edith Wharton 

[Cousin Betty] Honore de Balzac 

[Lady Windermere’s Fan] Oscar Wilde 

[Lincoln’s First Inaugural Address] Abraham Lincoln 

Emma 

Essays and Lectures 

Essays of Francis Bacon 

Essays, First Series 

Essays, Second Series 

Ethan Frome 

Cousin Betty 

Lady Windermere’s Fan 

Lincoln’s First Inaugural Address

[Lincoln’s Second Inaugural Address] Abraham Lincoln 

[Little Men] Louisa May Alcott 

[Little Women] Louisa May Alcott 

[Macbeth] William Shakespeare 

[Main Street] Sinclair Lewis 

[Mansfield Park] Jane Austen 

[Memoirs of Sherlock Holmes] Sir Arthur Conan Doyle 

[Middlemarch] George Eliot 

[Moby Dick] Herman Melville 

Lincoln’s Second Inaugural Address 

Little Men 

Little Women 

Macbeth 

Main Street 

Mansfield Park 

Memoirs of Sherlock Holmes 

Middlemarch 

Moby Dick

[Moll Flanders] Daniel Defoe 

[Much Ado About Nothing] William Shakespeare 

[Night and Day] Virginia Woolf 

[Northanger Abbey] Jane Austen 

[Nostromo] Joseph Conrad 

[O Pioneers!] Willa Cather 

[Of Human Bondage] W. Somerset Maugham 

[Oliver Twist] Charles Dickens 

[On the Decay of the Art of Lying] Mark Twain 

Moll Flanders 

Much Ado About Nothing 

Night and Day 

Northanger Abbey 

Nostromo 

O Pioneers! 

Of Human Bondage 

Oliver Twist 

On the Decay of the Art of Lying

On the Duty of Civil Disobedience 

[On the Duty of Civil Disobedience] Henry David Thoreau 

[Othello, Moor of Venice] William Shakespeare 

[Ozma of Oz] L. Frank Baum 

[Pandora] Henry James 

[Paradise Lost] John Milton 

[Persuasion] Jane Austen 

[Poems of Edgar Allan Poe] Edgar Allan Poe 

[Pollyanna] Eleanor H. Porter 

[Pride and Prejudice] Jane Austen 

Othello, Moor of Venice 

Ozma of Oz 

Pandora 

Paradise Lost 

Persuasion 

Poems of Edgar Allan Poe 

Pollyanna 

Pride and Prejudice

[Pygmalion] George Bernard Shaw 

Pygmalion 

Raffles: Further Adventures of the Amateur Cracksman 

[Raffles: Further Adventures of the Amateur Cracksman] E. W. Hornung 

[Rebecca Of Sunnybrook Farm] Kate Douglas Wiggin 

[Robinson Crusoe] Daniel Defoe 

[Romeo and Juliet] William Shakespeare 

[Rose in Bloom] Louisa May Alcott 

[Sense and Sensibility] Jane Austen 

[Silas Marner] George Eliot 

[Tarzan of the Apes] Edgar Rice Burroughs 

Rebecca Of Sunnybrook Farm 

Robinson Crusoe 

Romeo and Juliet 

Rose in Bloom 

Sense and Sensibility 

Silas Marner 

Tarzan of the Apes

[Tess of the d’Urbervilles] Thomas Hardy 

[The Adventures of Huckleberry Finn] Mark Twain 

[The Adventures of Pinocchio] Carlo Collodi 

[The Adventures of Tom Sawyer] Mark Twain 

[The Aeneid] Virgil 

[The Age of Innocence] Edith Wharton 

[The Amateur Cracksman] E. W. Hornung 

[The Analysis of Mind] Bertrand Russell 

[The Ballad of Reading Gaol] Oscar Wilde 

Tess of the d’Urbervilles 

The Adventures of Huckleberry Finn 

The Adventures of Pinocchio 

The Adventures of Tom Sawyer 

The Aeneid 

The Age of Innocence 

The Amateur Cracksman 

The Analysis of Mind 

The Ballad of Reading Gaol

[The Bells] Edgar Allan Poe 

[The Call of the Wild] Jack London 

[The Cask of Amontillado] Edgar Allan Poe 

[The Categories] Aristotle 

[The Chessmen of Mars] Edgar Rice Burroughs 

[The Comedy of Errors] William Shakespeare 

[The Count of Monte Cristo] Alexandre Dumas 

[The Emerald City of Oz] L. Frank Baum 

[The Fall of the House of Usher] Edgar Allan Poe 

The Bells 

The Call of the Wild 

The Cask of Amontillado 

The Categories 

The Chessmen of Mars 

The Comedy of Errors 

The Count of Monte Cristo 

The Emerald City of Oz 

The Fall of the House of Usher

[The Four Million] O. Henry 

[The Gambler] Fyodor Dostoevsky 

[The Gettysburg Address] Abraham Lincoln 

[The Gods of Mars] Edgar Rice Burroughs 

[The Happy Prince and Other Tales] Oscar Wilde 

The Four Million 

The Gambler 

The Gettysburg Address 

The Gods of Mars 

The Happy Prince and Other Tales 

The History of Tom Jones, a foundling 

[The History of Tom Jones, a foundling] Henry Fielding 

The History of Troilus and Cressida 

[The History of Troilus and Cressida] William Shakespeare 

The Hound of the Baskervilles 

[The Hound of the Baskervilles] Sir Arthur Conan Doyle 

[The Hunchback of Notre Dame] Victor Hugo 

The Hunchback of Notre Dame

[The Hunting of the Snark] Lewis Carroll 

[The Idiot] Fyodor Dostoevsky 

[The Iliad] Homer 

[The Importance of Being Earnest] Oscar Wilde 

[The Innocence of Father Brown] G. K. Chesterton 

[The Innocents Abroad] Mark Twain 

[The Invisible Man] H. G. Wells 

[The Island of Doctor Moreau] H. G. Wells 

[The Jungle Book] Rudyard Kipling 

The Hunting of the Snark 

The Idiot 

The Iliad 

The Importance of Being Earnest 

The Innocence of Father Brown 

The Innocents Abroad 

The Invisible Man 

The Island of Doctor Moreau 

The Jungle Book

[The Last Days of Pompeii] Edward Bulwer-Lytton 

[The Last of the Mohicans] James Fenimore Cooper 

[The Legend of Sleepy Hollow] Washington Irving 

The Last Days of Pompeii 

The Last of the Mohicans 

The Legend of Sleepy Hollow 

The Life and Adventures of Nicholas Nickleby 

[The Life and Adventures of Nicholas Nickleby] Charles Dickens 

The Life and Death of King Richard III 

[The Life and Death of King Richard III] William Shakespeare 

[The Life of King Henry V] William Shakespeare 

[The Lost Continent] Edgar Rice Burroughs 

[The Lost World] Sir Arthur Conan Doyle 

[The Man in the Iron Mask] Alexandre Dumas 

The Life of King Henry V 

The Lost Continent 

The Lost World 

The Man in the Iron Mask

The Man Upstairs and Other Stories 

[The Man Upstairs and Other Stories] P. G. Wodehouse 

[The Man Who Knew Too Much] G. K. Chesterton 

[The Man with Two Left Feet] P. G. Wodehouse 

[The Masque of the Red Death] Edgar Allan Poe 

[The Merchant of Venice] William Shakespeare 

[The Merry Adventures of Robin Hood] Howard Pyle 

[The Merry Wives of Windsor] William Shakespeare 

[The Moon and Sixpence] W. Somerset Maugham 

[The Moonstone] Wilkie Collins 

The Man Who Knew Too Much 

The Man with Two Left Feet 

The Masque of the Red Death 

The Merchant of Venice 

The Merry Adventures of Robin Hood 

The Merry Wives of Windsor 

The Moon and Sixpence 

The Moonstone

[The Murders in the Rue Morgue] Edgar Allan Poe 

[The Mysterious Affair at Styles] Agatha Christie 

[The Mystery of Edwin Drood] Charles Dickens 

[The Mystery of the Yellow Room] Gaston Leroux 

[The Odyssey] Homer 

The Murders in the Rue Morgue 

The Mysterious Affair at Styles 

The Mystery of Edwin Drood 

The Mystery of the Yellow Room 

The Odyssey 

The Origin of Species means of Natural Selection 

[The Origin of Species] means of Natural Selection] Charles Darwin 

[The Phantom of the Opera] Gaston Leroux 

[The Picture of Dorian Gray] Oscar Wilde 

[The Pit and the Pendulum] Edgar Allan Poe 

The Phantom of the Opera 

The Picture of Dorian Gray 

The Pit and the Pendulum

[The Portrait of a Lady] Henry James 

[The Purloined Letter] Edgar Allan Poe 

[The Raven] Edgar Allan Poe 

[The Red Badge of Courage] Stephen Crane 

[The Republic] Plato 

The Portrait of a Lady 

The Purloined Letter 

The Raven 

The Red Badge of Courage 

The Republic 

The Return of Sherlock Holmes 

[The Return of Sherlock Holmes] Sir Arthur Conan Doyle 

The Rime of the Ancient Mariner 

[The Rime of the Ancient Mariner] Samuel Taylor Coleridge 

[The Scarecrow of Oz] L. Frank Baum 

[The Scarlet Letter] Nathaniel Hawthorne 

The Scarecrow of Oz 

The Scarlet Letter

[The Sonnets] William Shakespeare 

The Sonnets 

The Strange Case of Dr. Jekyll and Mr. Hyde 

[The Strange Case of Dr. Jekyll and Mr. Hyde] Robert Louis Stevenson 

[The Taming of the Shrew] William Shakespeare 

[The Tell-Tale Heart] Edgar Allan Poe 

[The Tempest] William Shakespeare 

[The Tenant of Wildfell Hall] Anne Bronte 

[The Three Musketeers] Alexandre Dumas 

[The Time Machine] H. G. Wells 

[The Tin Woodman of Oz] L. Frank Baum 

The Taming of the Shrew 

The Tell-Tale Heart 

The Tempest 

The Tenant of Wildfell Hall 

The Three Musketeers 

The Time Machine 

The Tin Woodman of Oz

[The Tragedy of Coriolanus] William Shakespeare 

[The Tragedy of King Lear] William Shakespeare 

The Tragedy of Coriolanus 

The Tragedy of King Lear 

The Tragedy of King Richard the Second 

[The Tragedy of King Richard the Second] William Shakespeare 

[The Voyage Out] Virginia Woolf 

[The War in the Air] H. G. Wells 

[The War of the Worlds] H. G. Wells 

[The Wind in the Willows] Kenneth Grahame 

[The Wisdom of Father Brown] G. K. Chesterton 

[The Wonderful Wizard of Oz] L. Frank Baum 

The Voyage Out 

The War in the Air 

The War of the Worlds 

The Wind in the Willows 

The Wisdom of Father Brown 

The Wonderful Wizard of Oz

[Three Men in a Boat] Jerome K. Jerome 

[Through the Looking Glass] Lewis Carroll 

[Thus Spake Zarathustra] Friedrich Nietzsche 

[Thuvia, Maid of Mars] Edgar Rice Burroughs 

[To Helen] Edgar Allan Poe 

[Treasure Island] Robert Louis Stevenson 

[Twelfth Night] William Shakespeare 

Three Men in a Boat 

Through the Looking Glass 

Thus Spake Zarathustra 

Thuvia, Maid of Mars 

To Helen 

Treasure Island 

Twelfth Night 

Twenty Thousand Leagues Under the Seas 

[Twenty Thousand Leagues Under the Seas] Jules Verne 

[Twenty Years After] Alexandre Dumas 

Twenty Years After

[Typee] Herman Melville 

[Ulalume] Edgar Allan Poe 

[Uneasy Money] P. G. Wodehouse 

Typee 

Ulalume 

Uneasy Money 

Up From Slavery: An Autobiography 

[Up From Slavery: An Autobiography] Booker T. Washington 

[Vanity Fair] William Makepeace Thackeray 

[Walden] Henry David Thoreau 

[War and Peace] Leo Tolstoy 

[Warlord of Mars] Edgar Rice Burroughs 

[White Fang] Jack London 

Vanity Fair 

Walden 

War and Peace 

Warlord of Mars 

White Fang

Wuthering Heights 



Index 

A Christmas Carol, 1 

A Connecticut Yankee in King Arthur’s Court, 1 

A Dream Within a Dream, 1 

A Midsummer Night’s Dream, 1 

A Princess of Mars, 1 

A Room With a View, 1 

A Study in Scarlet, 1 

A Tale of Two Cities, 1 

A Thief in the Night, 2 

A Treatise on Government, 2 

A Woman of No Importance, 2 

A Woman of Thirty, 2 

Adventures of Sherlock Holmes, 2 

Aesop’s Fables, 2 

Agnes Grey, 2 

Alice’s Adventures in Wonderland, 2 

All’s Well That Ends Well, 2 

Allan Quatermain, 3 

Allan’s Wife, 3 

An Ideal Husband, 3 

An Occurrence at Owl Creek Bridge, 3 

Andersen’s Fairy Tales, 3 

Anna Karenina, 3 

Annabel Lee, 3 

Anne of Green Gables, 3 

Antony and Cleopatra, 3 

Apology, 4 

Around the World in Eighty Days, 4 

As You Like It, 4 

At the Earth’s Core, 4 

Barnaby Rudge, 4 

Berenice, 4 

Black Beauty, 4 

Bleak House, 4 

Buttered Side Down, 4 

Cousin Betty, 5, 6 

Crime and Punishment, 5 

Daisy Miller, 5 

David Copperfield, 5 

Dead Men Tell No Tales, 5 

Discourse on the Method, 5 

Dorothy and the Wizard in Oz, 5 

Dracula, 5 

Eight Cousins, 5 

Emma, 6 

Essays and Lectures, 6 

Essays of Francis Bacon, 6 

Essays, First Series, 6 

Essays, Second Series, 6 

Ethan Frome, 6 

Lady Windermere’s Fan, 6 

Lincoln’s First Inaugural Address, 6 

Lincoln’s Second Inaugural Address, 7 

Little Men, 7 

Little Women, 7 

24 

Macbeth, 7 

Main Street, 7 

Mansfield Park, 7 

Memoirs of Sherlock Holmes, 7 

Middlemarch, 7 

Moby Dick, 7 

Moll Flanders, 8 

Much Ado About Nothing, 8 

Night and Day, 8 

Northanger Abbey, 8 

Nostromo, 8 

O Pioneers, 8 

Of Human Bondage, 8 

Oliver Twist, 8 

On the Decay of the Art of Lying, 8 

On the Duty of Civil Disobedience, 9 

Othello, Moor of Venice, 9 

Ozma of Oz, 9 

Pandora, 9 

Paradise Lost, 9 

Persuasion, 9 

Poems of Edgar Allan Poe, 9 

Pollyanna, 9 

Pride and Prejudice, 9 

Pygmalion, 10 

Raffles: Further Adventures of the Amateur Cracksman, 

10 

Rebecca Of Sunnybrook Farm, 10 

Robinson Crusoe, 10 

Romeo and Juliet, 10 

Rose in Bloom, 10 

Sense and Sensibility, 10 

Silas Marner, 10 

Tarzan of the Apes, 10 

Tess of the d’Urbervilles, 11 

The Adventures of Huckleberry Finn, 11 

The Adventures of Pinocchio, 11 

The Adventures of Tom Sawyer, 11 

The Aeneid, 11 

The Age of Innocence, 11 

The Amateur Cracksman, 11 

The Analysis of Mind, 11 

The Ballad of Reading Gaol, 11 

The Bells, 12 

The Call of the Wild, 12 

The Cask of Amontillado, 12 

The Categories, 12 

The Chessmen of Mars, 12 

The Comedy of Errors, 12 

The Count of Monte Cristo, 12 

The Emerald City of Oz, 12 

The Fall of the House of Usher, 12 

The Four Million, 13 

The Gambler, 13 

The Gettysburg Address, 13

The Gods of Mars, 13 

The Happy Prince and Other Tales, 13 

The History of Tom Jones, a foundling, 13 

The History of Troilus and Cressida, 13 

The Hound of the Baskervilles, 13 

The Hunchback of Notre Dame, 13 

The Hunting of the Snark, 14 

The Idiot, 14 

The Iliad, 14 

The Importance of Being Earnest, 14 

The Innocence of Father Brown, 14 

The Innocents Abroad, 14 

The Invisible Man, 14 

The Island of Doctor Moreau, 14 

The Jungle Book, 14 

The Last Days of Pompeii, 15 

The Last of the Mohicans, 15 

The Legend of Sleepy Hollow, 15 

The Life and Adventures of Nicholas Nickleby, 15 

The Life and Death of King Richard III, 15 

The Life of King Henry V, 15 

The Lost Continent, 15 

The Lost World, 15 

The Man in the Iron Mask, 15 

The Man Upstairs and Other Stories, 16 

The Man Who Knew Too Much, 16 

The Man with Two Left Feet, 16 

The Masque of the Red Death, 16 

The Merchant of Venice, 16 

The Merry Adventures of Robin Hood, 16 

The Merry Wives of Windsor, 16 

The Moon and Sixpence, 16 

The Moonstone, 16 

The Murders in the Rue Morgue, 17 

The Mysterious Affair at Styles, 17 

The Mystery of Edwin Drood, 17 

The Mystery of the Yellow Room, 17 

The Odyssey, 17 

The Origin of Species means of Natural Selection, 17 

The Phantom of the Opera, 17 

The Picture of Dorian Gray, 17 

The Pit and the Pendulum, 17 

The Portrait of a Lady, 18 

The Purloined Letter, 18 

The Raven, 18 

The Red Badge of Courage, 18 

The Republic, 18 

The Return of Sherlock Holmes, 18 

The Rime of the Ancient Mariner, 18 

The Scarecrow of Oz, 18 

The Scarlet Letter, 18 

The Sonnets, 19 

The Strange Case of Dr. Jekyll and Mr. Hyde, 19 

The Taming of the Shrew, 19 

The Tell-Tale Heart, 19 

The Tempest, 19 

The Tenant of Wildfell Hall, 19 

The Three Musketeers, 19 

The Time Machine, 19 

The Tin Woodman of Oz, 19 

The Tragedy of Coriolanus, 20 

The Tragedy of King Lear, 20 

The Tragedy of King Richard the Second, 20 

The Voyage Out, 20 

The War in the Air, 20 

The War of the Worlds, 20 

The Wind in the Willows, 20 

The Wisdom of Father Brown, 20 

The Wonderful Wizard of Oz, 20 

Three Men in a Boat, 21 

Through the Looking Glass, 21 

Thus Spake Zarathustra, 21 

Thuvia, Maid of Mars, 21 

To Helen, 21 

Treasure Island, 21 

Twelfth Night, 21 

Twenty Thousand Leagues Under the Seas, 21 

Twenty Years After, 21 

Typee, 22 

Ulalume, 22 

Uneasy Money, 22 

Up From Slavery: An Autobiography, 22 

Vanity Fair, 22 

Walden, 22 

War and Peace, 22 

Warlord of Mars, 22 

White Fang, 22 

Wuthering Heights, 23

ENTRY SINGLE VARIABLE CALCULUS I 

[ENTRY SINGLE VARIABLE CALCULUS I] Authors: Oliver Knill: 2001 Literature: not yet 

Abel’s partial summation formula 

[Abel’s partial summation formula] is a discrete version of the partial integration formula: with An = �n k=1 ak 

one has �n k=m akbk = �n k=m Ak(bk − bk+1) + Anbn+1 − Am−1bm. 

Abel’s test 

[Abel’s test]: if an is a bounded monotonic sequence and bn is a convergent series, then the sum � 

n anbn 

converges. 

absolute value 

The [absolute value] of a real number x is denoted by |x| and defined as the maximum of x and −x. We can 

also write |x| = + √ x 2 . The absolute value of a complex number z = x + iy is defined as � x 2 + y 2 . 

accumulation point 

An [accumulation point] of a sequence an of real numbers is a point a which the limit of a subsequence ank of 

an. A sequence an converges if and only if there is exactly one accumulation point. Example: The sequence 

an = sin(πn) has two accumulation points, a = 1 and a = −1. The sequence an = sin(πn)/n has only the 

accumulation point a = 0. It converges. 

Achilles paradox 

The [Achilles paradox] is one of Zenos paradoxon. It argues that motion can not exist: ”set up a race between 

Achilles A and tortoise T . At the initial time t0 = 0, A is at the spot s = 0 while T is at position s1 = 1. Lets 

assume A runs twice as fast. The reace starts. When A reaches s1 at time t1 = 1, its opponent T has already 

advanced to a point s2 = 1 + 1/2. Whenever A reaches a point sk at time tk, where T has been at time tk−1, 

T has already advanced further to location sk+1. Because an infinite number of timesteps is necessary for A 

to reach T , it is impossible that A overcomes T .” The paradox exploits a misunderstanding of the concept of 

summation of infinite series. At the finite time t = � ∞ 

n=1 (tn − tn−1) = 2, both A and T will be at the same 

spot s = limn→∞ sn = 2. 

The [addition formulas] for trigonometric functions are 

addition formulas 

cos(a + b) = sin(a) cos(b) + cos(a) sin(b) 

sin(a + b) = cos(a) cos(b) − sin(a) sin(b)

alternating series 

An 

� 

[alternating series] is a series in which terms are alternatively positive and negative. An example is 

∞ 

n=1 an = �∞ n=1 (−1)n /n = −1 + 1/2 − 1/3 + 1/4 − .... An alternating series with an → 0 converges by 

the alternating series test. 

alternating series test 

Leibniz’s [alternating series test] assures that an alternating series � 

n an with |an| → 0 is a convergent series. 

acute 

An angle is [acute], if it is smaller than a right angle. For example α = π/3 = 60 ◦ is an acute angle. The angle 

α = 2π/3 = 120 ◦ is not an acute angle. The right angle α = π/2 = 90 ◦ does not count as an acute angle. The 

angle α = −π/6 = −30 ◦ is an acute angle. 

antiderivative 

The [antiderivative] of a function f is a function F (x) such that the derivative of F is f that is if d/dxF (x) = 

f(x). The antiderivative is not unique. For example, every function F (x) = cos(x) + C is the antiderivative of 

f(x) = sin(x). Every function F (x) = x n+1 /(n + 1) + C is the antiderivative of f(x) = x n . 

Arithmetic progression 

[Arithmetic progression] A sequence of numbers an for which bn = an+1 − an is constant, is called an arithmetic 

progression. For example, 3, 7, 11, 15, 19, ... is an arithmetic progression. The sequence 0, 1, 2, 4, 5, 6, 7 is not an 

arithmetic progression. 

arrow paradox 

The [arrow paradox] is a classical Zeno paradox with conclusion that motion can not exist: ”an object occupies 

at each time a space equal to itself, but something which occupies a space equal to itself can not move. Therefore, 

the arrow is always at rest.” 

asymptotic 

Two real functions are called [asymptotic] at a point a if limx→a f(x)/g(x) = 1. For example, f(x) = sin(x) and 

g(x) = x are asymptotic at a = 0. The point a can also be infinite: for example, f(x) = x and g(x) = √ x 2 + 1 

are asymptotic at a = ∞.

Bernstein polynomials 

The [Bernstein polynomials] of a continuous function f on the unit interval 0 ≤ x ≤ 1 are defined as Bn(x) = 

� n 

k=1 f(k/n)xk (1 − x) n−k n!/(k!(n − k!). 

Binomial coefficients 

[Binomial coefficients] The coefficients B(n, k) of the polynomial (x + 1) n for integer n are called Binomial 

coefficients. Explicitly one has B(n, k) = n!/(k!(n − k)!), where k! = k(k − 1)!, 0! = 1 is the factorial of k. The 

function B(n, k) can be defined for any real numbers n, k by writing n! = Γ(n + 1), where Γ is the Gamma 

function. If p is a positive real number and k is an integer, one has one has B(p, k) = p(p − 1)...(p − k + 1)/k!. 

For example, B(1/2, 0) = B(1/2, 1) = 1/2, B(1/2, 2) = −1/8. Indeed, (1 + x) 1/2 = 1 + x/2 − x 2 /8 + .... 

Binominal theorem 

The [Binominal theorem] tells that for a real number |z| < 1 and real number p, one has (1 + z) p = 

� ∞ 

k=0 B(p, k)zk , where B(p, k) is called the Binomial coefficient. If p is a positive integer, then (1 + z) p is 

a polynomial. For example: 

(1 + z) 4 = 1 + 4z + 6z 2 + 4z 3 + z 4 . 

If p is a noninteger or negative, then (1 + z) p is an infinite sum. For example 

(1 + z) −1/2 = 1 − x/2 + 3x 2 /8 − 5x 3 /16 + ... 

bisector 

A [bisector] is a straight line that bisects a given angle or a given line segment. For example, the y-axis x = 0 

in the plane bisects the line segment connecting (−1, 0) with (1, 0). The line x = y bisects the angle � (CAB) 

where C = (0, 1), A = (0, 0), B = (1, 0) at the point A. 

Bolzano’s theorem 

[Bolzano’s theorem] also called intermediate value theorem says that a continuous function on an interval (a, b) 

takes each value between f(a) and f(b). For example, the function f(x) = sin(x) takes any value between −1 

and 1 because f is continuous and f(−π/2) = −1 and f(π/2) = 1. 

Fermat principle 

The [Fermat principle] tells that if f is a function which is differentiable at z and f(x) > f(z) for all points in 

an interval (z − a, z + a) with a > 0, then f ′ (z) = 0.

fundamental theorem of calculus 

The [fundamental theorem of calculus]: if f is a differentiable function on a ≤ x ≤ b where a 

numbers, then f(b) − f(a) = � b 

a f ′ (x) dx. 

[integration rules]: 

• � af(x) dx = a � f(x) dx. 

• � f(x) + g(x) dx = � f(x) dx + � g(x) dx. 

integration rules 

• � fg dx = fG − � f ′ G, where G ′ = g. This is called integration by parts. 

intermediate value theorem 

The [intermediate value theorem] also called Bolzno theorem assures that a continuous function on an interval 

a ≤ z ≤ b takes each each value between f(a) and f(b). For example f(x) = cos(x) + cos(3x) + cos(5x) takes 

any value between [−3, 3] on [0, π] because f(0) = 3 and f(π) = −3. 

Cauchy’s convergence condition 

[Cauchy’s convergence condition]: a sequence an converges, if and only if it is a Cauchy sequence that is if for 

every constant c > 0, we can find n, such that for all k > n, m > n one has |am − ak| < c. For example, the 

sequence an = 1/ log(n) converges because for all c > 0, and the integer n closest to e 2c + 1 one has for k > n 

and m > n |am − ak| ≤ c. 

Cauchy’s convergence test 

[Cauchy’s convergence test]. Given a series �∞ the series is a convergent series. If r > 1, then the series diverges. 

k=1 ak with positive summands ak. If r = limn→∞ a 1/n 

n 

< 1 then

continuous 

A function f is called [continuous] at a point x if for every open interval V around f(x) there exists an open 

interval U around x such that f(U) is a subset of V . A function f is continuous in a set Y if it is continous at 

every point in Y . This definition is equivalent to: for every sequence xn → x, the sequence f(xn) converges to 

f(x). Examples: 

• Any polynomial like x 5 + 5x 3 + 3x is continuous on the entire line. 

• The sum and product of continuous functions is continuous. 

• the composition of two continuous functions is continuous. 

Discontinuities can happen in different ways: the function can become infinite like f(x) = 1/x at 0 or tan(x) 

at x = π/2, the function can jump like f(x) = sign(x) which is 1 if x > 0, −1 if x < 0 and 0 if x = 0. A 

function can also become too oscillatory at a point like f(x) = sin(1/x) at x = 0. Note that f(x) = x sin(1/x) 

is continuous on the entire real line. There are functions which are discontinuous at every point. An example 

is f(x) = 1 if x is rational and f(x) = −1 if x is irrational. 

Note that be restricting the domain of a function, one an make it continuous. For example: f(x) = 1/x is 

continuous on the positive real axes. 

converges 

A function f(x) [converges] to a value z at x if the function g which agrees with f away from x and satisfies 

g(x) = z is continouous at x. The value z is called the limit of f at x. For example the function f(x) = 

(1 − x 2 )/(1 − x) has the limit z = 2 at x = 1. The function g(x) which is defined bo be f(x) for x �= 1 and 

g(1) = 2 is indeed continuous. One writes z = limy→x f(y). One has 

• limx→z(f(x) + g(x)) = limx→z f(x) + limx→z g(x). 

• limx→z(f(x)g(x)) = limx→z f(x) + g(x). 

• limx→z f(g(x)) = f(limx→z g(x)). 

A series an is a [convergent series], if the partial sum sequence bn = � n 

k=1 ak converges to a finite limit a. 

absolutely convergent series 

A series � 

n an is called an [absolutely convergent series] if � 

n |an| is a convergent series. 

series 

Summing up a sequence is called a [series]. An important example is the geometric series 1 + 1/2 + 1/4 + 1/8 + 

1/16 + ..., which sums up to 2. An other example is the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... which 

has no finite limit. 

change of variables 

The [change of variables] in integration theory is the formula � f(x)dx = � f(g(u))g ′ (u)du if x = g(u). For 

example, � √ 1 − x 2 dx becomes with x = g(u) = sin(u) and dx = g ′ (u)du = cos(u)du the integral � cos 2(u) du.

differentiable 

A function f is called [differentiable] at z if there exists a function g which is continuous at z such that 

f(x) = f(z) + (x − z)g(x). The derivative of f at z is g(z) and also denoted f ′ (x). By solving for g(x) and 

letting x → z one can write g(z) = limx→z(f(x) − f(z))/(x − z). The quotient is called the differential quotient. 

• The sum of two at z differentiable functions is differentiable at z and (f + g) ′ (z) = f ′ (z) + g ′ (z). This is 

called the sum rule. 

• The prduct of two at z differentiable functions is differentiable at z and (fg) ′ = f ′ g + fg ′ . This is called 

the product rule. 

• The composition of two differentiable functions is differentiable and (f ◦ g) ′ = (f ′ ◦ g)g ′ . This is called the 

chain rule. 

Functions can be continuous without being differentiable. For example f(x) = |x| is continuous at 0 but not 

differentiable at 0. There are functions which are continuous everywhere but not differentiable at most points. 

An example is the Weierstrass function f(x) = � ∞ 

k=1 cos(k2 x)/k 2 . 

Extended mean value theorem 

[Extended mean value theorem]. If f(x) and g(x) are differentiable on the interval (a, b) and are continuous on 

the closed interval I = {a ≤ x ≤ b} then there exists a point x ∈ I for which 

f ′ (x)/g ′ (x) = (f(b) − f(a))/(g(b) − g(a)) . 

Proof. Otherwise one would have one of the following two possibilies: 

f ′ (x)(g(b) − g(a)) < g ′ (x)(f(b) − f(a)) for all x in (a, b) or 

f ′ (x)(g(b) − g(a)) < g ′ (x)(f(b) − f(a)) for all x in (a, b). 

Integration of these expressions using the fundamental theorem of calculus gives 

(f(b) − f(a))(g(b) − g(a)) < (g(b) − g(a))(f(b) − f(a)) or 

(f(b) − f(a))(g(b) − g(a)) > (g(b) − g(a))(f(b) − f(a)) which both are not possible. 

The special case g(x) = x is called the mean value theorem. 

factorial 

The [factorial] of a positive integer n is defined recursively by n! = n(n − 1)! and 0! = 1. For example, 5! = 120. 

The factorial function can be extended to the real line and is then called the Γ function: n! = Γ(n + 1), where 

Γ(z) = 

which is finite everywhere except at z = 0, −1, −2, .... 

� ∞ 

t 

0 

z−1 e −t dt . 

limit 

The [limit] of a sequence of numbers an is a number a such that an converges to a in the following sense: for 

every c > 0 there exists an integer m such that |an − a| < c for n > m. Limits can be defined in any metric 

space and more generally in any topological space.

maximum-value theorem 

The [maximum-value theorem] assures that a continuous function on an interval a < z 

that interval. 

parabola 

A [parabola] is the graph of the function f(x) = x 2 . More general parabolas can be obtained as graphs of 

f(x) = a(x − b) 2 + c where a, b, c are constants or curves obtained by rotating such a curve in the plane. For 

example the set of points in the plane satisfying x = y 2 form a parabola. Parabolas are examples of conic 

sections, intersections of a plane with a cone. 

L’Hopital rule 

The [L’Hopital rule] tells that if f and g are differentiable functions at x and f(x) = g(x) and g ′ (x) �= 0, then 

limx→z f(x)/g(x) = limx→z f ′ (x)/g ′ (x). 

For example, limx→0 sin(3x)/x = limx→0 3 cos(3x) = 3. The rule essentially tells that one can replace the 

functions by their linear approximation near a point to find the limit. The proof follows immediatly from the 

definition of differentiability: there exist continuous functions F, G such that f(x) = f(z) + (x − z)F (x) and 

g(x) = g(z) + (x − z)G(x). Because G(z) �= 0, the function F (x)/G(x) is continuous at z with value F (z)/G(z). 

Now: limx→z f(x)/g(x) = limx→z(f(z) + (x − z)F (x))/(g(z) + (x − z)G(x)) == limx→z f(z)/g(z). 

[Hopital rule] see L’Hopital rule. 

Hopital rule 

hyperbola 

A [hyperbola] is curve in the plane which can be described as the graph of the function f(x) = 1/x. Also 

translated, scaled and rotated versions of this curve is called a hyperbola. For example, the set of points (x, y) 

in the plane which satisfy (x − 1) 2 − (y − 2) 2 = 5 is a hyperbola. 

Mean value theorem 

[Mean value theorem]. If f(x) is a continuous function on an interval I = {a ≤ x ≤ b} which is differentiable 

on the open interval (a, b), then there exists a point x ∈ I for which f ′ (x) = C = (f(b) − f(a))/(b − a). Proof. 

Otherwise, f ′ (x) < C on (a, b) or f ′ (x) > C on (a, b). Integration gives using the fundamental theorem of 

calculus 

especially 

f(x) − f(a) = � x 

a f ′ (t)dt < C(x − a) or 

f(x) − f(a) = � x 

a f ′ (t)dt > C(x − a) 

f(x) − f(a) = � b 

a f ′ (t)dt < C(b − a) or 

f(x) − f(a) = � b 

a f ′ (t)dt > C(b − a) which is a contradiction 

The mean value theorem is a special case of the extended mean value theorem.

Rolle’s theorem 

[Rolle’s theorem] If f(x) is a continuous function on the interval I = {a ≤ x ≤ b} which is differentiable on the 

open interval (a, b) and f(a) = f(b), then there exists a point x ∈ (a, b), for which f ′ (x) = 0. 

Proof. f takes both its maximum and minimum on I. If the maximum is equal to the minimum, then f(x) is 

constant on I, otherwise, either the minium or the maximum is a point x in (a, b). At that point f ′ (x) = 0. 

qed. Rolle’s theorem is a special case of the mean value theorem 

rule of three 

The [rule of three] ia a rough rule of thumb when solving calculus problems or teaching calculus: 

Look at a calculus problem graphically, numerically and analytically. 

In other words, one should try to understand a calculus problem geometrically, algebraically and computationally. 

For example, the notion of the derivative of a function of one variable can be understood geometrically as 

a slope, can be understood through algebraic manipulations like (x n ) ′ = nx n−1 or computationally by plugging 

in numbers or doing things on a computer. 

Weierstrass function 

A [Weierstrass function] is an example of a function which is continuous but almost nowhere differentiable. An 

example is f(x) = � ∞ 

k=1 cos(k2 x)/k 2 . 



Index 

Abel’s partial summation formula, 1 

Abel’s test, 1 

absolute value, 1 

absolutely convergent series, 5 

accumulation point, 1 

Achilles paradox, 1 

acute, 2 

addition formulas, 1 

alternating series, 2 

alternating series test, 2 

antiderivative, 2 

Arithmetic progression, 2 

arrow paradox, 2 

asymptotic, 2 

Bernstein polynomials, 3 

Binomial coefficients, 3 

Binominal theorem, 3 

bisector, 3 

Bolzano’s theorem, 3 

Cauchy’s convergence condition, 4 

Cauchy’s convergence test, 4 

change of variables, 5 

continuous, 5 

converges, 5 

differentiable, 6 

Extended mean value theorem, 6 

factorial, 6 

Fermat principle, 3 

fundamental theorem of calculus, 4 

Hopital rule, 7 

hyperbola, 7 

integration rules, 4 

intermediate value theorem, 4 

L’Hopital rule, 7 

limit, 6 

maximum-value theorem, 7 

Mean value theorem, 7 

parabola, 7 

Rolle’s theorem, 8 

rule of three, 8 

series, 5 

Weierstrass function, 8 

9

ENTRY MULTIVARIABLE CALCULUS 

[ENTRY MULTIVARIABLE CALCULUS] Author: Oliver Knill: March 2000 -March 2004 Literature: Standard 

glossary of multivariable calculus course as taught at the Harvard mathematics department. 

acceleration 

The [acceleration] of a parametrized curve r(t) = (x(t), y(t), z(t)) is defined as the vector r ′′ (t). It is the rate of 

change of the velocity r ′ (t). It is significant, because Newtons law relates the acceleration r ′′ (t) of a mass point 

of mass m with the force F acting on it: mr ′′ (t) = F (r(t)) . This ordinary differential equation determines 

completely the motion of the particle. 

advection equation 

The [advection equation] ut = cux is a linear partial differential equation. Its general solution is u(t,x)=f(x+ct), 

where f(x)=u(0,x). The advection equation is also called transport equation. In higher dimensions, it generalizes 

to the gradient flow ut = cgrad(u). 

Archimedes spiral 

The [Archimedes spiral] is the plane curve defined in polar coordinates as r(t) = ct, where c is a constant. In 

Euclidean coordinates, it is given by the parametrization r(t) = (ct cos(t), ct sin(t)). 

axis of rotation 

The [axis of rotation] of a rotation in Euclidean space is the set of fixed points of that rotation. 

Antipodes 

Two points on the sphere of radius r are called [Antipodes] (=anti-podal points) if their Euclidean distance is 

maximal 2r. If the sphere is centered at the origin, the antipodal point to (x, y, z) is the point (−x, −y, −z).

The [boundary] of a geometric object. Examples: 

boundary 

• The boundary of an interval I = {a ≤ x ≤ b} is the set with two points {a, b}. For example, {0 ≤ x ≤ 1 

the boundary {0, 1}. 

• The boundary of a region G in the plane is the union of curves which bound the region. The unit disc 

has as a boundary the unit circle. The entire plane has an empty boundary. 

• The boundary of a surface S in space is the union of curves which bound the surface. For example: A 

semisphere has as the boundary the equator. The entire sphere has an empty boundary. 

• The boundary of a region G in space is the union of surfaces which bound the region. For example, the 

unit ball has the unit sphere as a boundary. A cube has as a boundary the union of 6 faces. 

• The boundary of a curve r(t), t ∈ [a, b] consists of the two points r(a), r(b). 

The boundary can be defined also in higher dimensions where surfaces are also called manifolds. The dimension 

of the boundary is always one less then the dimension of the object itself. In cases like the half cone, the tip of 

the cone is not considered a part of the boundary. It is a singular point which belongs to the surface. While the 

boundary can be defined for far more general objects in a mathematical field called ”topology”, the boundaries 

of objects occuring in multivariable calculus are assumed to be of dimension one less than the object itself. 

Burger’s equation 

The [Burger’s equation] ut = uut is a nonlinear partial differential equation in one dimension. It is a simple 

model for the formation of shocks. 

Cartesian coordinates 

[Cartesian coordinates] in three-dimensional space describe a point P with coordinates x, y and z. Other possible 

coordinate systems are cylindrical coordinates and spherical coordinates. Going from one coordinate system to 

an other is called a coordinate change. 

Cavalieri principle 

[Cavalieri principle] tells that if two solids have equal heights and their sections at equal distances have have 

areas with a given ratio, then the volumes of the solids have the same ratio. 


A [change of variables] is defined by a coordinate transformation. Examples are changes between cylindrical 

coordinates, spherical coordinates or Cartesian coordinates. Often one uses also rotations, allowing to use a 

convenient coordinate system, like for example, when one puts a coordinate system so that a surface of revolution 

has as the symmetry axes the z-axes.

circle 

A [circle] is a curve in the plane whose distance from a given point is constant. The fixed point is called the 

center of the circle. The distance is the radius of the circle. One can parametrize a circle by r(t) = (cos(t),sin(t) 

or given as an implicit equation g(x, y) = x 2 +y 2 = 1. The circle is an example of a conic section, the intersection 

of a cone with a plane to which ellipses, hyperbola and parabolas belong to. 

cone 

A [cone] in space is the set of points x 2 + y 2 = z 2 in space. Also translates, scaled and rotated versions of this 

set are still called a cone. For example 2x 2 + 3y 2 = 7z 2 is an elliptical cone. 

conic section 

A [conic section] is the intersection of a cone with a plane. Hyperbola, ellipses and parabola lines and pairs of 

intersecting lines are examples of conic sections. 

continuity equation 

The [continuity equation] is the partial differential equation ρt + div(ρv) = 0, where ρ is the density of the 

fluid and v is the velocity of the fluid. The continuity equation is the consequence of the fact that the negative 

change of mass in a small ball is equal to the amount of mass which leaves the ball. The later is the flux of the 

current j = vρ through the surface and by the divergence theorem the integral of div(j). 

cos theorem 

The [cos theorem] relates the length of the edges a,b,c in a triangle ABC with one of the angles α: a 2 = 

b 2 + c 2 − 2bc cos(α) Especially, if α = π/2, it becomes the theorem of Pythagoras. 

critical point 

A [critical point] of a function f(x, y) is a point (x0, y0), where the gradient ∇f(x0, y0) vanishes. Critical points 

are also called stationary points. For functions of two variables f(x, y), critical points are typically maxima, 

minima or saddle points realized by f(x, y) = −x 2 − y 2 , f(x, y) = x 2 + y 2 or f(x, y) = x 2 − y 2 . 

chain rule 

The [chain rule] expresses the derivative of the composition of two functions in terms of the derivatives of the 

functions. It is (fg) ′ (x) = f ′ (g(x))g ′ (x). For example, if r(t) is a curve in space and F a function in three 

variables, then (d/dt)f(r(t)) = grad(f)·r ′ (t). Example. If T and S are maps on the plane, then (T S) ′ = T ′ (S)S ′ , 

where T’ is the Jacobean of T and S’ is the Jacobean of S.


A [change of variables] on a region R in Euclidean space is given by an invertible map T : R → T (R). The change 

of variables formula � 

� 

f(x) dx = 

T (R) R f(T x)det(T ′ (x)) dx allows to evaluate integrals of a function f of several 

variables on a complicated region by integrating on a simple region R. In one dimensions, the change of variable 

formula is the formula for substitution. Example: (2D polar coordinates) T (r, φ) = (x cos(θ), y sin(θ). with 

det(T’)=r maps the rectangle [0, s] × [0, 2π] into the disc. Example of 3D spherical coordinates are T (r, θ, φ) = 

(r cos(θ) sin(φ), r sin(θ) sin(φ), r cos(φ)), det(T ′ ) = r2 sin(φ) maps the rectangular region (0, s) × (0, 2π0 × (0, π) 

onto a sphere of radius s. 

curl 

The [curl] of a vector field F = (P, Q, R) in space is the vector field (Ry − Qz, Pz − Rx, Qx − Py). It measures 

the amount of circulation = vorticity of the vector field. The curl of a vector field F=(P,Q) in the plane is the 

scalar field (Qx − Py). It measures the vorticity of the vector field in the plane. 

curvature 

The [curvature] of a parametrized curve r(t) = (x(t),y(t),z(t)) is defined as k(t) = |r ′ (t) × r ′′ (t)|/|r ′ (t)| 3 . 

Examples: 

• The curvature of a line is zero. 

• The curvature of a circle of radius r is 1/r. 

curve 

A [curve] in space is the image of a map X : t− > r(t) = (x(t), y(t), z(t)), where x(t), y(t), z(t) are three 

piecewise smooth functions. For general continuous maps x(t), y(t), z(t), the length or the velocity of the curve 

would no more be defined. 

cross product 

The [cross product] of two vectors v = (v1, v2, v3) and w = (w1, w2, w3) is the vector (v2w3 − w2v3, v3w1 − 

w3v1, v1w2 − w2v1). 

curve 

A [curve] in three-dimensional space is the image of a map r(t) = (x(t), y(t), z(t)), where x(t), y(t), z(t) are 

three continuous functions. A curve in two-dimensional space is the image of a map r(t) = (x(t), y(t)).

cylinder 

A [cylinder] is a surface in three dimensional space such that its defining equation f(x,y,z)=0 does not involve one 

of the variables. For example, z = 2 sin(y) defines a cylinder. A cylinder usually means the surface x 2 + y 2 = r 

or a translated rotated version of this surface. 

derivative 

The [derivative] of a function f(x) of one variable at a point x is the rate of change of the function at this 

point. Formally, it is defined as limdx→0(f(x + dx) − f(x))/dx. One writes f’(x) for the derivative of f. The 

derivative measures the slope of the graph of f(x) at the point. If the derivative exists for all x, the function is 

called differentiable. Functions like sin(x) or cos(x) are differentiable. One has for example f ′ (x) = cos(x) if 

f(x) = sin(x). An example of a function which is not differentiable everywhere is f(x)=—x—. The derivative 

at 0 is not defined. 

cylindrical coordinates 

[cylindrical coordinates] in three dimensional space describe a point P by the coordinates r = (x 2 + y 2 + z 2 ) 1/2 , 

phi=arctan(y/x),z, where P=(x,y,z) are the Cartesian coordinates of P. Other coordinate systems are Cartesian 

coordinates or spherical coordinates. 

The [determinant] of a matrix A = 

� a b 

c d 

determinant 

⎛ 

� 

is ad − bc. The determinant of a matrix A = ⎝ 

a 

d 

b 

e 

c 

f 

g h i 

aei + bfg + cdh − ceg − fha − ibd. The determinant is relevant when changing variables in integration. 

directional derivative 

The [directional derivative] of f(x,y,z) in the direction v is the dot product of the gradient of f with v. It measures 

the rate of change of f at a point P when moving trough the point (x,y,z) with velocity v. 

distance 

The [distance] of two points P=(a,b,c) and Q=(u,v,w) in three dimensional Euclidean space is the square root 

of (a − u) 2 + (b − v) 2 + (c − w) 2 . The distance of two points P=(a,b) and Q = (u, v) in the plane is the square 

root of (a − u) 2 + (b − v) 2 . 

distance 

The [distance] between two nonparallel lines in three dimensional Euclidean space is given by the forumla 

d = |(v × w) · u|/|(v × w)|, where v and w are arbitrary nonzero vectors in each line and u is an arbitrary vector 

connecting a point on the first line to a point of the second line. 

⎞ 

⎠ is

divergence 

The [divergence] of a vector field F=(P,Q,R) is the scalar field div(F ) = Px +Qy +Rz. The value div(F )(x, y, z) 

measures the amount of expansion of the vector field at the point (x, y, z). 

dot product 

The [dot product] of two vectors v = (v1, v2, v3) and w = (w1, w2, w3) is the scalar v1w1 + v2w2 + v3w3. 

ellipse 

An [ellipse] is the set of points in the plane which satisfy an equation (x − a) 2 /A 2 + (y − b) 2 /B 2 = 1. It is 

inscribed in a rectangle of length A and width B centered at (a, b). Ellipses can also be defined as the set of 

points in the plance whose sum of the distances to two points is constants. The two fixed points are called the 

foci of the ellipse. The line through the foci of a noncircular ellipse is called the focal line, the points where 

focal axes and a noncircular ellipse cross, are called vertices of the ellipse. The major axis of the ellipse is the 

line segment connecting the two vertices, the minor axis is the symmetry line of the ellipse which mirrors the 

two focal points or the two vertices. Ellipses are examples of conic sections, the intersection of a cone with a 

plane. 

ellipsoid 

An [ellipsoid] is the set of points in three dimensional Euclidean space, which satisfy an equation (x − a) 2 /A 2 + 

(y − b) 2 /B 2 + (z − c) 2 /C 2 = 1. It is inscribed in a box of length, width and height A,B,C centered at (a, b, c). 

equation of motion 

The [equation of motion] of a fluid is the partial differential equation ρDv/dt = −grad(p) + F , where F 

are external forces like gravity rho g, or magnetic force j × B and Dv/dt is the total time derivative Dv/dt = 

vt+vgrad(v). The term - grad(p) is the pressure force. Together with an incompressibility assumption div(v)=0, 

these equations of motion are called Navier Stokes equations. 

flux integral 

The [flux integral] of a vector field F through a surface S=X(R) is defined as the double integral of X(F).n over 

R, where n = Xu × Xv is the normal vector of the surface as defined through the parameterization X(u, v). 

[Fubinis theorem] tells that � b � d 

a c f(x, y) dxdy = � d 

c 

Fubinis theorem 

� b 

f(x, y) dydx. 

a

gradient 

The [gradient] of a function f at a point P=(x,y,z) is the vector (fx(x, y, z), fy(x, y, z), fz(x, y, z)) where fx 

denotes the partial derivative of f with respect to x. 

Hamilton equations 

The [Hamilton equations] to a function f(x, y) is the system of ordinary differential equations x ′ (t) = 

fy(x, y), y ′ (t) = −fx(x, y). which is called Hamilton system. Solution curves of this system are located on 

level curves f(x, y) = c because by the chain rule one has d/dtf(x(t), y(t)) = fxx ′ + fyy ′ = fxfy − fxfy = 0. 

The preservation of f is in physics called energy conservation. 

heat equation 

The [heat equation] is the Partial differential equation ut = mu∆(u), where mu is a constant, and ∆u is the 

Laplacian of u. The heat equation is also called the diffusion equation. 

The [Hessian] is the determinant of the Hessian matrix. 

Hessian 

Hessian matrix 

The [Hessian matrix] of a function f(x,y,z) at a point (u, v, w) is the 3x3 matrix f ′′ ⎛ 

⎞ 

(u, v, w) = H(u, v, w) = 

⎝ 

fxx fxy fxz 

fyx fyy fyz 

fzx fzy fzz � 

fxx fxy 

H(u, v, w) = 

derivative test. 

⎠. The Hessian matrix of a function f(x, y) at a point (u,v) is the 2x2 matrix f ′′ = 

fyx fyy 

� 

. The Hessian matrix is useful to classify critical points of f(x, y) using the second 

hyperbola 

A [hyperbola] is a plane curve which can be defined as the level curve g(x, y) = x 2 /a 2 + y 2 /b 2 = 1 or given as 

a parametrized curve r(t) = (a cosh(t), b sinh(t)). A hyperbola can geometrically also be defined as the set of 

points whose distances from two fixed points in the plane is constant. The two fixed points are called the focal 

points of the hyperbola. The line through the focal points of a hyperbola is called the focal axis. The points, 

where the focal axis and the hyperbola cross are called vertices. A hyperbola is an example of a conic section, 

the intersection of a cone with a plane.

hyperboloid 

A [hyperboloid] is the set of points in three dimensional Euclidean space, which satisfy an equation (x−u) 2 /a 2 − 

(y − v) 2 /b 2 − (z − w) 2 /c 2 = I, where I = 1 or I = −1. For a=b=c=1, the hyperboloid is obtained by rotating 

a hyperbola x 2 − y 2 = 1 around the x-axes. It is two-sided for I=-1 (the intersection of the plane z=c with the 

hyperboloid is then empty) and one-sided for I=1. 

incompressible 

A vector field F is called [incompressible] if its divergence is zero div(F ) = 0. The notation has its origins from 

fluid dynamics, where velocity fields F of fluids, gases or plasma often are assumed to be incompressible. If a 

vector field is incompressible and is a velocity field, then the corresponding flow preserves the volume. 

continuity equation 

The [continuity equation] ρt + div(i) = 0 links density ρ and velocity field i. It is an infinitesimal�description � � 

which is equivalent to the preservation of mass by the theorem of Gauss. The change of mass M(t) ρ dV R 

inside a region R in space is the minus the flux of mass through the boundary S of R. 

interval 

An [interval] is a subset of the real line defined by two points a,b. One can write I = {a ≤ x ≤ b} for a closed 

interval, I = {a < x < b} for an open interval and I = {a ≤ x < b}, I = {a < x ≤ b} for half open intervals. If 

a = −∞ and b = ∞, then the interval is the entire real line. If a = 0, b − ∞, then I = (a, b) is the set of positive 

real numbers. Intervals can be characterized as the connected sets in the real line. 

integral 

An [integral] of f(x) over an interval I on the line is the limit (1/n) �n i=1 f(i/n) for n → ∞ over the integers 

and the sum is taken over all i such that i/n is in I. An integral of f(x, y) over a region R in the plane is the 

limit (1/n2 ) � 

(i/n,j/n)∈R f(i/n, j/n) for n → ∞. Such an integral is also called double integral. Often, double 

integrals can be evaluated by iterating two one-dimensional integrals. An integral of f(x,y,z) over a domain R 

in space is the limit (1/n3 ) � 

(i/n,j/n,k/n)∈R f(i/n, j/n, k/n) for n to infinity. Such an integral is also called a 

triple integral. Often, triple integrals can be evaluated by iterating three one-dimensional integrals. 

intercept 

An [intercept] is the intersection of a surface with a coordinate axes. Like traces, intercepts are useful for 

drawing surfaces by hand. For example, the two sheeted hyperboloid x 2 + y 2 − z 2 = −1 has the intercepts 

x 2 − z 2 = −1 and y 2 − z 2 = −1 (hyperbola) and an empty intercept with the z axes.

jerk 

The [jerk] of a parametrized curve r(t)=(x(t),y(t),z(t)) is defined as r”’(t). It is the rate of change of the 

acceleration. By Newtons law, the jerk measures the rate of change of the force acting on the body. 

Lagrange multiplier 

A [Lagrange multiplier] is an additional variable introduced for solving extremal problems under constraints. 

To extremize f(x, y, z) on a surface g(x, y, z) = 0 then an extremum satisfies the equations f ′ = Lg ′ , g = 0, 

where L is the Lagrange multiplier. These are four equations for four unknowns x,y,z,l. Additionally, one has 

to check for solutions of g ′ (x, y, z) = 0. 

Example. If we want to extremize F (x, y, z) = −x log(x) − y log(y) − z log(z) under the constraint G(x, y, z) = 

x + y + z = 1, we solve the equations −1 − log(x) = λ1 −1 − log(y) = λ1 −1 − log(z) = λ1 x + y + z = 1, the 

solution of which is x = y = z = 1/3. 

Lagrange method 

The [Lagrange method] to solve extremal problems under constraints: 

1) in order that a function f of several variables is extremal on a constraint set g = c, we either have ∇g = 0 

or the point is a solution to the Lagrange equations ∇f = λ∇g, g = c. 

2) in order to extremize a function f of several variables under the contraint set g = c, h = d, we have to solve 

the Lagrange equations ∇f = λ∇g + µ∇h, g = c, h = d or solve ∇g = ∇h = 0. 

Laplacian 

The [Laplacian] of a function f(x, y, z) is defined as ∆(f) = fxx+fyy+fzz. One can write it as ∆ = divgrad(f). 

Functions for which the Laplacian vanish are called harmonic. Laplacian appear often in PDE’s Examples: the 

Laplace equation ∆(f) = 0, the Poisson equation ∆(f) = ρ, the Heat equation ft = µ∆(f) or the wave equation 

ftt = c 2 ∆f. The [length] of a curve r(t)=(x(t),y(t),z(t)) from t=a to t=b is the integral of the speed —r’(t)— 

over the interval a,b. Example. the length of the curve r(t)=(cos(t),sin(t)) from t=0 to t = π is π because the 

speed |r ′ (t)| is 1. 

length 

The [length] of a vector v = (a, b, c) is the square root of v · v = a 2 + b 2 + c 2 . An other word for length is norm. 

If a vector has length 1, it is called normalized or a unit vector. 

level curve 

A [level curve] of a function f(x, y) of two variables is the set of points which satisfy the equation f(x, y) = c. 

For example, if f(x, y) = x 2 −y 2 , then its level curves are hyperbola. Level curves are orthogonal to the gradient 

vector field grad(f).

level surface 

A [level surface] of a scalar function f(x, y, z) is the set of points which satisfy f(x, y, z) = c. For example, if 

f(x, y, z) = x 2 + y 2 + 3z 2 , then its level surfaces are ellipsoids. Level surfaces are orthogonal to the gradient 

field grad(f). 

linear approximation 

The [linear approximation] of a function f(x,y,z) at a point (u, v, w) is the linear function L(x, y, z) = f(u, v, w)+ 

∇f(u, v, w) · (x − u, y − v, z − w). 

line 

A [line] in three-dimensional space is a curve in space given by r(t) = P + tv, where P is a point in space and v 

is a vector in space. The representation r(t)=P+tv is called a parameterization of the line. Algebraically, a line 

can also be given as the intersection of two planes: ax+by +cz = d, ux+vy +wz = q. The corresponding vector 

v in the line is the cross product of (a, b, c) and (u, v, w). A point P = (x, y, z) on the line can be obtained by 

fixing one of the coordinates, say z=0 and solving the system ax + by = d, ux + vy = q for the unknowns x and 

y. 

line integral 

The [line integral] of a vector field F (x, y) along a curve C : r(t) = (x(t), y(t)), t ∈ [a, b] in the plane is defined 

as � � b 

F · ds = F (r(t)) · r ′ (t) dt , 

C 

a 

where r ′ (t) = (x ′ (t), y ′ (t)) is the velocity. The definition is similar in three dimensions where F (x, y, z) is a 

vector field and C : r(t) = (x(t), y(t), z(t)), t ∈ [a, b] is a curve in space. 

Maxwell equations 

The [Maxwell equations] are a set of partial differential equations which determine the electric field E and 

magnetic field B, when the charge density ρ and the current density j are given. There are 4 equations: 

div(B) = 0 no magnetic monopoles 

curl(E) = −Bt/c Faradays law, change of magnetic flux produces voltage 

curl(B) = Et/c + (4π/c)j Ampere’s law, current or E change produce magnetism 

div(E) = 4πρ Gauss law, electric charges produce an electric field 

nabla 

[nabla] is a mathematical symbol used when writing the gradient ∇f of a function f(x, y, z). Nabla looks like 

an upside down ∆. Etymologically, the name has the meaning of an Egyption harp.

nabla calculus 

The [nabla calculus] introduces the vector ∇ = (∂x, ∂y, ∂z). It satisfies ∇(f) = grad(f), ∇xF = curl(F ), 

∇ · F = div(F ). Using basic vector operation rules and differentiation rules like ∇(fg) = (∇f)g + f(∇g) one 

can verify identities: like for example 

div(curl)F = 0, curl(grad)f = 0, curl(curlF ) = grad(divF ) − ∆(F ), div(E × F ) = F · curl(E) − E · curl(F ). 

nonparallel 

Two vectors v and w are called [nonparallel] if they are not parallel. Two vectors in space are parallel if and 

only if their cross product v × w is nonzero. 

normal vector 

A [normal vector] to a parametrized surface X(u, v) = (x(u, v), y(u, v), z(u, v)) at a point P=(x,y,z) is the vector 

XuxXv. It is orthogonal to the tangent plane spanned by the two tangent vectors Xu and Xv. 

normalized 

A vector is called [normalized] if its length is equal to 1. For example, the vector (3/5, 4/5) is normalized. The 

vector (2, 1) is not normalized. 

octant 

An [octant] is one of the 8 regions when dividing three dimensional space with coordinate planes. It is the 

analogue of quadrant in two dimensions. 

open set 

An [open set] R in the plane or in space is a set for which every point P is contained in a small disc U which 

is still contained in R. The disc x 2 + y 2 < 1 is an example of an open set. The set x 2 + y 2 ≤ 1 is not open 

because the point (1, 0) for example has no neighborhood disc contained in R. 

open 

A set is called [open], if it is an open set. It means that every point in the set is contained in a neighborhood 

which still is in the set. The complement of open sets are called closed.

ordinary differential equation 

An [ordinary differential equation] (ODE) is an equation for a function or curve f(t) which relates derivatives 

f,f’,f”.... of f. An example is f’=c f which has the solution f(t) = Ce ( ct), where C is a constant. Only 

derivatives with respect to one variable may appear in an ODE. In most cases, the variable t is associated with 

time. Examples: 

f ′ = cf population model c > 0. 

f ′ = −cf radioactive decay c > 0 

f ′ = cf(1 − f) logistic equation 

f ′′ = −cf harmonic oscillator 

f ′′ = F (f) general form of Newton equations 

By increasing the dimension of the phase space, every ordinary differential equation can be written as a first 

order autonomous system x ′ = F (x). For example, f ′′ = −f can be written with the vector x = (x1, x2) = (f, f ′ ) 

as (x ′ 1, x ′ 2) = (f ′ , f ′′ ) = (f ′ , f) = (x ′ 2, −x ′ 2). There is a 2 × 2 matrix such that x ′ = Ax. 

orthogonal 

Two vectors v and w are called [orthogonal] if v · w = 0. An other word for orthogonal is perpendicular. The 

zero vector 0 is orthogonal to any other vector. 

parabola 

A [parabola] is a plane curve. It can be defined as the set of points which have the same distance to a line and 

a point. The line is called the directrix, the point is called the focus of the parabola. One can parametrize a 

parabola as r(t) = (t, t 2 ). It is also possible to give a parabola as a level curve g(x, y) = y − x 2 = 0 of a function 

of two variables. A parabola is an example of a conic section, to which also circles, ellipses and hyperbola 

belong. 

parallelogram 

A [parallelogram] E can be defined as the image of the unit square under a map T (s, t) = sv + tw, where u 

and v are vectors in the plane. One says, E is spanned by the vectors v and w. The area of a parallelogram is 

|v × w|. 

parallelepiped 

A [parallelepiped] E can be defined as the image of the unit cube under a linear map T (r, s, t) = ru+sv+tv, where 

u,v,w are vectors in space. One says, E is spanned by the vectors u, v and w. The volume of a parallelepiped 

is |u · (v × w)|. 

perpendicular 

Two vectors v and w are called [perpendicular] if v · w = 0. An other word for perpendicular is orthogonal. The 

zero vector v = 0 is perpendicular to any other vector.

quadratic approximation 

The [quadratic approximation] of a function f(x,y,z) at a point (u, v, w) is the quadratic function Q(x, y, z) = 

L(x, y, z) + [H(u, v, w)(x − u, y − v, z − w)] · (x − u, y − v, z − w)/2, where H(u, v, w) is the Hessian matrix of f 

at (u, v, w) and where L(x, y, z) is the linear approximation of f(x, y, z) at (u, v, w). For example, the function 

f(x, y) = 3 + sin(x + y) + cos(x + 2y) has the linear approximation L(x, y) = 4 + x + y and the quadratic 

approximation Q(x, y) = 4 + x + y + (x + 2y) 2 /2. 

quadrant 

A [quadrant] is one of the 4 regions when dividing the two dimensional space using coordinate axes. It is the 

analogue of octant in three dimensions. For example, the set {x > 0, y > 0} is the open upper right quadrant. 

The set {x ≥ 0, y ≥ 0} is the closed upper right quadrant. 

parallel 

Two vectors v and w are called [parallel] if there exists a real number λ such that v = λw. Two vectors in space 

are parallel if and only if their cross product v × w is zero. 

A [parametrized surface] is defined by a map 

parametrized surface 

X(u, v) = (x(u, v), y(u, v), z(u, v)) 

from a region R in the uv-plane to xyz-space. Examples 

• Sphere: X(u, v) = (r cos(u) sin(v), r sin(u) sin(v), r cos(v)) R = [0, 2π) × [0, π], u and v are called Euler 

angles. 

• Plane X(u, v) = P + uU + vV , where P is a point, U,V are vectors and R is the entire plane. 

• Surface of revolution is parametrized by X(u, v) = (f(v) cos(u), f(v) sin(u), v) where u is an angle measuring 

the rotation round the z axes and f(v) is a nonnegative function giving the distance to the z-axes 

at the height v. 

• A graph of a function f(x, y) is parametrized by X(u, v) = (u, v, f(u, v)). 

• A torus is parametrized by X(u, v) = (a + b cos(v)) cos(u), (a + b cos(v)) sin(u), sin(v)) on R = [0, 2π) × 

[0, 2π).

parametrized curve 

A [parametrized curve] in space is defined by a map r(t) = (x(t), y(t), z(t)) from an interval I to space. Examples 

are 

• Circle in the xy-plane r(t) = (cos(t), sin(t), 0) with t ∈ [0, 2π]. 

• Helix r(t) = (cos(t), sin(t), t) with t ∈ [a, b]. 

• Line r(t) = P + tV , where V is a vector and P is a point and −∞ < t < ∞. 

• A line segment connecting P with Q r(t) = P + t(Q − P ), where t ∈ [0, 1]. 

partial derivative 

The [partial derivative] of a function of several variables f is the derivative with respect to one variable assuming 

the other variables are constants. One writes for example fy(x, y, z) for the partial derivative of f(x,y,z) with 

respect to y. 

partial differential equation 

A [partial differential equation] is an equation for a function of several variables in which partial derivatives 

with respect to different variables appear. Examples: 

ut = cux 

Advection equation 

ut = µ∆(u) Heat equation 

utt = c 2 ∆(u) Wave equation 

utt = c 2 ∆(u) − m 2 u Klein Gordon equation 

∆(u) = 0 Laplace equation 

∆(u) = ρ Poisson equation 

ut + uxxx + 6uux = 0 KdV equation 

ut = uux 

Burger equation 

div(B) = div(E) = 0 Bt = −ccurl(E) Et = ccurl(B) Maxwell equation (vacuum) 

ihut = h2 /2m∆u + V u Schroedinger equation 

curl(A) = F Vector potential equation 

plane 

A [plane] in three dimensional space is the set of points (x, y, z) which satisfy an equation ax + by + cz = d. A 

parametrization of a plane is given by the map (s, t) ↦→ X(u, v) = sv + tw, where v, w are two vectors. If three 

points P1, P2, P3 are given in space, then X(s, t) = P1 + s(P2 − P1) + t(P3 − P1) is a parametrisation of the 

plane which contains all three points. 

polar coordinates 

[polar coordinates] in the plane describe a point P=(x,y) with the coordinate (r, t) where r = (x 2 + y 2 ) 1/2 is 

the distance to the origin and t is the angle between the line OP and the x axes. The angle t = arctan(y/x) ∈ 

(−π/2, π/2] has to be augmented by π if x < 0 or x = 0, y < 0. The Cartesian coordinates of P are obtained 

from the Polar coordinates as x = r cos(t), y = r sin(t).

potential 

A function U(x, y, z) is called a [potential] to a vector field F (x, y, z) if grad(U) = F at all points. The vector 

field F is then called conservative or a potential field. Not every vector field is conservative. If curl(F ) = 0 

everywhere in space, then F has a potential. 

projection 

The [projection] of a vector v onto a vector w is the vector w(v · w)/|w| 2 . The scalar projection is the length of 

the projection. 

right handed 

A coordinate system in space is [right handed] if it can be rotated into the situation such that if the z axes 

points to the observer of the xy plane, then a 90 degree rotation brings the x axes to the y axes. Otherwise 

the coordinate system is called left handed. If u is a vector on the positive x axes, v is a vector on the positive 

y axes and w is a vector on the positive z axes, then the coordinate system is right handed if and only if the 

triple product u · (v × w) is positive. 

second derivative test 

The [second derivative test]. If the determinant of the Hessian matrix det(f ′′ (x, y)) < 0 then (x, y) is a saddle 

point. If f ′′ (x, y) > 0 and and fxx(x, y) < 0 then (x, y) is a local maximum. If det(f ′′ (x, y)) < 0 and 

fxx(x, y) > 0 then (x, y) is a local minimum. 

Space 

[Space] is usually used as an abbreviation for three dimensional Euclidean space. In a wider sense, it can mean 

linear space a vector space in which on can add and scale. 

speed 

The [speed] of a curve r(t) = (x(t), y(t), z(t)) at time t is the length of the velocity vector r ′ (t) = 

(x ′ (t), y ′ (t), z ′ (t)). 

sphere 

A [sphere] is the set of points in space, which have a given distance r from a point P=(a,b,c). It is the set 

(x − a) 2 + (y − b) 2 + (z − c) 2 = r 2 . For a=b=c=0,r=1 one obtains the unit sphere: x 2 + y 2 + z 2 = 1. Spheres can 

be define in any dimenesions. A sphere in two dimensions is a circle. A sphere in 1 dimension is the union of two 

points. The unit sphere in 4 dimensions is the set of points (x, y, z, w) ∈ R 4 which satisfy x 2 + y 2 + z 2 + w 2 = 1 

Spheres can be defined in any space equiped with a distance like d((x, y), (u, v)) = |x − u| + |y − v| in the plane.

superformula 

The [superformula] describes a class of curves with a few parameters m, n1, n2, n3, a, b. It is the polar graph 

r(t) = (| cos(mt/4)| n1 /a + | sin(mt/4)| n2 /b) −1/n3 . 

It had been proposed by the Belgian Biologist Johan Gielis in 1997. 

superposition 

The principle of [superposition] tells that the sum of two solutions of a linear partial differential equation (PDE) 

is again a solution of the PDE. For example, f(x, y) = sin(x − t) and g(x, y) = e x−t are both solutions to the 

transport equation ft(t, x) + fx(t, x) = 0. Therefore also the sum sin(x − t) + e x−t is a solution. For nonlinear 

partial differential equations the superposition principle is no more true which is one of the reasons for the 

difficulty with dealing with nonlinear systems. 

surface 

A [surface] can either be described as a parametrized surface or implicitely as a level surface g(x, y, z) = 0. In 

the first case, the surface is given as the image of a map X : (u, v) ↦→ (x(u, v), y(u, v), z(u, v)) where u,v ranges 

over a parameter domain R in the plane. In the second case, the surface is determined by a function of three 

variables. Sometimes, one can describe a surface in both ways like in the following examples: 

Sphere: X(u, v) = (r cos(u) sin(v), r sin(u) sin(v), r cos(v)), g(x, y, z) = x 2 + y 2 + z 2 = r 2 

Graphs: X(u, v) = (u, v, f(u, v)), g(x, y, z) = z − f(x, y) = 0 

Planes: X(u, v) = P + uU + vV , g(x, y, z) = ax + by + cz = d, (a, b, c) = UxV . 

Surface of revolution: X(u, v) = (f(v) cos(u), f(v) sin(u), v), g(x, y, z) = f((x 2 + y 2 ) ( 1/2)) − z = 0 

surface of revolution 

A [surface of revolution] is a surface which is obtained by rotating a curve around a fixed line. If that line 

is the z-axes, the surface can be given in cylindrical coordinates as r = f(z). A parametrization is X(t, z) = 

(f(z) cos(t), f(z) sin(t), z). 

� 2π 

0 

surface area 

� � 

The [surface area] of surface S = X(R) is defined as the integral of R |Xu × Xv(u, v)| dudv. For example, 

for X(u, v) = (r cos(u) sin(v), r sin(u) sin(v), r cos(v)) on R = {0 ≤ u ≤ 2π, 0 ≤ v ≤ π}, where S = X(R) 

is the sphere of radius r, one has Xu × Xv = r sin(v)X and |Xu × Xv| = sin(v)r2 � π 

0 

. The surface area is 

r2 sin(v) dudv = 4πr2 . 

surface integral 

A [surface integral] of a function f(x, y, z) over a surface S = X(R) is defined as the integral of f(X(u, v))|Xu × 

Xv(u, v)| over R. In the special case when f(x, y, z) = 1, the surface integral is the surface area of the surface 

S.

tangent plane 

The [tangent plane] to an implicitely defined surface g(x, y, z) = c at the point (x0, y0, z0) is the plane ax + by + 

cz = d, where (a, b, c) = ∇f(x0, y0, z0) is the gradient of g at (x0, y0, z0) and d = ax0 + by0 + cz0. 

tangent line 

The [tangent line] to an implicitely defined curve g(x, y) = c at the point (x0, y0) is the line ax + by = d, where 

(a, b) is the gradient of g(x, y) at the point (x0, y0) and d = ax0 + by0. 

theorem of Clairot 

The [theorem of Clairot] assures that one can interchange the order of differentiation when taking partial 

derivatives. More precicely, if f(x, y) is a function of two variables for which both fxy = fyx are continuous, 

then fxy = fyx. 

theorem of Gauss 

The [theorem of Gauss] states that the flux of a vector field F through the boundary S of a solid R in threedimensional 

space is the integral of the divergence div(F) of F over the region R: 

� � � 

� � 

div(F ) dV = F · dS . 

R 

S 

theorem of Green 

The [theorem of Green] states that the integral of the curl(F ) = Qx − Py of a vector field F = (P, Q) over a 

region R in the plane is the same as the line integral of F along the boundary C of R. 

� � 

� 

curl(F ) dA = F ds . 

R 

The boundary C is traced in such a way that the region is to the left. The boundary has to be piecewise smooth. 

The theorem of Green can be derived from the theorem of Stokes. 

[Green’s theorem] see theorem of Green. 

C 

Green’s theorem 

Green’s theorem 

The determinant of the Jacobean matrix is often called Jacobean or Jacobean determinant.

Jacobean matrix 

[Jacobean matrix] If T (u, v) = (f(u, v), g(u, � v)) is a transformation � from a region R to a region S in the plane, 

fu(u, v) fv(u, v) 

the Jacobean matrix dT is defined as 

. It is the linearization of T near (u, v). Its 

gu(u, v) gv(u, v) 

determinant called the Jacobean determiant measures the area change of a small area element dA = dudv when 

maped by T . For example, if T (r, θ) = (r cos(θ), r sin(θ)) = (x, � y) is the coordinate �transformation 

which maps 

cos(θ) sin(θ) 

R = {r ≥ 0, θ ∈ [0, 2π)} to the plane, then dT is the matrix 

which has determinant r. 

−r sin(θ) r cos(θ) 

theorem of Stokes 

The [theorem of Stokes] states that the flux of a vector field F in space through a surface S is equal to the line 

integral of F along the boundary C of S: 

� � 

� 

curl(F ) · dS = F ds . 

S 

three dimensional space 

The [three dimensional space] consists of all points (x, y, z) where x, y, z ranges over the set of real numbers. 

To distinguish it from other three-dimensional spaces, one calls it also Euclidean space. 

torus 

A [torus] is a surface in space defined as the set of points which have a fixed distance from a circle. It can be 

parametrized by X(u, v) = (a + b cos(v)) cos(u), (a + b cos(v)) sin(u), sin(v)) on R = [0, 2π) × [0, 2π), where a, b 

are positive constants. 

trace 

The [trace] of a surface in three dimensional space is the intersection of the surface with one of the coordinate 

planes x=0 or y=0 or z=0. Traces help to draw a surface when given the task to do so by hand. Other marking 

points are intercepts, the intersection of the surface with the coordinate axes. 

triangle 

A [triangle] in the plane or in space is defined by three points P, Q, R. If v = P Q, w = P R, then |v × w|/2 is 

the area of the triangle. 

C 

triple product 

The [triple product] between three vectors u, v, w in space is defined as the scalar u · (v × w). The absolute 

value |u · (v × w)| is the volume of the paralelepiped spanned by u, v and w.

[triple dot product] (see triple product). 

triple dot product 

unit sphere 

The [unit sphere] is the sphere x 2 + y 2 + z 2 = 1. It is an example of a two-dimensional surface in three 

dimensional space. 

unit tangent vector 

The [unit tangent vector] to a parametrized curve r(t)=(x(t),y(t),z(t)) is the normalized velocity vector T (t) = 

r ′ (t)/|r ′ (t)|. Together with the normal vector N(t) = T ′ (t)/|T ′ (t)| and the binormal vector B(t) = T (t)xN(t), 

it forms a triple of mutually orthogonal vectors. 

vector 

A [vector] in the plane is defined by two points P, Q. It is the line segment v pointing from P to Q. If P = (a, b) 

and Q = (c, d) then the coordinates of the vector are v = (c − a, d − b). Points P in the plane can be identified 

by vectors pointing from 0 to P. A vector in space is defined by two points P,Q in space. If P = (a, b, c) and 

Q = (d, e, f), then the coordinates of the vector are v = (d − a, e − b, f − c). Points P in space can be identified 

by vectors pointing from 0 to P . Two vectors which can be translated into each other are considered equal. 

Remarks. 

• One could define vectors more precisely as affine vectors and introduce an equivalence relation among 

them: two vectors are equivalent if they can be translated into each other. The equivalence classes are the 

vectors one deals with in calculus. Since the concept of equivalence relation would unnessesarily confuse 

students, the more fuzzy definition above is prefered. 

• One should avoid definitions like ”Vectors are objects which have length and direction” given in some 

Encyclopedias. The zero vector (0, 0, 0) is an example of an object which has length but no direction. It 

nevertheless is a vector. 

vector field 

A [vector field] in the plane is a map F (x, y) = (P (x, y), Q(x, y)) which assigns to each point (x, y) in the 

plane a vector F (x, y). An example of a vector field in the plane is F (x, y) = (−y, x). An other example is 

the gradient field F (x, y) = ∇f(x, y) where f(x, y) is a function. A vector field in space is a map F (x, y, z) = 

(P (x, y, z), Q(x, y, z), R(x, y, z)) which assigns to each point (x, y, z) in space a vector F (x, y, z). An example is 

the vector field F (x, y, z) = (x 2 , yz, x − y). An other example is the gradient field F (x, y, z) = ∇f(x, y, z) of a 

function f(x, y, z). 

velocity 

The [velocity] of a parametrized curve r(t)=(x(t),y(t),z(t)) at time t is the vector r ′ (t) = (x ′ (t), y ′ (t), z ′ (t)). It 

is tangent to the curve at the point r(t).

volume 

The [volume] of a body G is defined as the integral of the constant function f(x,y,z)=1 over the body G. 

wave equation 

The [wave equation] is the partial differential equation utt = c 2 ∆(u), where ∆(u) is the Laplacian of u. 

Light in vacuum satisfies the wave equation. This can be derived from the Maxwell equations: the identity 

∆(B) = grad(div(B) − curl(curl(B)) gives together with div(B) = 0 and curl(B) = Et/c the relation 

∆(B) = −d/dtcurl(E)/c which leads with the Maxwell equation Bt = −ccurl(E) to the wave equation 

∆B = Btt/c 2 . The equation Ett = c 2 ∆E is derived in the same way. 

zero vector 

The [zero vector] is the vector for which all components are zero. In the plane it is v = (0, 0), in space it is 

v = (0, 0, 0). The zero vector is a vector. It has length 0 and no direction. Definitions like ”a vector is a quantity 

which has both length and direction” are misleading. 



Index 

acceleration, 1 

advection equation, 1 

Antipodes, 1 

Archimedes spiral, 1 

axis of rotation, 1 

boundary, 2 

Burger’s equation, 2 

Cartesian coordinates, 2 

Cavalieri principle, 2 

chain rule, 3 

change of variables, 2, 4 

circle, 3 

cone, 3 

conic section, 3 

continuity equation, 3, 8 

cos theorem, 3 

critical point, 3 

cross product, 4 

curl, 4 

curvature, 4 

curve, 4 

cylinder, 5 

cylindrical coordinates, 5 

derivative, 5 

determinant, 5 

directional derivative, 5 

distance, 5 

divergence, 6 

dot product, 6 

ellipse, 6 

ellipsoid, 6 

equation of motion, 6 

flux integral, 6 

Fubinis theorem, 6 

gradient, 7 

Green’s theorem, 17 

Hamilton equations, 7 

heat equation, 7 

Hessian, 7 

Hessian matrix, 7 

hyperbola, 7 

hyperboloid, 8 

incompressible, 8 

integral, 8 

intercept, 8 

interval, 8 

Jacobean matrix, 18 

jerk, 9 

Lagrange method, 9 

Lagrange multiplier, 9 

Laplacian, 9 

length, 9 

21 

level curve, 9 

level surface, 10 

line, 10 

line integral, 10 

linear approximation, 10 

Maxwell equations, 10 

nabla, 10 

nabla calculus, 11 

nonparallel, 11 

normal vector, 11 

normalized, 11 

octant, 11 

open, 11 

open set, 11 

ordinary differential equation, 12 

orthogonal, 12 

parabola, 12 

parallel, 13 

parallelepiped, 12 

parallelogram, 12 

parametrized curve, 14 

parametrized surface, 13 

partial derivative, 14 

partial differential equation, 14 

perpendicular, 12 

plane, 14 

polar coordinates, 14 

potential, 15 

projection, 15 

quadrant, 13 

quadratic approximation, 13 

right handed, 15 

second derivative test, 15 

Space, 15 

speed, 15 

sphere, 15 

superformula, 16 

superposition, 16 

surface, 16 

surface area, 16 

surface integral, 16 

surface of revolution, 16 

tangent line, 17 

tangent plane, 17 

theorem of Clairot, 17 

theorem of Gauss, 17 

theorem of Green, 17 

theorem of Stokes, 18 

three dimensional space, 18 

torus, 18 

trace, 18 

triangle, 18 

triple dot product, 19

triple product, 18 

unit sphere, 19 

unit tangent vector, 19 

vector, 19 

vector field, 19 

velocity, 19 

volume, 20 

wave equation, 20 

zero vector, 20

ENTRY LINEAR ALGEBRA 

[ENTRY LINEAR ALGEBRA] Author: Oliver Knill: Spring 2002-Spring 2004 Literature: Standard glossary 

of multivariable calculus course as taught at the Harvard mathematics department. 

adjacency matrix 

The [adjacency matrix] of a graph is a matrix Aij, where Aij = 1 whenever there is an edge from node i to 

node j in the graph. Otherwise, Aij ⎛ ⎞ 

= 0. Example: the graph with three nodes with the shape of a V has the 

0 1 0 

adjacency matrix A = ⎝ 1 0 1 ⎠, where node 2 is conneced to both node 1 and 3 and node 1 and 3 are not 

0 1 0 

connected to each other. 

affine transformation 

An [affine transformation] is the composition of a linear transformation with a shift like for example: T (x, y) = 

(2x + y, 3x + 4y) + (2, 3). 

Algebra 

[Algebra] was originally the art of solving equations and systems of equations. The word comves from the Arabic 

”al-jabr” meaning ”restauration”. The term was used by Mohammed al-Khowarizmi, who worked in Bhagdad. 

algebraic multiplicity 

The [algebraic multiplicity] of a root y of a polynomial p is the maximal integer k for which p(x) = (x−y) k q(x). 

The algebraic multiplicity is bigger or equal than the geometric multiplicity. 

angle 

The [angle] between two vectors v and w is arccos((x · y)/(||x||||y||), where x ˙y is the dot product between x and 

y and ||x|| = √ x · x is the length of x. The inverse of cos gives two angles in [0, 2π]. One usually choses the 

smaller angle. 

argument 

The [argument] of a complex number z = x + iy is φ if z = re iφ . The argument is determined only up to 

addition of 2π. It can be determined as φ = arctan(y/x) + A, where A = 0 if x > 0 or x = 0, y > 0 and A = π 

if x < 0 or x = 0 and y < 0. For example, arg(i) = π/2 and arg(−i) = 3pi/2. The argument is the imaginary 

part of log(z) because log(re iphi ) = log(r) + iφ.

associative law 

The [associative law] is (AB)C = A(BC). It is an identity which some mathematical operations satisfy. For 

example, the matrix multiplication satisfies the associative law. One says also, that the operation is associative. 

An example of a product which is not associative is the cross product v × w: if i, j, k are the standard basis 

vectors, then i × (i × j) = i × k = −j and (i × i) × j = 0 × j = 0. 

augmented matrix 

The [augmented matrix] of a linear equation Ax = b is the n × (n + 1) matrix � A b � . One considers the 

augmented matrix when solving a linear system Ax = b. The reduced row echelon form rref � A b � contains 

the solution vector x in the last column, if a solution exists. More generally, a matrix which contains a given 

matrix as a submatrix is called an augmented matrix. 

basis 

A [basis] of a linear space is a finite set of vectors v1, ..., vn, which are linearly independent and which span the 

linear space. If the basis contains n vectors, the vector space has dimension n. 

basis theorem 

The [basis theorem] states that d linearly independent vectors in a vector space of dimension d forms a basis. 

block matrix 

A [block matrix] is a matrix A, where the only non-zero elements are contained in a sequence of smaller square 

matrices arranged ⎛ along the main ⎞ diagonal of A. Such matrices are also called block diagonal matrices. The 

1 2 0 0 0 

⎜ 3 2 0 0 0 ⎟ 

matrix A = ⎜ 0 0 5 0 0 ⎟ 

⎟. 

is an example of a block-diagonal matrix containing a two 2x2, and a 1x1 

⎝ 0 0 0 6 7 ⎠ 

0 0 0 8 9 

matrix in its diagonal. 

Cauchy-Schwarz inequality 

The [Cauchy-Schwarz inequality] tells that |x · y| is smaller or equal to ||x|| ||y||. Equality holds if and only if x 

and y are parallel vectors. 

Cayley-Hamilton theorem 

The [Cayley-Hamilton theorem] assures that every square matrix A satisfies p(A) = 0, where p(x) = det(A − x) 

is the characteristic polynomial of A and the right hand side 0 is the zero matrix.

change of basis 

A [change of basis] from an old basis 

vj to a new basis 

wj is described by an invertible matrix S which relates the coordinates (a1, ..., an) of a vector a = � 

i aivi 

in the old v-basis with the coordinates (b1, ..., bn) of the same vector b = � 

i biwi in the new w-basis. The 

relation of the coordinates is b = Sa In that case, one has vj = � 

j ST ijwj, where ST is the transpose of S. 

For example if v1 = (1, 0), v2 = (0, 1), w1 = (3, 4)w2 = (2, �3), then�a = (a1, a2) = (5, 7) in the v-basis has the 

3 2 

coordinates b = (b1, b2) = (1, 1) in the w-basis. With S = 

and S 

4 3 

T � � 

3 4 

= 

we have b = Sa and 

2 3 

w1 = 3v1 + 4v2, w2 = 2v1 + 3v2. 

characteristic matrix 

The [characteristic matrix] of a square matrix A is the matrix A(x) = (xI − A) , where I is the identity matrix. 

The characteristic matrix is a function of the free variable x. 

characteristic polynomial 

The [characteristic polynomial] of a matrix A is the polynomial p(x) = det(xI − A), where I is the identity 

matrix. It has the form p(x) = x n − tr(A)x ( n − 1) + ... + (−1) n det(A) where tr(A) is the trace of A and det(A) 

is the determinant of the matrix A. The eigenvalues of A are the roots of the characteristic polynomial of A. 

Cholesky factoriztion 

The [Cholesky factoriztion] of a symmetric and positive definite matrix A is A = R T R, where R is upper 

triangular with positive diagonal entries. 

circulant matrix 

A [circulant matrix] is a square matrix, where the entries in each diagonal are constant. ⎛ 

0 

If S is⎞the shift 

1 0 

matrix which has 1 in the side diagonal and 0 everywhere else like in the 3x3 case: S = ⎝ 0 0 1 ⎠, then a 

1 0 0 

circular matrix can be written as A = a0 + a1S + ... + an−1Sn−1 . A general 3x3 circulant matrix has the form 

A = a + bS + cS2 ⎛ ⎞ 

a b c 

which is S = ⎝ c a b ⎠. 

b c a 

classical adjoint 

The [classical adjoint] adj(A) of a n×n matrix A is the n×n matrix whose entry aij is aij = (−1) ( i+j)det(Aji), 

where Aji is a minor of A. The classical adjoint plays a role in Cramer’s rule A ( − 1) = adj(A)/det(A). The 

name ”adjoint” comes from the fact that we have a change indices like in the adjoint. However, the classical 

adjoint has nothing to do with the adjoint.

codomain 

The [codomain] of a linear transformation T : X → Y is the target space Y . The name has its origin from 

naming X the domain of A. 

cofactor 

A [cofactor] Cij of a n × n matrix A is the determinant of the (n − 1) × (n − 1) matrix obtained by removing 

row i and column j from A and multiplying the result with (−1) i+j . 

coefficient 

A [coefficient] of a matrix A is an entry Aij in the i’th row and the j’th column. For a real matrix, all entries 

are real numbers, for a complex matrix, the entries can be complex numbers. 

⎜ 

A [column] of a matrix is one of the vectors ⎜ 

⎝ 

in the image of the transformation x ↦→ A(x). 

The matrix A defining a linear equation Ax=b or 

⎛ 

A11 

A21 

A31 

. . . 

Am1 

column 

⎞ 

⎛ 

⎟ 

⎠ , 

⎜ 

⎝ 

column 

A1n 

A2n 

A3n 

. . . 

Amn 

⎞ 

A11x1 + . . . A1nxn = b1 

. . . = . . . 

Am1x1 + . . . Amnxn = bm 

⎟ of a m × n matrix A. Column vectors are 

⎠ 

is called the [coefficient matrix] of the system. The augmented matrix is the m × (n + 1) matrix � A b � , 

where b forms an additional column. 

column picture 

The [column picture] of a linear equation Ax = b is that the vector b becomes a linear combination of the 

columns of A. The linear equation is solvable if the vector b is in the column space of A. 

column space 

The [column space] of a matrix A is the linear space spanned by the columns of A.

commuting matrices 

Two [commuting matrices] A, B satisfy AB = BA. In that case, if A is diagonalizable, then also B is diagonalizable 

and both A and B share the same n eigenvectors. 

commutative law 

The [commutative law] A ∗ B = B ∗ A for some operation ∗ is an identity which holds for certain operations like 

the addition of matrices. Other operations like the multiplication of matrices does not satisfy the commutative 

law. One says: matrix multiplication is not commutative. 

complex conjugate 

The [complex conjugate] of a complex number z = x + iy is the complex number x − iy. It has the same length 

|z| as z. 

Complex numbers 

[Complex numbers] form an extension of the real numbers. They are obtained by introducing i = (−1) 1/2 and 

extending the rules of addition (a + ib) + (c + id) = (a + c) + i(b + d) and multiplication (a + ib)(c + id) = 

(ac − bd) + i(ad + bc). The absolute value r = |x + iy| is the length of the vector (a, b). The argument of 

z, φ = arg(z) is defined as the angle in [0, 2π) between the x axes and the vector (a, b). Using these polar 

coordinates one can see the Euler identity z = r exp(iφ) = r cos(φ) + ir sin(φ). 

consistent 

A system of linear equations Ax = b is called [consistent], if there exists for every vector b a solution vector x 

to the equation Ax=b. If the system has no solution, the system is called inconsistent. 

continuous dynamical system 

A [continuous dynamical system] is defined by an ordinary differential equation d/dtu = f(u) where u = u(t) is 

a vector valued function and f(u) is a vector field. If f(u) is linear, the equation has the form d/dtu = Au. The 

name ”continuous” comes from the fact that the time variable t is taken in the continuum. This distinguishes 

the system from discrete dynamical systems u(t+1) = f(u(t)) determined by a map f and where t is an integer. 

For linear continuous dynamical systems, the origin 0 is invariant. The origin is called asymptotically stable 

if x(t) → 0 for all initial conditions x(0). For continuous dynamical systems ut = Au, this is equivalent with 

the requirement that all eigenvalues of A have a negative real part. In two dimensions, where the trace and 

the determinant determine the eigenvalues, linear stability is characterized by det(A) > 0, tr(A) < 0 (stability 

quadrant).

covariance matrix 

A [covariance matrix] A of two finite dimensional random variables x, y with expectation E[x] = E[y] = 0 is 

defined as Aij = E[xiyj], where E[x] = (x1 + ... + xn)/n is the mean or expectation of x. The covariance matrix 

is always symmetric. If the covariance matrix is diagonal, the random variables x, y are called uncorrelated. 

Cramer’s rule 

[Cramer’s rule] tells that a solution x of a linear equation Ax = b can be obtained as xi = det(Ai)/det(A), 

where Ai is the matrix obtained by replacing the column i of A with the vector b. 

de Moivre formula 

The [de Moivre formula] is (cos(x)+i sin(x)) n = cos(nx)+i sin(nx). It is useful to derive trigonometric identities 

like cos(x) 3 − 3 sin(x) 2 cos(x) = cos(3x). 

determinant 

The [determinant] of a n × n square matrix A is the sum over all products A[1, π(1)]...A[n, π(n)](−1) π , where 

π runs over all permutations of {1, 2, ..., n} and (−1) π is the sign of the permutation π. Example: ⎛ for a 2 × ⎞2 

� � 

a b c 

a b 

matrix A = 

the determinant is det(A) = ad − bc. Example: For a 3x3 matrix A = ⎝ d e f ⎠, 

c d 

g h i 

the determinant is det(A) = aei + bfg + cdh − ceg − bfg − cdh. Properties of the determinant are det(AB) = 

det(A)det(B), det(AT ) = det(A), det(A−1 ) = 1/det(A). 

differential equation 

A [differential equation] is an equation for a function f in one or several variables which involves derivatives 

with respect to these variables. An ordinary differential equation is a differential equation, where derivatives 

appear only with respect to one variable. By adding new variables if necessary (for example for t, or derivatives 

ut, utt etc, one can write an ordinary differential equation always in the form xt = f(x). 

dilation 

A [dilation] is a linear transformation x → bx. Dilations scale each vector v by the factor b but leave the 

direction of v invariant. 

dimension 

The [dimension] of a vector space X is the number of basis vectors in a basis of X.

distributive law 

The [distributive law] is A ∗ (B + C) = A ∗ B + A ∗ C. The set of matrices with matrix multiplication * and 

addition + is an example where the distributive law applies. 

dot product 

The [dot product] v · w of two vectors v and w is the sum of the products viwi of their components vi, wi. For 

complex vectors, the dot product is defined as � 

i viwi. Examples: 

• (3, 2, 1) · (1, 2, −1) = 6. 

• if v · w = 0, then the vectors are orthogonal. 

• the length of the vector |v| is the square root of v · v. 

• v · w = |v||w| cos(α), where α is the angle between v and w. 

• if A, B are two n × n matrices, then (AB)ij is the dot product of the i’th row of A with the j’th column 

of B 

echelon matrix 

The [echelon matrix] of a matrix A is a matrix rref(A), where the pivot in each row comes after the pivot in 

the previous row. The pivot is the first nonzero entry in each row. The echelon matrix is also called a matrix 

in reduced row echelon form. 

eigenbasis 

An [eigenbasis] to a matrix A is a basis which consists of eigenvectors of A. 

eigenvalue 

An [eigenvalue] � � λ of a matrix A is a number for which there exists a vector v such that Av = λv. Example: 

3 2 

A = 

has the eigenvector v = (0, 1) with eigenvalue x = 4. 

0 4 

eigenvector 

An [eigenvector] � v of a� matrix A is a nonzero vector v for which Av = λv with some number λ (called eigenvalue). 

−1 1 

Example: 

has the eigenvector v = (1, 1) with eigenvalue λ = 0. 

1 1

Elimination 

[Elimination] is a process which reduces a matrix A to its echelon matrix rref(A). See row reduced echelon 

form. 

ellipsoid 

An [ellipsoid] can be written as the set of points x which satisfy x ˙ Ax = 1, where A is a positive definit matrix. 

The axes vi of the ellipse are the eigenvectors of A have the length 1/xi , where xi are the eigenvalues of A. 

entry 

An [entry] or coefficient of a matrix is the number or the variable A(i, j) of a matrix. 

expansion factor 

The [expansion factor] of a linear map is the absolute value of the determinant of A. It is the volume of the 

parallelepiped obtained as the image of the unit cube under A. 

exponential 

The [exponential] exp(A) of a matrix A is defined as the sum exp(A) = 1 + A + A 2 /2! + A 3 /3! + .... The linear 

system of differential equations x ′ = Ax for x(t) has the solution x(t) = exp(At)x(0). 

factorization 

The [factorization] of a polynomial p(x) is the representation p(x) = a(λ1 − x)...(λn − x), where lambdai are 

the n roots of the polynomials whose existence is assured by the fundamental theorem of algebra. 

Fourier coefficients 

The [Fourier coefficients] of a 2π periodic function f(x) on [−π, π] is cn = (1/2π) � p 

if(x)exp(−inx). One 

−pi 

has f(x) = � 

n cn exp(inx). By writing f(x) = g(x) + h(x), where g(x) = [f(x) + f(−x)]/2 is even and 

h(x) = [f(x) − f(−x)]/2 is odd one can obtain real versions: the even function can be written as a cos-series 

g(x) = sum∞ n=0an cos(nx), where an = (2/π) � π 

0 g(x) cos(nx)dx for n > 0 and a0 = (1/π) � π 

g(x) dx. The odd 

0 

function can be written as the sin-series h(x) = sum∞ n=1bn sin(nx), where bn = (2/π) � π 

h(x) sin(nx)dx. The 

0 

complex Fourier coefficients cn are coordinates of f(x) with respect to the orthonormal basis exp(inx). The 

real Fourier coefficients are the coordinates of f(x) with respect to orthogonal basis 1, cos(nx), sin(nx), n > 0. 

The [Fourier series] of a function f is f(x) = �∞ n=−∞ cn exp(inx) or f(x) = � 

n an cos(nx) for even functions 

or f(x) = sum ∞ n=1 bn sin(nx) for odd functions.

fundamental theorem of algebra 

The [fundamental theorem of algebra] states that a polynomial p(x) = x n + ...a1x + a0 of degree n has exactly 

n roots. 

Gauss Jordan elimination 

The [Gauss Jordan elimination] is a method for solving linear equations. It was already known by the Chinese 

2000 years ago. Gauss called it ”eliminatio vulgaris”. The method does linear combinations of the rows of a 

n × (n + 1) matrix until the system is solved. Example: 

2 x + 4y = 2 

3 x + y = 12 

x + 2y = 1 

3 x + y = 13 

x + 2y = 1 

- 5y = 10 

x + 2y = 1 

y = -2 

x = 5 

y = -2 

First, the top equation was scaled, then three times the first equation was subtracted from the second equation. 

Then the the second equation was scaled. Finally, twice the the second equation was subtracted from the first. 

geometric multiplicity 

The [geometric multiplicity] of an eigenvalue λ is the dimension of ker(λ − A). The geometric multiplicity is 

smaller or equal to the algebraic multiplicity. 

Gibbs phenomenon 

The [Gibbs phenomenon] � describes the error when doing a Fourier approximation of the discontinuous Heavyside 

−1if − π ≤ x ≤ 0 

function f(x) = 

The Fourier approximation is sn(x) = (4/π) 

1if0 < x < π 

�n k=1 sin((2k−1)t)/(2k−1). 

The derivative d/dxsn can be computed: (differentiate and sum up the geometric series): (2/π) sin(2nx)/ sin(x) 

which vanishes at x = ±π/2n. These are the extrema for sn. Now, sn(π/2n) = (4/π) � sin((2k − 

1)(π/2n))/((2k−1)(π/2n)) is a Riemann sum approximation for (2/π) � π 

0 sin(t)/tdt = π(1−π2 /(3!3)+π4 /(5!5)− 

...) = 1.1793... This overshoot is called the Gibbs phenomenon. It was first discoverd by Wilbraham in 1848 

then by Gibbs in 1899. The human eye can recognize the Gibbs phenomenon as ”ghosts” on a TV screen, unless 

it is corrected for. 

Gramm-Schmidt process 

The [Gramm-Schmidt process] is an algorithm which constructs from a basis v1, ..., vn an orthonormal basis 

w1, ..., wn. The procedure goes by induction: if w1, ..., wk−1 are orthonormal, then the next vector wk is 

wk = uk/||uk||, where uk = vk − (w1 · vk)w1 − (w2 · vk)w2 · · · − (wk−1 · v + k)wk−1.

graph 

A [graph] is a set of n nodes connected by m edges. It is completely defined by its adjacency matrix. In a 

directed graph, the nodes are oriented. Examples: 

• a complete graph has all nodes connected. There are n(n − 1)/2 edges. Its adjacency matrix is E − I, 

where E is the n × n matrix with all entries equal to 1 and I is the identity matrix. 

• a tree ⎛has 

m = n⎞ − 1 edges and no closed loops. An example is the graph with the adjecency matrix 

0 1 0 

A = ⎝ 1 0 1 ⎠. 

0 1 0 

� 

0 

• The directed graph with two edges 1 → 2 has the adjacency matrix A = 

0 

1 

0 

� 

. 

Hankel matrix 

A [Hankel matrix] is a square matrix A, where the entries are constant along the antidiagonal. ⎛ In other ⎞ words, 

a b c 

each entry Aij depends only on i + j. A general 3 × 3 Hankel matrix is of the form A = ⎝ b c d ⎠. 

c d e 

heat equation 

The [heat equation] is the linear partial differential equation ut = µuxx. The heat equation on a finite interval 

[0, π] with boundary conditions u(0, t) = 0, u(π, t) = 0 and initial conditions u(x, 0) = f(x) can be solved 

with the Fourier series u(x, t) = � 

n>0 an sin(nx)e−µn2t , where an = (2/π) � p 

if(x) sin(nx) dx are the Fourier 

0 

coefficients. 

Hermitian matrix 

A [Hermitian matrix] satisfies A ∗ = A T = A, where A T is the transpose of A and A is the complex conjugate 

matrix, where all entries are replaced by their complex conjugates. 

Hessenberg matrix 

A [Hessenberg matrix] A is an upper triangular matrix with only⎛one extra nonzero ⎞ adjacent diagonal below 

a b c 

the diagonal. Example: a general 3 × 3 Hessenberg matrix is A = ⎝ d e f ⎠. 

0 g h

Hilbert matrix 

A [Hilbert matrix] is a symmetric square matrix, where Aij = 1/(i + j − 1). It is an example of a Hankel matrix 

and positive definit. Hilbert matrices are examples of matrices which⎛are difficult to invert, ⎞ because their 

1 1/2 1/3 

determinant is small. For example, for n = 3, the Hilbert matrix A = ⎝ 1/2 1/3 1/4 ⎠ has determinant 

1/3 1/4 1/5 

1/2160. A [hyperplane] in n-dimensional space V is a (n − 1)-dimensional linear subspace of V . 

identity matrix 

The [identity matrix] is the matrix I which has 1 in the diagonal and zero ⎛ everywhere ⎞ else. The identity matrix 

1 0 0 

I satisfies IA = A for any matrix A. The 3x3 identity matrix is A = ⎝ 0 1 0 ⎠. 

0 0 1 

incidence matrix 

The [incidence matrix] of a directed graph with n nodes and m edges is a m×n matrix which has a row for each 

edge connecting nodes i and j with entries −1 and 1 in columns i, j. ⎛Example. 

The directed ⎞ graph 1 ⇒ 2 ⇐ 3, 

−1 1 0 0 

1 → 4 with 3 edges and 4 nodes has the 3 × 4 incidence matrix A = ⎝ 0 1 −1 0 ⎠. 

−1 0 0 1 

inconsistent 

A system of linear equations Ax = b is called [inconsistent] if the system has no solutions. 

indefinite matrix 

An [indefinite matrix] is a matrix, whith eigenvalues of different sign. A positive definite matrix is an example 

of a matrix which is not indefinite. 

Independent vectors 

[Independent vectors]. If no linear combination a1v1 + ... + anvn is zero unless all ai are zero, then the vectors 

v1, ..., vn are called independent. If A is the matrix which contains the vectors vi as columns, then the kernel of 

A is trivial. A basis consists of independent vectors. 

Independent vectors 

The linear transformation corresponding to the identity matrix is called the [identity transformation].

image 

The [image] of a linear transformation T : X → Y, T (x) = Ax is the subset of all vectors y = Ax, xinX in 

Y. The image is denoted by im(T ) or im(A) and is a subset of the codomain Y of T . The image is also 

called the range. The dimension of the image of T is equal to the rank of A and the dimension satisfies 

dim(ker(A)) + dim(im(A)) = n, where n is the dimension of the linear space X. 

index 

The [index] of a linear map T is defined as ind(A) = dim(kerA)−dim(cokerA), where coker(A) is the orthogonal 

complement of the image of A. Examples are: 

• The index of a n × n matrix A is dim(kerA) − dim(cokerA) = 0. 

• The index of the differential operator Df = f ′ acting on smooth functions on the real line is 1 − 0 = 1 

because D has a one dimensional kernel (the constant functions) and a zero dimensional cokernel (all 

functions can be obtained as the image of D). The index of D n is n. 

• The index of the differential operator Df = f ′ acting on smooth functions on the circle is 1 − 1 = 0 

because D has a one dimensional kernel (the constant functions) and a one-dimensional cokernel (the 

constant functions, one can not find a periodic function g such that g’=1). 

• The Atiyah-Singer index theorem relates topological properties of a surface M with the index of a ”Dirac 

operator” T on it. The previous two examples exemply that. T = D is a Dirac operator and the topology 

of the circle or the line are different. 

inverse 

The [inverse] of a square matrix A is a matrix B satisfying � AB � = I and BA = I where I is�the identity� matrix. 

a b 

d −b 

For example, the inverse of the transformation A = 

is the transformation B = 

/(ad − 

c d 

−c a 

bc). 

invertible 

A square matrix A is [invertible] if there exists a matrix B such that AB = I. A matrix A is invertible if and 

only if the determinant of A is different from zero. 

Jordan normal form 

The [Jordan normal form] J = S −1 AS of a square matrix A is a block matrix J = diag(J1, ..., Jk), where each 

block is of the form Jk = xkIk + Nk, where xk is an eigenvalue of A, Ik is an identity matrix and Nk is a matrix 

with 1 in the first sidediagonal. If all eigenvalues of A are different, then the Jordan normal form is a diagonal 

matrix.

kernel 

The [kernel] of a linear transformation T : X → Y , T (x) = Ax is the linear space {xinXsuchthatAx = 0}. 

The kernel is denoted by ker(T ) or ker(A) and is a subset of the domain X of T . The dimension of the kernel 

dim(ker(A)) and the dimension of the image dim(im(A)) are related by dim(ker(A)) + dim(im(A)) = n, where 

n is the dimension of the linear space X. The kernel of a transformation is computed by building rref(A), the 

reduced row echelon form of A. Echelon = ”series of steps”. Every vector in the kernel of rref(A) is also in the 

kernel of A. 

Leontief 

Wassily [Leontief]. A Russian-born US economist who was working also at Harvard University. Leontief was 

a winner of the 1973 Economics Nobel prize for the development of the input-output method and for its 

application to important economic problems. Linear algebra students find the following problem in textbooks: 

two industries A and B produce output with value x and y (in millions of dollars). Assume that the consumer 

demand is a for the product of A and b for the product of B. Assume also an industry demand: p x is transfered 

from A to B, and q y is transfered from B to A. For which x and y are both the industry and conumer demand 

satisfied? The problem is equivalent to solving the linear system 

x − qy = a 

−px + y = b 

Laplace equation 

The [Laplace equation] in a region G is the linear partial differential equation uxx + uyy = 0. A solution 

is determined if u(x, y) is prescribed on the boundary of G. On the square [0, π] × [0, π] with boundary 

conditions 0 except at the side y = π, where one has u(x, π) = f(x), one can find a solution via Fourier series: 

u(x, y) = � 

n>0 an sin(nx) sinh(ny)/sinh(nπ), where an = (2/π) � π 

f(x) sin(nx) dx. The case with general 

0 

boundary conditions can be solved by adding corresponding solutions u(x, y), u(y, x), u(x, π − y), u(π − y, x) for 

the other 3 sides of the square. 

Laplace expansion 

The [Laplace expansion] is a formula for the determinant of A: det(A) = (−1) i+1 ai1det(Ai1) + ... + 

(−1) i+n aindet(Ain). 

leading one 

A [leading one] is an entry of a matrix in reduced row echelon form which is contained in a row with this element 

as the first nonzero entry. 

leading variable 

A [leading variable] is a variable which corresponds to a leading one in rref(A).

least-squares solution 

A vector x ∈ R n is called a [least-squares solution] of the system Ax = b where A is a m × n matrix, if ||b − Ay|| 

is less or equal then ||b − Ax|| for all y ∈ R n . If x is the least-squares solution of Ax = b then Ax is the 

orthogonal projection of b onto the image im(A). The explicit formula is x = (A T A) ( − 1)A T b and derived from 

that A T (Ax − b) = 0 which itself just means that Ax − b is orthogonal to the image of A. 

length 

The [length] of a vector v is ||v|| = (v · v) 1/2 = (v 2 1 + ... + v 2 n) ( 1/2). The length of a complex number x + iy is 

the length of (x, y). The length of a vector depends on the basis, usually it is understood with respect to the 

standard basis. 

linear combination 

A [linear combination] of n vectors v1, ..., vn is a vector a1v1 + ... + anvn. 

Linearly dependent vectors 

[Linearly dependent vectors]. If there exist a1, ..., an which are not all zero such that a1v1 + ... + anvn = 0, then 

the vectors v1, ..., vn are called linearly dependent. 

Linearity 

[Linearity] is a property of maps between linear spaces: it means that lines are maped into lines and the image 

of the sum of two vectors is the same as sum of the images. For example: T (x, y, z) = (2x + z, y − x) is linear. 

T (x, y) = (x 2 − y, x) is nonlinear. 

linear dynamical system 

A [linear dynamical system] is defined by a linear map x ↦→ Ax. The orbits of the dynamical system are 

x, Ax, A 2 x, .... 

linear space 

A [linear space] is the same as a vector space. It is a set which is closed under addition and multiplication with 

real numbers. 

linear combination 

A sum a1v1 + ... + anvn is called a [linear combination] of the vectors v1, ..., vn.

linear subspace 

A [linear subspace] of a vector space V is a subset of V which is also a vector space. In particular, it is closed 

under addition, scalar multiplication and contains a neutral element. 

linear system of equations 

A [linear system of equations] is an equation of the form Ax=b, where A is a m × n matrix, x is a n-vector and 

b is a m-vector. There are three possibilities: 

• consistent with one solution: no row vector (0...0�1) in rref(A|b). There is exactly one solution if there is 

a leading one in each column of rref(A). 

• consistent with infinitely many solutions: there are columns with no leading one. 

• Inconsistent with no solutions: there is a row (0...0�1) in rref(A�b). 

logarithm 

The [logarithm] log(z) of a complex number z = x + iy �= 0 is defined as log |z| + iarg(z), where arg(z) is the 

argument of z. The imaginary part of the logarithm is only defined up to a multiple of 2π. 

Markov matrix 

A [Markov matrix] is a square matrix, where all entries are nonnegative and the sum of each column is 1. One 

of the eigenvalues of a Markov matrix is 1 because A T has the eigenvector (1, 1, ..., 1). If all entries of a Markov 

matrix are positive, then A k v converges to the eigenvector v with eigenvalue 1. This vector is called the ”steady 

state” vector. 

matrix 

A [matrix] is a rectangular ⎛ array of numbers. ⎞ The following 3 × 4 matrix for example consists of three rows 

2 8 4 2 

and four columns: A = ⎝ 1 2 1 3 ⎠. The first index addresses the row, the second the column of the 

2 −1 1 2 

matrix. A n × m matrix maps the m-dimensional space to the n-dimensional space. 

Matrix multiplication 

[Matrix multiplication] is an operation defined between a (n × m) matrix A and a (m × p) matrix B. (AB)ij 

is the dot product between the i’th row of A with the j’th column of B. Example: (n = 2, m = 3, p = 4) 

� � 

2 1 0 

1 0 1 

⎛ 

⎞ 

2 1 0 0 � � 

⎝ 0 0 1 2 ⎠ 

4 2 1 2 

= 

. 

3 4 1 1 

1 3 1 1

minor 

A [minor] of a matrix A is a matrix A(i, j) which is obtained from A by deleting row i and column j. 

nilpotent matrix 

A [nilpotent matrix] is a matrix A for which some power Ak ⎛ ⎞ 

is the zero matrix. A nilpotent matrix has only 

0 1 1 

zero eigenvalues. he matrix A = ⎝ 0 0 2 ⎠ for example satisfies A 

0 0 0 

3 = 0 and is therefore nilpotent. 

non-leading coefficient 

A [non-leading coefficient] is an entry in the row reduced echelon form of a matrix A which is nonzero and which 

comes after the leading 1. The relevance of this definition comes from the fact that the number of columns with 

non-leading coefficients is the dimension of the kernel of the map. 

normal equation 

The [normal equation] to the linear equation Ax=b is the consistent system A T Ax = A T b. 

normal matrix 

A matrix A is called a [normal matrix], if AA T = A T A. A normal matrix has orthonormal possibly complex 

eigenvectors. 

null space 

The [null space] of a matrix A is the same as the kernal of A. It is spanned by the solutions Av = 0. The 

dimension of the null space is n − r, where n is the number of columns of A and r is the rank of A. 

ordinary differential equation 

An [ordinary differential equation] is a differential equation, where derivatives appear only with respect to one 

variable. By adding new variables if necessary (for example for t, or derivatives ut, utt etc. one can write such 

an equation always in the form xt = f(x). An ordinary differential equation defines a continuous dynamical 

system. The initial condition x(0) determines the trajectories x(t). An ordinary differential equation of the 

form ut = Au, where A is a matrix is called a linear ordinary differential equation.

orthogonal 

Two vectors v,w are [orthogonal] if their dot product v · w vanishes. 

orthogonal basis 

An [orthogonal basis] is a basis such that all vectors in the basis are orthogonal. 

orthonormal basis 

An [orthonormal basis] is a basis such that all vectors are orthogonal and normed. 

orthonormal complement 

The [orthonormal complement] of a linear subspace V in R n is the set of vectors which are orthogonal to V . 

orthogonal projection 

The [orthogonal projection] onto a linear space V is proj V (x) = (x · v1)v1 + ... + (x · vn)vn, where the vj form 

an orthonormal basis in V . Despite the name, an orthogonal projection is not an orthogonal transformation. It 

has a kernel. In an eigenbasis, a projection has the form (x, y) → (x, 0). 

Euclidean space 

[ Euclidean space ] is the linear space of all vectors = 1 × n matrices. R 0 is the space 

0. The space R 1 is the real linear space of all real numbers, the R 2 is the plane, the R 3 the Euclidean three 

dimensional space. 

parallel 

Two vectors v and w are called [parallel] if v both are nonzero and one is a multiple of the other. 

parallelepiped 

A set in R n is a [parallelepiped] E if it is the linear image A(Q) of the unit cube Q. The volume of a n-dimensional 

parallelepiped E in R n satisfies vol(E) = |det(A)|, in general, vol(E) = (det(A T A)) 1/2 .

partial differential equation 

A [partial differential equation] (PDE) is an equation for a multi-variable function which involves partial derivatives. 

It is called linear if (u + v) and rv are solutions whenever u and v are solutions. Examples of linear 

PDEs: 

Examples of nonlinear PDE: 

ut = cux 

transport equation 

ut = buxx 

heat equation 

utt = auxx 

wave equation 

uxx + uyy = 0 Laplace equation 

uxx + uyy = f(x, y) Poisson equation 

ihut = −uxx + V (x) Schroedinger equation 

ut + uux = auxx 

ut + uux = −uxxx 

Burger equation 

Korteweg de Vries equation 

utt − uxx = sin(x) Sine Gordon equation 

utt − uxx = f(x) Nonlinear wave equation 

ihut = −uxx − |x| 2 x Nonlinear Schroedinger equation 

ut + ux(x, t) 2 /2 + V (x) = 0 Hamilton Jacobi equation 

permutation matrix 

A [permutation matrix] A is a square matrix with entries Aij = Iiπ(j) where π is a permutation of 

1, ..., n and where I is the identiy matrix. There are n! permutation ⎛ matrices. ⎞ Example: for n = 3 the 

0 1 0 

permutation π(1, 2, 3) = (2, 1, 3) defines the permutation matrix A = ⎝ 1 0 0 ⎠. 

0 0 1 

pivot 

A [pivot] d is the first nonzero diagonal entry when a row is used in Gaussian elimination. 

pivot column 

A column of a matrix is called a [pivot column] if the corresponding column of rref(A) contains a leading one. 

The pivot columns are important because they form a basis for the image of A. 

polar decomposition 

The [polar decomposition] of a matrix A is A = OB, where O is orthogonal and where B is positive semidefinit. 

positive definite matrix 

A [positive definite matrix] is a symmetric matrix which satisfies v · Av > 0, for every nonzero vector v.

power 

The n’th [power] of a matrix A is defined as An = AA ( n − 1) = AA....A. The eigenvalues of An are λn i , where 

λi are the eigenvalues of A. 

QR decomposition 

The [QR decomposition] of a matrix A is obtained during the Gramm-Schmitt orthogonalization process. It is 

A = QR, where Q is an orthogonal matrix and where R is an upper triangular matrix. 

rank 

The [rank] of a linear matrix A is the set of leading 1’s in the matrix rref(A). 

orientation 

The [orientation] of n vectors v1, ..., vn in the n-dimensional Euclidean space is defined as the sign of det(A), 

where A is the matrix with vi in the columns. 

orthogonal 

A square matrix A is [orthogonal] if it preserves length: ||Av|| = ||v|| for all vectors v. 

perpendicular 

Two vectors v and w are called [perpendicular] if their dot product vanishes: v · w = 0. A synonym of 

perpendicular is orthogonal. 

projection matrix 

A [projection matrix] is a matrix P which satisfies P 2 = P = P T . It has eigenvalues 1 or 0. The image is a linear 

subspace S. The vectors in S are eigenvectors to the eigenvalues 1. The vectors in the orthogonal complement 

of S are eigenvectors to the eigenvalue 0. If A is the matrix which contains the basis of S as the columns, then 

P = A(A T A) −1 A T is the projection onto S. 

pseudoinverse 

The [pseudoinverse] of a (m × n) matrix A is the (n × m) matrix A + that maps the image of A to the image of 

A T . The kernel of A + is the kernel of A T and the rank of A + is equal to the rank of A. A + A is the projection 

on the image of A T and AA + is the projection on the image of A. Especially, if A is an invertible (n × n) 

matrix, then A + is the inverse of A. The pseudoinverse is also called Moore-Penrose inverse.

ank 

The [rank] of a matrix A is the dimension of the image of A. 

Rayleigh quotient 

The [Rayleigh quotient] of a symmetric matrix A is defined as the function q(v) = (v · Av)/(v · v). The maximal 

value of q(v) is the maximal eigenvalue of A and the minimal value of q(v) is the minimal eigenvalue of A. 

rotation 

A [rotation] is a linear transformation which preserves the angle between two vectors as well as their lengths. 

A rotation in three dimensional space is determined by the axis of rotation as well as the rotation angle. 

rotation matrix 

A [rotation matrix] is the matrix belonging to a rotation. A rotation matrix is an example of an orthogonal 

matrix. Example: In two dimensions, a rotation matrix has the form A = 

rotation-dilation matrix 

� cos(t) sin(t) 

− sin(t) cos(t) 

� 

p 

A [rotation-dilation matrix] is a 2x2 matrix of the form A = 

q 

−q 

p 

� 

. It has the eigenvalues p ± iq. The 

action of A represents the complex multiplication with the complex number p+iq in the complex plane. 

A [row] of a matrix is formed by horizontal lines A1j, j = 1, ..n of a m × n matrix A. 

row 

shear 

A [shear] is a linear transformation in the plane which has in a suitable basis the form T (x, y) = (x, y + ax). 

More generally, in n dimensions, one can define as shear along a m-dimensional plane. If a basis is chosen so 

that the plane has the from (x, 0) then a shear is T (x, y) = (x, y + ax). Shears have determinant 1 and preserve 

therefore volume. 

singular 

A square matrix A is called [singular] if it has no inverse. A matrix A is singular if and only if det(A) = 0. 

� 

.

singular value decomposition 

The [singular value decomposition] (SVD) of a matrix writes a matrix A in the form A = UDV T , where U, V 

are orthogonal and D is diagonal. The first r columns of U form an orthonormal basis of the image of A and the 

first r columns of V form an orthonormal basis of the image of A T . The last columns of U form an orthonormal 

basis of the kernel of A T and the last columns of V form a basis of the kernel of A. 

skew symmetric 

A matrix A is [skew symmetric] if it is minus its transpose that is if AT = −A. The eigenvalues of a skewsymmetric 

matrix are purely imaginary. The eigenvectors are orthogonal. If A is skew symmetric, then B = 

exp(At) is an orthogonal matrix, because BT � � 

0 1 

B = exp(−At)exp(At) = 1. For example A = 

gives 

−1 0 

the rotation matrix exp(At). 

span 

The [span] of a set of vectors v1, ...vn is the set of all linear combinations of v1, ..., vn. 

spectral theorem 

The [spectral theorem] tells that a real symmetric matrix A can be diagonalized A = UDU T , where U is an 

orthonormal matrix containing an orthonormal eigenbasis in the columns and where D is a diagonal matrix 

D = diag(x1, ..., xn), where xi are the eigenvalues of A. 

square matrix 

A [square matrix] is a matrix which has the same number of rows than columns. 

asymptotically stable 

A linear dynamical system is called [asymptotically stable] if A n x → 0 for all initial values x, where A n is the 

n’th power of the matrix A. This is equivalent to the fact that all eigenvalues λ of A satisfy |λ| < 1. 

Stability triangle 

[Stability triangle]. A discrete dynamical system in the plane is asymptotically stable if and only if the trace 

and determinant are in the stability triangle |tr(A)| − 1 < det(A) < 1. A rotation-dilation A is asymptotically 

stable if and only if det(A) < 1.

educed row echelon form 

The [reduced row echelon form] rref(A) of a m x n matrix A is the end product of Gauss-Jordan elimination. 

The matrix rref(A) has the following properties: 

• if a row has nonzero entries, then the first nonzero entry is 1, called leading 1. 

• if a columns contains a leading 1, then all other entries in that column are 0. 

• if a row contains a leading 1, then every row above contains a leading 1 further left. 

The algorithm to produce rref(A) from A is obtained by putting the cursor to the upper left corner and repeating 

the following steps until nothing changes anymore 

1. if the cursor entry is zero swap the cursor row with the first row below that has a nonzero entry in that 

column 

2. divide the cursor row by the cursor entry to make the cursor entry = 1 

3. eliminate all other entries in cursor column by subtracting suitable multiples of the cursor row from the 

other row 

4. move the cursor down one row and and to the right one column. If the cursor entry is zero and all entries 

below are zero, move the cursor to the next column. 

5. repeat 4 if as long as necessary and move then to 1 

reflection 

A [reflection] is a linear transformation T different from the identity transformation which satisfies T 2 = 1. The 

eigenvalues of T are −1 or 1. In an eigenbasis, the reflection has the form T (x, y) = (x, −y). The determinant 

of a reflection is 1 if and only if the dimension � of the eigenspace � to -1 is even. For example, a reflection at a 

cos(2x) sin(2x) 

line in the plane has the matrix A = 

which has determinant −1. A reflection at the 

sin(2x) −cos(2x) 

origin in the plane is −I with determinant 1. 

root 

A [root] of a polynomial p(x) is a complex value z such that p(z) = 0. According to the fundamental theorem 

of algebra, a polynomial of degree n has exactly n roots. 

symmetric 

A matrix A is [symmetric] if it is equal to its transpose. The spectral theorem for symmetric matrices tells that 

they have real eigenvalues and symmetric matrices can always be diagonalized with an orthogonal matrix S. 

span 

The [span] of a finite set of vectors v1, ..., vn is the set of all possible linear combinations c1v1 + c2v2 + ... + cnvn 

where ci are real numbers. For example, if v1 = (1, 0, 0) and v2 = (0, 1, 0), then the span of v1, v2 in three 

dimensional space is the xy-plane. The span is a linear space.

spectral theorem 

The [spectral theorem] for a symmetric matrix A assures that A can be diagonalized: there exists an orthogonal 

matrix S such that A ( − 1)AS is diagonal and contains the eigenvalues of A in the diagonal. 

standard basis 

The [standard basis] of the n-dimensional Euclidean space consists of the columns of the identity matrix I. 

symmetric 

A matrix A is called [symmetric] if A T = A. A symmetric matrix has to be a square matrix. Real symmetric 

matrices can be diagonalized. 

Toepitz matrix 

A [Toepitz matrix] is a square matrix A, where the entries are constant along the diagonal. In other ways Aij 

⎛ ⎞ 

c d e 

depends only on j − i. Example: a 3 × 3 Toeplitz matrix is of the form A = ⎝ b c d ⎠. 

a b c 

trace 

The [trace] of a matrix A is the sum of the diagonal entries of A. The trace is independent of the basis and is 

equal to the sum of the eigenvalues of A. 

transpose 

The [transpose] A T of a matrix A is the matrix with entries Aij if A has the entries Aji. The rank of A T is 

equal to the rank of A. For square matrices, the eigenvalues of A T and A agree because A and A T have the 

same eigenvalues. Transposition satisfies (A T ) T = A, (AB) T = B T A T and (A ( − 1)) T = (A T ) ( − 1). 

triangle inequality 

The [triangle inequality] tells that in a linear space, ||v + w|| ≤ ||v|| + ||w||. One has equality if and only if the 

vectors v and w are orthogonal.

eigenvalues of a two times two matrix 

� a b 

c d 

� 

The [eigenvalues of a two times two matrix] A = 

are λ1 = tr(A)/2 + ((tr(A)/2) 2 − det(A)) 1/2 and 

λ2 = tr(A)/2 − ((tr(A)/2) 2 − det(A)) 1/2 � � 

� � 

λi − d 

. The eigenvectors are vi = 

if c �= 0 or vi = 

if 

c 

b �= 0. (If b = 0, c = 0, then the standard vectors are eigenvector.) 

unit vector 

b 

λi − a 

A [unit vector] is a vector of length 1. A given nonzero vector can be made a unit vector by scaling: v/||v|| is 

a unit vector. 

Vandermonde Matrix 

A [Vandermonde Matrix] is a square matrix with entries Aij = x j−1 

i , where x1, ..., xn are some real numbers. 

The determinant of a Vandermonde Matrix is � 

j>i (xj − xi). Example: x1 = 2, x2 = 3, x3 = −1 defines the 

⎛ 

⎞ 

1 2 4 

Vandermonde Matrix A = ⎝ 1 3 9 ⎠ which has the determinant det(A) = (3 − 2)(−1 − 2)(−1 − 3) = 12. 

1 −1 1 

vector 

A [vector] is a matrix with one column. The entries of the vector are called coefficients. 

vector space 

A [vector space] X is a set equipped with addition and scalar multiplication. A vector space is also called a 

linear space. The addition operation is a group: 

The scalar multiplication satisfies: 

f + g = g + f Commutativity 

(f + g) + h = f + (g + h) Associativity 

f + 0 = 0 Existence of a neutral element 

f + x = 0 Existence of a unique inverse 

r(f + g) = rf + rg Distributivity 

(r + s)f = rf + sf Distributivity 

r(sf) = (rs)f Associativity 

1f = f One element

wave equation 

The [wave equation] is the linear partial differential equation utt = c2uxx where c is a constant. The wave 

equation on a finite interval 0 < x < 1 with boundary conditions u(0, t) = 0, u(π, t) = 0 and initial conditions 

u(x, 0) = f(x), ut(x, 0) = g(x) can be solved with the Fourier series: u(x, t) = � 

n > 0an sin(nx) cos(nct) + 

bn sin(nx) sin(nct) where an = (2/π) � π 

0 f(x) sin(nx) dx, and bn = (2/π)intπ 0 g(x) sin(nx) dx/(cn) are Fourier 

coefficients. 

zero matrix 



Index 

Euclidean space , 17 

adjacency matrix, 1 

affine transformation, 1 

Algebra, 1 

algebraic multiplicity, 1 

angle, 1 

argument, 1 

associative law, 2 

asymptotically stable, 21 

augmented matrix, 2 

basis, 2 

basis theorem, 2 

block matrix, 2 

Cauchy-Schwarz inequality, 2 

Cayley-Hamilton theorem, 2 

change of basis, 3 

characteristic matrix, 3 

characteristic polynomial, 3 

Cholesky factoriztion, 3 

circulant matrix, 3 

classical adjoint, 3 

codomain, 4 

coefficient, 4 

cofactor, 4 

column, 4 

column picture, 4 

column space, 4 

commutative law, 5 

commuting matrices, 5 

complex conjugate, 5 

Complex numbers, 5 

consistent, 5 

continuous dynamical system, 5 

covariance matrix, 6 

Cramer’s rule, 6 

de Moivre formula, 6 

determinant, 6 

differential equation, 6 

dilation, 6 

dimension, 6 

distributive law, 7 

dot product, 7 

echelon matrix, 7 

eigenbasis, 7 

eigenvalue, 7 

eigenvalues of a two times two matrix, 24 

eigenvector, 7 

Elimination, 8 

ellipsoid, 8 

entry, 8 

expansion factor, 8 

exponential, 8 

factorization, 8 

Fourier coefficients, 8 

fundamental theorem of algebra, 9 

26 

Gauss Jordan elimination, 9 

geometric multiplicity, 9 

Gibbs phenomenon, 9 

Gramm-Schmidt process, 9 

graph, 10 

Hankel matrix, 10 

heat equation, 10 

Hermitian matrix, 10 

Hessenberg matrix, 10 

Hilbert matrix, 11 

identity matrix, 11 

image, 12 

incidence matrix, 11 

inconsistent, 11 

indefinite matrix, 11 

Independent vectors, 11 

index, 12 

inverse, 12 

invertible, 12 

Jordan normal form, 12 

kernel, 13 

Laplace equation, 13 

Laplace expansion, 13 

leading one, 13 

leading variable, 13 

least-squares solution, 14 

length, 14 

Leontief, 13 

linear combination, 14 

linear dynamical system, 14 

linear space, 14 

linear subspace, 15 

linear system of equations, 15 

Linearity, 14 

Linearly dependent vectors, 14 

logarithm, 15 

Markov matrix, 15 

matrix, 15 

Matrix multiplication, 15 

minor, 16 

nilpotent matrix, 16 

non-leading coefficient, 16 

normal equation, 16 

normal matrix, 16 

null space, 16 

ordinary differential equation, 16 

orientation, 19 

orthogonal, 17, 19 

orthogonal basis, 17 

orthogonal projection, 17 

orthonormal basis, 17 

orthonormal complement, 17 

parallel, 17

parallelepiped, 17 

partial differential equation, 18 

permutation matrix, 18 

perpendicular, 19 

pivot, 18 

pivot column, 18 

polar decomposition, 18 

positive definite matrix, 18 

power, 19 

projection matrix, 19 

pseudoinverse, 19 

QR decomposition, 19 

rank, 19, 20 

Rayleigh quotient, 20 

reduced row echelon form, 22 

reflection, 22 

root, 22 

rotation, 20 

rotation matrix, 20 

rotation-dilation matrix, 20 

row, 20 

shear, 20 

singular, 20 

singular value decomposition, 21 

skew symmetric, 21 

span, 21, 22 

spectral theorem, 21, 23 

square matrix, 21 

Stability triangle, 21 

standard basis, 23 

symmetric, 22, 23 

Toepitz matrix, 23 

trace, 23 

transpose, 23 

triangle inequality, 23 

unit vector, 24 

Vandermonde Matrix, 24 

vector, 24 

vector space, 24 

wave equation, 25 

zero matrix, 25

MATHEMATICIANS 

[MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates 

grabbed from mactutor in 2000. 

[Abbe] Abbe Ernst (1840-1909) 

Abbe 

Abel 

[Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and analysis, 

in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at 

the age of 19. 

[AbrahamMax] Abraham Max (1875-1922) 

[Ackermann] Ackermann Wilhelm (1896-1962) 

[AdamsFrank] Adams J Frank (1930-1989) 

[Adams] Adams John Couch (1819-1892) 

[Adelard] Adelard of Bath (1075-1160) 

[Adler] Adler August (1863-1923) 

AbrahamMax 

Ackermann 

AdamsFrank 

Adams 

Adelard 

Adler

[Adrain] Adrain Robert (1775-1843) 

[Aepinus] Aepinus Franz (1724-1802) 

[Agnesi] Agnesi Maria (1718-1799) 

Adrain 

Aepinus 

Agnesi 

Ahlfors 

[Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at 

Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. 

[Ahmes] Ahmes (1680BC-1620BC) 

[Aida] Aida Yasuaki (1747-1817) 

[Aiken] Aiken Howard (1900-1973) 

[Airy] Airy George (1801-1892) 

[Aitken] Aitken Alec (1895-1967) 

Ahmes 

Aida 

Aiken 

Airy 

Aitken

[Ajima] Ajima Naonobu (1732-1798) 

[Akhiezer] Akhiezer Naum Ilich (1901-1980) 

[Albanese] Albanese Giacomo (1890-1948) 

[Albert] Albert of Saxony (1316-1390) 

[AlbertAbraham] Albert A Adrian (1905-1972) 

[Alberti] Alberti Leone (1404-1472) 

[Albertus] Albertus Magnus Saint (1200-1280) 

[Alcuin] Alcuin of York (735-804) 

[Alexander] Alexander James (1888-1971) 

Ajima 

Akhiezer 

Albanese 

Albert 

AlbertAbraham 

Alberti 

Albertus 

Alcuin 

Alexander

[AlexanderArchie] Alexander Archie (1888-1958) 

[Aleksandrov] Alexandroff Pave (1896-1982) 

[AleksandrovAleksandr] Alexandroff Alexander 

[Ampere] Ampère André-Marie (1775-1836) 

[Amsler] Amsler Jacob (1823-1912) 

AlexanderArchie 

Aleksandrov 

AleksandrovAleksandr 

Ampere 

Amsler 

Anaxagoras 

[Anaxagoras] Anaxagoras of Clazomenae (499BC-428BC) 

[Anderson] Anderson Oskar (1887-1960) 

[Andreev] Andreev Konstantin (1848-1921) 

[Angeli] Angeli Stephano degli (1623-1697) 

Anderson 

Andreev 

Angeli

[Anstice] Anstice Robert (1813-1853) 

[Anthemius] Anthemius of Tralles (474-534) 

[Antiphon] Antiphon the Sophist (480BC-411BC) 

[Apastamba] Apastamba (600BC-540BC) 

[Apollonius] Apollonius of Perga (262BC-190BC) 

[Appell] Appell Paul (1855-1930) 

[Arago] Arago Francois (1786-1853) 

[Arbogast] Arbogast Louis (1759-1803) 

[Arbuthnot] Arbuthnot John (1667-1735) 

Anstice 

Anthemius 

Antiphon 

Apastamba 

Apollonius 

Appell 

Arago 

Arbogast 

Arbuthnot

[Archimedes] Archimedes of Syracuse (287BC-212BC) 

[Archytas] Archytas of Tarentum (428BC-350BC) 

[Arf] Arf Cahit (1910-1997) 

[Argand] Argand Jean (1768-1822) 

[Aristaeus] Aristaeus the Elder (360BC-300BC) 

[Aristarchus] Aristarchus of Samos (310BC-230BC) 

[Aristotle] Aristotle (384BC-322BC) 

[Arnauld] Arnauld Antoine (1612-1694) 

[Aronhold] Aronhold Siegfried (1819-1884) 

Archimedes 

Archytas 

Arf 

Argand 

Aristaeus 

Aristarchus 

Aristotle 

Arnauld 

Aronhold

[Artin] Artin Emil (1898-1962) 

[AryabhataII] Aryabhata II 

[AryabhataI] Aryabhata I (476-550) 

[Atiyah] Atiyah Michael 

[Atwood] Atwood George (1745-1807) 

[Auslander] Auslander Maurice (1926-1994) 

[Autolycus] Autolycus of Pitane (360BC-290BC) 

Artin 

AryabhataII 

AryabhataI 

Atiyah 

Atwood 

Auslander 

Autolycus 

Bezout 

[Bezout] Bézout Etienne (1730-1783) French geometer and analyst. 

[Bocher] Bocher Maxime (1867-1918) 

Bocher

[Burgi] Bürgi Joost (1552-1632) 

[Babbage] Babbage Charles (1791-1871) 

[Bachet] Bachet Claude (1581-1638) 

[Bachmann] Bachmann Paul (1837-1920) 

[Backus] Backus John 

[Bacon] Bacon Roger (1219-1292) 

[Baer] Baer Reinhold (1902-1979) 

[Baire] Baire René-Louis (1874-1932) 

[BakerAlan] Baker Alan 

Burgi 

Babbage 

Bachet 

Bachmann 

Backus 

Bacon 

Baer 

Baire 

BakerAlan

[Baker] Baker Henry (1866-1956) 

[Ball] Ball Walter W Rouse (1850-1925) 

[Balmer] Balmer Johann (1825-1898) 

Baker 

Ball 

Balmer 

Banach 

[Banach], Stefan, (1892-1945) Polish mathematician who founded functional analysis. 

[Banneker] Banneker Benjamin (1731-1806) 

[BanuMusaMuhammad] Banu Musa Jafar (810-873) 

[BanuMusa] Banu Musa brothers 

[BanuMusaal-Hasan] Banu Musa al-Hasan (810-873) 

[BanuMusaAhmad] Banu Musa Ahmad (810-873) 

Banneker 

BanuMusaMuhammad 

BanuMusa 

BanuMusaal-Hasan 

BanuMusaAhmad

[Barbier] Barbier Joseph Emile (1839-1889) 

[Bari] Bari Nina (1901-1961) 

[Barlow] Barlow Peter (1776-1862) 

[Barnes] Barnes Ernest (1874-1953) 

[Barocius] Barozzi Francesco (1537-1604) 

[Barrow] Barrow Isaac (1630-1677) 

[Bartholin] Bartholin Erasmus (1625-1698) 

[Batchelor] Batchelor George (1920-2000) 

[Bateman] Bateman Harry (1882-1946) 

Barbier 

Bari 

Barlow 

Barnes 

Barocius 

Barrow 

Bartholin 

Batchelor 

Bateman

[Battaglini] Battaglini Guiseppe (1826-1894) 

[Al-Battani] Battani Abu al- (850-929) 

[Baudhayana] Baudhayana (800BC-740BC) 

Battaglini 

Al-Battani 

Baudhayana 

Bayes 

[Bayes] Thomas, (1702-1761). English probability theorist and theologian. 

[Beaugrand] Beaugrand Jean (1595-1640) 

[Bell] Bell Eric Temple (1883-1960) 

[Bellavitis] Bellavitis Giusto (1803-1880) 

[Beltrami] Beltrami Eugenio (1835-1899) 

[Bendixson] Bendixson Ivar Otto (1861-1935) 

Beaugrand 

Bell 

Bellavitis 

Beltrami 

Bendixson

[Benedetti] Benedetti Giovanni (1530-1590) 

[Bergman] Bergman Stefan (1895-1977) 

[Berkeley] Berkeley George (1685-1753) 

[Bernays] Bernays Paul Isaac (1888-1977) 

[BernoulliDaniel] Bernoulli Daniel (1700-1782) 

[BernoulliJohann] Bernoulli Johann (1667-1748) 

[BernoulliNicolaus] Bernoulli Nicolaus 

Benedetti 

Bergman 

Berkeley 

Bernays 

BernoulliDaniel 

BernoulliJohann 

BernoulliNicolaus 

BernoulliJacob 

[BernoulliJacob] Bernoulli Jakob (1654-1705) Swiss analyst, probability theorist and physicist. 

Bernstein 

[Bernstein] Sergei Natanovich (1880-1968). Russian analyst.

[BernsteinFelix] Bernstein Felix (1878-1956) 

[Bers] Bers Lipa (1914-1993) 

[Bertini] Bertini Eugenio (1846-1933) 

[Bertrand] Bertrand Joseph (1822-1900) 

[Berwald] Berwald Lugwig (1883-1942) 

[Berwick] Berwick William (1888-1944) 

[Besicovitch] Besicovitch Abram (1891-1970) 

BernsteinFelix 

Bers 

Bertini 

Bertrand 

Berwald 

Berwick 

Besicovitch 

Bessel 

[Bessel] Friedrich Wilhelm, (1784-1846) calculated orbit of Halley’s orbit as 20 year old. Made accurate measurements 

of stellar positions. Professor of Astronomy at Koenigsberg. 

[Betti] Betti Enrico (1823-1892) 

Betti

[Beurling] Beurling Arne (1905-1986) 

[BhaskaraI] Bhaskara I (600-680) 

[BhaskaraII] Bhaskaracharya (1114-1185) 

[Bianchi] Bianchi Luigi (1856-1928) 

[Bieberbach] Bieberbach Ludwig (1886-1982) 

[Bienayme] Bienaymé Iréneé-Jules (1796-1878) 

[Binet] Binet Jacques (1786-1856) 

[Bing] Bing R (1940-1986) 

[Biot] Biot Jean-Baptiste (1774-1862) 

Beurling 

BhaskaraI 

BhaskaraII 

Bianchi 

Bieberbach 

Bienayme 

Binet 

Bing 

Biot

[Birkhoff] Birkhoff George (1884-1944) 

[BirkhoffGarrett] Birkhoff Garrett (1911-1996) 

[Al-Biruni] Biruni Abu al 

[BjerknesVilhelm] Bjerknes Vilhelm (1862-1951) 

[BjerknesCarl] Bjerknes Carl (1825-1903) 

[Black] Black Max (1909-1988) 

[Blaschke] Blaschke Wilhelm (1885-1962) 

[Blichfeldt] Blichfeldt Hans (1873-1945) 

[Bliss] Bliss Gilbert (1876-1951) 

Birkhoff 

BirkhoffGarrett 

Al-Biruni 

BjerknesVilhelm 

BjerknesCarl 

Black 

Blaschke 

Blichfeldt 

Bliss

[Bloch] Bloch André (1893-1948) 

[Bobillier] Bobillier Etienne (1798-1840) 

[Bochner] Bochner Salomon (1899-1982) 

[Boethius] Boethius Anicus (475-524) 

[Boggio] Boggio Tommaso (1877-1963) 

[Bohl] Bohl Piers (1865-1921) 

[BohrHarald] Bohr Harald (1887-1951) 

[BohrNiels] Bohr Niels (1885-1962) 

[Boltzmann] Boltzmann Ludwig (1844-1906) 

Bloch 

Bobillier 

Bochner 

Boethius 

Boggio 

Bohl 

BohrHarald 

BohrNiels 

Boltzmann

[BolyaiFarkas] Bolyai Farkas (1775-1856) 

[Bolyai] Bolyai Jänos (1802-1860) 

BolyaiFarkas 

Bolyai 

Bolza 

[Bolza], Oscar (1857-1943), German-born American analyst. 

[Bolzano] Bolzano Bernhard (1781-1848) 

[Bombelli] Bombelli Rafael (1526-1573) 

[Bombieri] Bombieri Enrico 

[Bonferroni] Bonferroni Carlo (1892-1960) 

[Bonnet] Bonnet Pierre (1819-1892) 

Bolzano 

Bombelli 

Bombieri 

Bonferroni 

Bonnet 

Boole 

[Boole], George (1815-1864), English Logician, who made also contributions to analysis and probability theory.

[Boone] Boone Bill (1920-1983) 

[Borchardt] Borchardt Carl (1817-1880) 

[Borda] Borda Jean (1733-1799) 

Boone 

Borchardt 

Borda 

Borel 

[Borel], Felix Edouard Justin Emile, (1871-1956) French measure theorist and probability theorist. 

[Borgi] Borgi Piero (1424-1484) 

[Born] Born Max (1882-1970) 

[Borsuk] Borsuk Karol (1905-1982) 

[Bortkiewicz] Bortkiewicz Ladislaus (1868-1931) 

[Bortolotti] Bortolotti Ettore (1866-1947) 

Borgi 

Born 

Borsuk 

Bortkiewicz 

Bortolotti

[Bosanquet] Bosanquet Stephen (1903-1984) 

[Boscovich] Boscovich Ruggero (1711-1787) 

[Bose] Bose Satyendranath (1894-1974) 

[Bossut] Bossut Charles (1730-1814) 

[Bouguer] Bouguer Pierre (1698-1758) 

[Boulliau] Boulliau Ismael (1605-1694) 

[Bouquet] Bouquet Jean Claude (1819-1885) 

[Bour] Bour Edmond (1832-1866) 

Bosanquet 

Boscovich 

Bose 

Bossut 

Bouguer 

Boulliau 

Bouquet 

Bour 

Bourbaki 

[Bourbaki], Nicolas (1939- ) Collective pseudonym of a group of mostly French mathematicians.

[Bourgain] Bourgain Jean 

[Boutroux] Boutroux Pierre Léon (1880-1922) 

[Bowditch] Bowditch Nathaniel (1773-1838) 

[Bowen] Bowen Rufus (1947-1978) 

[Boyle] Boyle Robert (1627-1691) 

[Boys] Boys Charles (1855-1944) 

[Bradwardine] Bradwardine Thomas (1290-1349) 

[Brahe] Brahe Tycho (1546-1601) 

[Brahmadeva] Brahmadeva (1060-1130) 

Bourgain 

Boutroux 

Bowditch 

Bowen 

Boyle 

Boys 

Bradwardine 

Brahe 

Brahmadeva

[Brahmagupta] Brahmagupta (598-670) 

[Braikenridge] Braikenridge William (1700-1762) 

[Bramer] Bramer Benjamin (1588-1652) 

[Brashman] Brashman Nikolai (1796-1866) 

[BrauerAlfred] Brauer Alfred (1894-1985) 

[Brauer] Brauer Richard (1901-1977) 

[Brianchon] Brianchon Charles (1783-1864) 

Brahmagupta 

Braikenridge 

Bramer 

Brashman 

BrauerAlfred 

Brauer 

Brianchon 

Briggs 

[Briggs] Briggs Henry (1561-1630) English mathematician producing tables of common logarithms up to 15 

digits. 

[Brillouin] Brillouin Marcel (1854-1948) 

Brillouin

[Bring] Bring Erland (1736-1798) 

[Brioschi] Brioschi Francesco (1824-1897) 

[Briot] Briot Charlese (1817-1882) 

[Brisson] Brisson Barnabé (1777-1828) 

[Britton] Britton John (1927-1994) 

[Brocard] Brocard Henri (1845-1922) 

[Brodetsky] Brodetsky Selig 

[Bromwich] Bromwich Thomas (1875-1929) 

[Bronowski] Bronowski Jacob (1908-1974) 

Bring 

Brioschi 

Briot 

Brisson 

Britton 

Brocard 

Brodetsky 

Bromwich 

Bronowski

[Brouncker] Brouncker William (1620-1684) 

Brouncker 

Brouwer 

[Brouwer] Brouwer Luitzen Egbertus Jan (1881-1966) Dutch matematician and philosopher. 

[Brown] Brown Ernest (1866-1938) 

[Browne] Browne Marjorie (1914-1979) 

[Bruno] Bruno Giuseppe (1828-1893) 

[Bruns] Bruns Heinrich (1848-1919) 

[Bryson] Bryson of Heraclea (450BC-390BC) 

[Buffon] Buffon Georges Comte de (1707-1788) 

[Bugaev] Bugaev Nicolay (1837-1903) 

Brown 

Browne 

Bruno 

Bruns 

Bryson 

Buffon 

Bugaev

[Bukreev] Bukreev Boris (1859-1962) 

[Bunyakovsky] Bunyakovsky Viktor (1804-1889) 

[Burchnall] Burchnall Joseph (1892-1975) 

[Burkhardt] Burkhardt Heinrich (1861-1914) 

[Burkill] Burkill John (1900-1993) 

[Burnside] Burnside William (1852-1927) 

[Caccioppoli] Caccioppoli Renato (1904-1959) 

[Cajori] Cajori Florian (1859-1930) 

[Calderon] Calderón Alberto (1920-1998) 

Bukreev 

Bunyakovsky 

Burchnall 

Burkhardt 

Burkill 

Burnside 

Caccioppoli 

Cajori 

Calderon

[Callippus] Callippus (370BC-310BC) 

[Campanus] Campanus of Novara (1220-1296) 

[Campbell] Campbell John (1862-1924) 

[Camus] Camus Charles (1699-1768) 

[Cannell] Cannell Doris (1913-2000) 

[Cantelli] Cantelli Francesco (1875-1966) 

[CantorMoritz] Cantor Moritz (1829-1920) 

Callippus 

Campanus 

Campbell 

Camus 

Cannell 

Cantelli 

CantorMoritz 

Cantor 

[Cantor] Cantor Georg (1845-1918) [Cantor] Georg, (1845-1918) German mathematician. Precise definition of 

infinite set. 

[Caramuel] Caramuel Juan (1606-1682) 

Caramuel

[Caratheodory] Carathéodory Constantin (1873-1950) 

Caratheodory 

Cardan 

[Cardan] Cardano Girolamo (1501-1576) Italian mathematicaian, physician and astrologer. First publication 

for the solution of the general cubic equation (solution found by Tartaglia). 

[Carlitz] Carlitz Leonard (1907-1999) 

[Carlyle] Carlyle Thomas (1795-1881) 

Carlitz 

Carlyle 

Carnot 

[Carnot] Carnot Lazare (1753-1823) French mathematician and politican best known for his work on the foundations 

of calculus and modern geometry. 

CarnotSadi 

[CarnotSadi] Carnot Sadi (1796-1832) French mathematical physisist working on the foundations of thermodynamics. 

Carnot’s work led directly to the discovery of the second law of thermodynamics. 

[Carslaw] Carslaw Horatio (1870-1954) 

[Cartan] Cartan Elie (1869-1951) 

[CartanHenri] Cartan Henri 

Carslaw 

Cartan 

CartanHenri

[Cartwright] Cartwright Dame Mary (1900-1998) 

[Casorati] Casorati Felice (1835-1890) 

[Cassels] Cassels John 

[Cassini] Cassini Giovanni (1625-1712) 

[Castel] Castel Louis (1688-1757) 

[Castelnuovo] Castelnuovo Guido (1865-1952) 

[Castigliano] Castigliano Alberto (1847-1884) 

[Castillon] Castillon Johann (1704-1791) 

[Catalan] Catalan Eugène (1814-1894) 

Cartwright 

Casorati 

Cassels 

Cassini 

Castel 

Castelnuovo 

Castigliano 

Castillon 

Catalan

[Cataldi] Cataldi Pietro (1548-1626) 

Cataldi 

Cauchy 

[Cauchy] Cauchy Augustin-Louis (1789-1857) French mathematician who introduced modern notions of continuity 

limit, convergence and differentiability, proved Cauchy’s theorem in group theory, contributed to the 

calculus of variations, probability theory and the study of differential equations. 

Cavalieri 

[Cavalieri] Cavalieri Bonaventura (1598-1647) Italian mathematician. Introduced method of indivisibles, a 

forerunner of integral calculus to determine the area enclosed by certain curves. 

Cayley 

[Cayley] Cayley Arthur (1821-1895) English mathematician working in the theory of matrices, abstract groups 

and algebraic geometry. 

[Cech] Cech Eduard (1893-1960) 

[Cesaro] Cesàro Ernesto (1859-1906) 

[CevaGiovanni] Ceva Giovanni (1647-1734) 

[CevaTommaso] Ceva Tommaso (1648-1737) 

[Chatelet] Chatelet Gabrielle du (1706-1749) 

Cech 

Cesaro 

CevaGiovanni 

CevaTommaso 

Chatelet

[Chandrasekhar] Chandrasekhar Subrah. (1910-1995) 

[Chang] Chang Sun-Yung Alice 

[Chaplygin] Chaplygin Sergi (1869-1942) 

[Chapman] Chapman Sydney (1888-1970) 

[Chasles] Chasles Michel (1793-1880) 

[Chebotaryov] Chebotaryov Nikolai (1894-1947) 

[Chebyshev] Chebyshev Pafnuty (1821-1894) 

[Chern] Chern Shiing-shen 

[Chernikov] Chernikov Sergei (1912-1987) 

Chandrasekhar 

Chang 

Chaplygin 

Chapman 

Chasles 

Chebotaryov 

Chebyshev 

Chern 

Chernikov

[Chevalley] Chevalley Claude (1909-1984) 

[Ch’in] Chiu-Shao Ch’in (1202-1261) 

[Chowla] Chowla Sarvadaman (1907-1995) 

[Christoffel] Christoffel Elwin (1829-1900) 

[Chrysippus] Chrysippus (280BC-206BC) 

[Chrystal] Chrystal George (1851-1911) 

[Chuquet] Chuquet Nicolas (1445-1500) 

Chevalley 

Ch’in 

Chowla 

Christoffel 

Chrysippus 

Chrystal 

Chuquet 

Church 

[Church] Church Alonzo (1903-1995) American mathematical logician. 

Clairaut 

[Clairaut] Clairaut Alexis (1713-1765) French mathematician and physicist who worked on the problem of 

geodesic flows, celestial mechanics and cubic curves.

[Clapeyron] Clapeyron Emile (1799-1864) 

[Clarke] Clarke Samuel (1675-1729) 

[Clausen] Clausen Thomas (1801-1885) 

[Clausius] Clausius Rudolf (1822-1888) 

[Clavius] Clavius Christopher (1537-1612) 

[Clebsch] Clebsch Alfred (1833-1872) 

[Cleomedes] Cleomedes Cleomedes (10AD-70) 

[Clifford] Clifford William (1845-1879) 

[Coates] Coates John 

Clapeyron 

Clarke 

Clausen 

Clausius 

Clavius 

Clebsch 

Cleomedes 

Clifford 

Coates

[Coble] Coble Arthur (1878-1966) 

[Cochran] Cochran William (1909-1980) 

[Cocker] Cocker Edward (1631-1675) 

[Codazzi] Codazzi Delfino (1824-1873) 

Coble 

Cochran 

Cocker 

Codazzi 

Cohen 

[Cohen] Paul Joseph (1934-) American mathematician who resolved the status of the continuum hypothesis. 

[Cole] Cole Frank (1861-1926) 

[Collingwood] Collingwood Edward (1900-1970) 

[Collins] Collins John (1625-1683) 

[Condorcet] Condorcet Marie Jean (1743-1794) 

Cole 

Collingwood 

Collins 

Condorcet

[Connes] Connes Alain 

[Conon] Conon of Samos (280BC-220BC) 

[ConwayArthur] Conway Arthur (1875-1950) 

[Conway] Conway John 

[Coolidge] Coolidge Julian (1873-1954) 

[Cooper] Cooper Lionel (1915-1977) 

[Copernicus] Copernicus Nicolaus (1473-1543) 

[Copson] Copson Edward (1901-1980) 

[Cosserat] Cosserat Eugène (1866-1931) 

Connes 

Conon 

ConwayArthur 

Conway 

Coolidge 

Cooper 

Copernicus 

Copson 

Cosserat

[Cotes] Cotes Roger (1682-1716) 

[Courant] Courant Richard (1888-1972) 

[Cournot] Cournot Antoine (1801-1877) 

[Couturat] Couturat Louis (1868-1914) 

[Cox] Cox Gertrude (1900-1978) 

[Coxeter] Coxeter Donald 

[Craig] Craig John (1663-1731) 

[CramerHarald] Cramér Harald (1893-1985) 

[Cramer] Cramer Gabriel (1704-1752) 

Cotes 

Courant 

Cournot 

Couturat 

Cox 

Coxeter 

Craig 

CramerHarald 

Cramer

[Crank] Crank John 

[Crelle] Crelle August (1780-1855) 

[Cremona] Cremona Luigi (1830-1903) 

[Crighton] Crighton David (1942-2000) 

[Cunha] Cunha Anastäcio da (1744-1787) 

[Cunningham] Cunningham Ebenezer (1881-1977) 

[Curry] Curry Haskell (1900-1982) 

[Cusa] Cusa Nicholas of (1401-1464) 

[Durer] Dürer Albrecht (1471-1528) 

Crank 

Crelle 

Cremona 

Crighton 

Cunha 

Cunningham 

Curry 

Cusa 

Durer

[Dandelin] Dandelin Germinal (1794-1847) 

[Danti] Danti Egnatio (1536-1586) 

[DantzigGeorge] Dantzig George 

[Darboux] Darboux Gaston (1842-1917) 

[Darwin] Darwin George (1845-1912) 

[Dase] Dase Zacharias (1824-1861) 

[Davenport] Davenport Harold (1907-1969) 

[Davidov] Davidov August (1823-1885) 

[Davies] Davies Evan Tom (1904-1973) 

Dandelin 

Danti 

DantzigGeorge 

Darboux 

Darwin 

Dase 

Davenport 

Davidov 

Davies

[Dechales] Dechales Claude (1621-1678) 

[Dedekind] Dedekind Richard (1831-1916) 

[Dee] Dee John (1527-1608) 

[Dehn] Dehn Max (1878-1952) 

[Delamain] Delamain Richard (1600-1644) 

[Delambre] Delambre Jean Baptiste (1749-1822) 

[Delaunay] Delaunay Charles (1816-1872) 

[Deligne] Deligne Pierre 

[Delone] Delone Boris (1890-1973) 

Dechales 

Dedekind 

Dee 

Dehn 

Delamain 

Delambre 

Delaunay 

Deligne 

Delone

[Delsarte] Delsarte Jean (1903-1968) 

[Democritus] Democritus of Abdera (460BC-370BC) 

[Denjoy] Denjoy Arnaud (1884-1974) 

[Deparcieux] Deparcieux Antoine (1703-1768) 

[Desargues] Desargues Girard (1591-1661) 

[Descartes] Descartes René (1596-1650) 

[Dickson] Dickson Leonard (1874-1954) 

[Dickstein] Dickstein Samuel (1851-1939) 

[Dieudonne] Dieudonné Jean (1906-1992) 

Delsarte 

Democritus 

Denjoy 

Deparcieux 

Desargues 

Descartes 

Dickson 

Dickstein 

Dieudonne

[Digges] Digges Thomas (1546-1595) 

[Dilworth] Dilworth Robert 

[Dinghas] Dinghas Alexander (1908-1974) 

[Dini] Dini Ulisse (1845-1918) 

[Dinostratus] Dinostratus (390BC-320BC) 

[Diocles] Diocles (240BC-180BC) 

[Dionis] Dionis du Séjour A (1734-1794) 

[Dionysodorus] Dionysodorus (250BC-190BC) 

[Diophantus] Diophantus of Alexandria (200-284) 

Digges 

Dilworth 

Dinghas 

Dini 

Dinostratus 

Diocles 

Dionis 

Dionysodorus 

Diophantus

[Dirac] Dirac Paul (1902-1984) 

[Dirichlet] Dirichlet Lejeune (1805-1859) 

[DixonArthur] Dixon Arthur Lee (1867-1955) 

[Dixon] Dixon Alfred (1865-1936) 

[Dodgson] Dodgson Charles (1832-1898) 

[Doeblin] Doeblin Wolfang (1915-1940) 

[Domninus] Domninus of Larissa (420-480) 

[Donaldson] Donaldson Simon 

[Doob] Doob Joseph 

Dirac 

Dirichlet 

DixonArthur 

Dixon 

Dodgson 

Doeblin 

Domninus 

Donaldson 

Doob

[Doppelmayr] Doppelmayr Johann (1671-1750) 

[Doppler] Doppler Christian (1803-1853) 

[Douglas] Douglas Jesse (1897-1965) 

[Dowker] Dowker Clifford (1912-1982) 

[Drach] Drach Jules (1871-1941) 

[Drinfeld] Drinfeld Vladimir 

[Dubreil] Dubreil Paul (1904-1994) 

[Dudeney] Dudeney Henry (1857-1931) 

[Duhamel] Duhamel Jean-Marie (1797-1872) 

Doppelmayr 

Doppler 

Douglas 

Dowker 

Drach 

Drinfeld 

Dubreil 

Dudeney 

Duhamel

[Duhem] Duhem Pierre (1861-1916) 

[Dupin] Dupin Pierre (1784-1873) 

[Dupre] Dupré Athanase (1808-1869) 

[Dynkin] Dynkin Evgenii 

[EckertJohn] Eckert J Presper (1919-1995) 

[EckertWallace] Eckert Wallace J (1902-1971) 

[Eckmann] Eckmann Beno 

[Eddington] Eddington Arthur (1882-1944) 

[Edge] Edge William (1904-1997) 

Duhem 

Dupin 

Dupre 

Dynkin 

EckertJohn 

EckertWallace 

Eckmann 

Eddington 

Edge

[Edgeworth] Edgeworth Francis (1845-1926) 

[Egorov] Egorov Dimitri (1869-1931) 

[Ehrenfest] Ehrenfest Paul (1880-1933) 

[Ehresmann] Ehresmann Charles (1905-1979) 

[Eilenberg] Eilenberg Samuel (1913-1998) 

[Einstein] Einstein Albert (1879-1955) 

[Eisenhart] Eisenhart Luther (1876-1965) 

[Eisenstein] Eisenstein Gotthold (1823-1852) 

[Elliott] Elliott Edwin (1851-1937) 

Edgeworth 

Egorov 

Ehrenfest 

Ehresmann 

Eilenberg 

Einstein 

Eisenhart 

Eisenstein 

Elliott

[Empedocles] Empedocles (492BC-432BC) 

[Engel] Engel Friedrich (1861-1941) 

[Enriques] Enriques Federigo (1871-1946) 

[Enskog] Enskog David (1884-1947) 

[Epstein] Epstein Paul (1871-1939) 

[Eratosthenes] Eratosthenes of Cyrene (276BC-197BC) 

[Erdelyi] Erdélyi Arthur (1908-1977) 

[Erdos] Erdös Paul (1913-1996) 

[Erlang] Erlang Agner (1878-1929) 

Empedocles 

Engel 

Enriques 

Enskog 

Epstein 

Eratosthenes 

Erdelyi 

Erdos 

Erlang

[Escher] Escher Maurits (1898-1972) 

[Esclangon] Esclangon Ernest (1876-1954) 

[Euclid] Euclid of Alexandria (325BC-265BC) 

[Eudemus] Eudemus of Rhodes (350BC-290BC) 

[Eudoxus] Eudoxus of Cnidus (408BC-355BC) 

Escher 

Esclangon 

Euclid 

Eudemus 

Eudoxus 

Euler 

[Euler] Euler Leonhard (1707-1783) Swiss mathematician, worked on practially all fields of mathematics. 

[Eutocius] Eutocius of Ascalon (480-540) 

[Evans] Evans Griffith (1887-1973) 

[Ezra] Ezra Rabbi Ben (1092-1167) 

Eutocius 

Evans 

Ezra

[FaadiBruno] Faà di Bruno Francesco (1825-1907) 

[Faber] Faber Georg 

[Fabri] Fabri Honoré (1607-1688) 

[FagnanoGiovanni] Fagnano Giovanni (1715-1797) 

[FagnanoGiulio] Fagnano Giulio (1682-1766) 

[Faltings] Faltings Gerd 

[Fano] Fano Gino (1871-1952) 

[Faraday] Faraday Michael (1791-1867) 

[Farey] Farey John (1766-1826) 

FaadiBruno 

Faber 

Fabri 

FagnanoGiovanni 

FagnanoGiulio 

Faltings 

Fano 

Faraday 

Farey

[Fatou] Fatou Pierre (1878-1929) 

[Faulhaber] Faulhaber Johann (1580-1635) 

[Fefferman] Fefferman Charles 

[Feigenbaum] Feigenbaum Mitchell 

[Feigl] Feigl Georg (1890-1945) 

[Fejer] Fejér Lipót (1880-1959) 

[Feller] Feller William (1906-1970) 

[Fermat] Fermat Pierre de (1601-1665) 

[Ferrar] Ferrar Bill (1893-1990) 

Fatou 

Faulhaber 

Fefferman 

Feigenbaum 

Feigl 

Fejer 

Feller 

Fermat 

Ferrar

[Ferrari] Ferrari Lodovico (1522-1565) 

[Ferrel] Ferrel William (1817-1891) 

[Ferro] Ferro Scipione del (1465-1526) 

[Feuerbach] Feuerbach Karl (1800-1834) 

[Feynman] Feynman Richard (1918-1988) 

[Fields] Fields John (1863-1932) 

[Finck] Finck Pierre-Joseph (1797-1870) 

[Fincke] Fincke Thomas (1561-1656) 

[FineHenry] Fine Henry (1858-1928) 

Ferrari 

Ferrel 

Ferro 

Feuerbach 

Feynman 

Fields 

Finck 

Fincke 

FineHenry

[Fine] Fine Oronce (1494-1555) 

[Finsler] Finsler Paul (1894-1970) 

[Fischer] Fischer Ernst (1875-1959) 

[Fisher] Fisher Sir Ronald (1890-1962) 

[Fiske] Fiske Thomas (1865-1944) 

[FitzGerald] FitzGerald George (1851-1901) 

[Flugge-Lotz] Flügge-Lotz Irmgard (1903-1974) 

[Flamsteed] Flamsteed John 

[Fomin] Fomin Sergei (1917-1975) 

Fine 

Finsler 

Fischer 

Fisher 

Fiske 

FitzGerald 

Flugge-Lotz 

Flamsteed 

Fomin

FontainedesBertins 

[FontainedesBertins] Fontaine des Bertins A (1704-1771) 

[Fontenelle] Fontenelle Bernard de (1657-1757) 

[Forsyth] Forsyth Andrew (1858-1942) 

[Burali-Forti] Forti Cesare Burali- (1861-1931) 

[Fourier] Fourier Joseph (1768-1830) 

[Fowler] Fowler Ralph (1889-1944) 

[Fox] Fox Charles (1897-1977) 

[Frechet] Fréchet Maurice (1878-1973) 

[Fraenkel] Fraenkel Adolf (1891-1965) 

Fontenelle 

Forsyth 

Burali-Forti 

Fourier 

Fowler 

Fox 

Frechet 

Fraenkel

[FrancaisJacques] Francais Jacques (1775-1833) 

[FrancaisFrancois] Francais Francois (1768-1810) 

[Francoeur] Francoeur Louis (1773-1849) 

[Frank] Frank Philipp (1884-1966) 

[Franklin] Franklin Philip (1898-1965) 

[FranklinBenjamin] Franklin Benjamin (1706-1790) 

[Frattini] Frattini Giovanni (1852-1925) 

[Fredholm] Fredholm Ivar (1866-1927) 

[Freedman] Freedman Michael 

FrancaisJacques 

FrancaisFrancois 

Francoeur 

Frank 

Franklin 

FranklinBenjamin 

Frattini 

Fredholm 

Freedman

[Frege] Frege Gottlob (1848-1925) 

[Freitag] Freitag Herta (1908-2000) 

[Frenet] Frenet Jean (1816-1900) 

[FrenicledeBessy] Frenicle de Bessy B (1605-1675) 

[Frenkel] Frenkel Jacov (1894-1952) 

[Fresnel] Fresnel Augustin (1788-1827) 

[Freudenthal] Freudenthal Hans (1905-1990) 

[Freundlich] Freundlich Finlay (1885-1964) 

[Friedmann] Friedmann Alexander (1888-1925) 

Frege 

Freitag 

Frenet 

FrenicledeBessy 

Frenkel 

Fresnel 

Freudenthal 

Freundlich 

Friedmann

[Friedrichs] Friedrichs Kurt (1901-1982) 

[Frisi] Frisi Paolo (1728-1784) 

[Frobenius] Frobenius Georg (1849-1917) 

[Fubini] Fubini Guido (1879-1943) 

[Fuchs] Fuchs Lazarus (1833-1902) 

[Fueter] Fueter Rudolph (1880-1950) 

[Fuller] Fuller R Buckminster (1895-1983) 

[Fuss] Fuss Nicolai (1755-1826) 

[Godel] Gödel Kurt (1906-1978) 

Friedrichs 

Frisi 

Frobenius 

Fubini 

Fuchs 

Fueter 

Fuller 

Fuss 

Godel

[Gopel] Göpel Adolph (1812-1847) 

[Galerkin] Galerkin Boris (1871-1945) 

[Galileo] Galileo Galilei (1564-1642) 

[Gallarati] Gallarati Dionisio 

[Galois] Galois Evariste (1811-1832) 

[Galton] Galton Francis (1822-1911) 

[Gassendi] Gassendi Pierre (1592-1655) 

[Gauss] Gauss Carl Friedrich (1777-1855) 

[Gegenbauer] Gegenbauer Leopold (1849-1903) 

Gopel 

Galerkin 

Galileo 

Gallarati 

Galois 

Galton 

Gassendi 

Gauss 

Gegenbauer

[Geiser] Geiser Karl (1843-1934) 

[Gelfand] Gelfand Israil 

[Gelfond] Gelfond Aleksandr (1906-1968) 

[Gellibrand] Gellibrand Henry (1597-1636) 

[Geminus] Geminus (10BC-60AD) 

[GemmaFrisius] Gemma Frisius Regnier (1508-1555) 

[Genocchi] Genocchi Angelo (1817-1889) 

[Gentzen] Gentzen Gerhard (1909-1945) 

[Gergonne] Gergonne Joseph (1771-1859) 

Geiser 

Gelfand 

Gelfond 

Gellibrand 

Geminus 

GemmaFrisius 

Genocchi 

Gentzen 

Gergonne

[Germain] Germain Sophie (1776-1831) 

[Gherard] Gherard of Cremona (1114-1187) 

[Ghetaldi] Ghetaldi Marino (1566-1626) 

[Gibbs] Gibbs J Willard (1839-1903) 

[GirardAlbert] Girard Albert (1595-1632) 

[GirardPierre] Girard Pierre Simon (1765-1836) 

[Glaisher] Glaisher James (1848-1928) 

[Glenie] Glenie James (1750-1817) 

[Gohberg] Gohberg Israel 

Germain 

Gherard 

Ghetaldi 

Gibbs 

GirardAlbert 

GirardPierre 

Glaisher 

Glenie 

Gohberg

[Goldbach] Goldbach Christian (1690-1764) 

[Goldstein] Goldstein Sydney (1903-1989) 

[Gompertz] Gompertz Benjamin (1779-1865) 

[Goodstein] Goodstein Reuben (1912-1985) 

[Gordan] Gordan Paul (1837-1912) 

[Gorenstein] Gorenstein Daniel (1923-1992) 

[Gosset] Gosset William (1876-1937) 

[Goursat] Goursat Edouard (1858-1936) 

[Govindasvami] Govindasvami (800-860) 

Goldbach 

Goldstein 

Gompertz 

Goodstein 

Gordan 

Gorenstein 

Gosset 

Goursat 

Govindasvami

[Graffe] Gräffe Karl (1799-1873) 

[Gram] Gram Jorgen (1850-1916) 

[Grandi] Grandi Guido (1671-1742) 

[Granville] Granville Evelyn 

[Grassmann] Grassmann Hermann (1808-1877) 

[Grave] Grave Dmitry (1863-1939) 

[Green] Green George (1793-1841) 

[Greenhill] Greenhill Alfred (1847-1927) 

[Gregory] Gregory James (1638-1675) 

Graffe 

Gram 

Grandi 

Granville 

Grassmann 

Grave 

Green 

Greenhill 

Gregory

[GregoryDuncan] Gregory Duncan (1813-1844) 

[GregoryDavid] Gregory David (1659-1708) 

[DeGroot] Groot Johannes de (1914-1972) 

[Grosseteste] Grosseteste Robert (1168-1253) 

[Grossmann] Grossmann Marcel (1878-1936) 

[Grothendieck] Grothendieck Alexander 

[Grunsky] Grunsky Helmut (1904-1986) 

[Guarini] Guarini Guarino (1624-1683) 

[Guccia] Guccia Giovanni (1855-1914) 

GregoryDuncan 

GregoryDavid 

DeGroot 

Grosseteste 

Grossmann 

Grothendieck 

Grunsky 

Guarini 

Guccia

[Gudermann] Gudermann Christoph (1798-1852) 

[Guenther] Guenther Adam (1848-1923) 

[Guinand] Guinand Andy (1912-1987) 

[Guldin] Guldin Paul (1577-1643) 

[Gunter] Gunter Edmund (1581-1626) 

[Hajek] Häjek Jaroslav (1926-1974) 

[Herigone] Hérigone Pierre (1580-1643) 

[Holder] Hölder Otto (1859-1937) 

[Hormander] Hörmander Lars 

Gudermann 

Guenther 

Guinand 

Guldin 

Gunter 

Hajek 

Herigone 

Holder 

Hormander

[Haar] Haar Alfréd (1885-1933) 

[Hachette] Hachette Jean (1769-1834) 

[Hadamard] Hadamard Jacques (1865-1963) 

[Hadley] Hadley John (1682-1744) 

[Hahn] Hahn Hans (1879-1934) 

[Hall] Hall Philip (1904-1982) 

[HallMarshall] Hall Marshall Jr. (1910-1990) 

[Halley] Halley Edmond (1656-1742) 

[Halmos] Halmos Paul 

Haar 

Hachette 

Hadamard 

Hadley 

Hahn 

Hall 

HallMarshall 

Halley 

Halmos

[Halphen] Halphen George (1844-1889) 

[Halsted] Halsted George (1853-1922) 

[Hamill] Hamill Christine 

[Hamilton] Hamilton William R (1805-1865) 

[HamiltonWilliam] Hamilton William (1788-1856) 

[Hamming] Hamming Richard W (1915-1998) 

[Hankel] Hankel Hermann (1839-1873) 

[HardyClaude] Hardy Claude (1598-1678) 

[Hardy] Hardy G H (1877-1947) 

Halphen 

Halsted 

Hamill 

Hamilton 

HamiltonWilliam 

Hamming 

Hankel 

HardyClaude 

Hardy

[Harish-Chandra] Harish-Chandra (1923-1983) 

[Harriot] Harriot Thomas (1560-1621) 

[Hartley] Hartley Brian (1939-1994) 

[Hartree] Hartree Douglas (1897-1958) 

[Hasse] Hasse Helmut (1898-1979) 

[Hausdorff] Hausdorff Felix (1869-1942) 

[Hawking] Hawking Stephen 

[Al-Haytham] Haytham Abu Ali al 

[Heath] Heath Thomas (1861-1940) 

Harish-Chandra 

Harriot 

Hartley 

Hartree 

Hasse 

Hausdorff 

Hawking 

Al-Haytham 

Heath

[Heaviside] Heaviside Oliver (1850-1925) 

[Heawood] Heawood Percy (1861-1955) 

[Hecht] Hecht Daniel (1777-1833) 

[Hecke] Hecke Erich (1887-1947) 

[Hedrick] Hedrick Earle (1876-1943) 

[Heegaard] Heegaard Poul (1871-1948) 

[Heilbronn] Heilbronn Hans (1908-1975) 

[Heine] Heine Eduard (1821-1881) 

[Heisenberg] Heisenberg Werner (1901-1976) 

Heaviside 

Heawood 

Hecht 

Hecke 

Hedrick 

Heegaard 

Heilbronn 

Heine 

Heisenberg

[Hellinger] Hellinger Ernst (1883-1950) 

[Helly] Helly Eduard (1884-1943) 

[Heng] Heng Zhang (78AD-139) 

[Henrici] Henrici Olaus (1840-1918) 

[Hensel] Hensel Kurt (1861-1941) 

[Heraclides] Heraclides of Pontus (387BC-312BC) 

[Herbrand] Herbrand Jacques (1908-1931) 

[Hermann] Hermann Jakob (1678-1733) 

[Hermite] Hermite Charles (1822-1901) 

Hellinger 

Helly 

Heng 

Henrici 

Hensel 

Heraclides 

Herbrand 

Hermann 

Hermite

[Heron] Heron of Alexandria (10AD-75) 

[HerschelCaroline] Herschel Caroline (1750-1848) 

[Herschel] Herschel John (1792-1871) 

[Herstein] Herstein Yitz (1923-1988) 

[Hesse] Hesse Otto (1811-1874) 

[Heyting] Heyting Arend (1898-1980) 

[Higman] Higman Graham 

[Hilbert] Hilbert David (1862-1943) 

[Hill] Hill George (1838-1914) 

Heron 

HerschelCaroline 

Herschel 

Herstein 

Hesse 

Heyting 

Higman 

Hilbert 

Hill

[Hille] Hille Einar (1894-1980) 

[Hindenburg] Hindenburg Carl (1741-1808) 

[Hipparchus] Hipparchus of Rhodes (190BC-120BC) 

[Hippias] Hippias of Elis (460BC-400BC) 

[Hippocrates] Hippocrates of Chios (470BC-410BC) 

[Hironaka] Hironaka Heisuke 

[Hirsch] Hirsch Kurt (1906-1986) 

[Hirst] Hirst Thomas (1830-1891) 

[Gnedenko] Hniedenko Boris (1912-1995) 

Hille 

Hindenburg 

Hipparchus 

Hippias 

Hippocrates 

Hironaka 

Hirsch 

Hirst 

Gnedenko

[Houel] Hoüel Jules (1823-1886) 

[Hobbes] Hobbes Thomas (1588-1679) 

[Hobson] Hobson Ernest (1856-1933) 

[Hodge] Hodge William (1903-1975) 

[Hollerith] Hollerith Herman (1860-1929) 

[Holmboe] Holmboe Bernt (1795-1850) 

[Honda] Honda Taira (1932-1975) 

[Hooke] Hooke Robert (1635-1703) 

[HopfEberhard] Hopf Eberhard (1902-1983) 

Houel 

Hobbes 

Hobson 

Hodge 

Hollerith 

Holmboe 

Honda 

Hooke 

HopfEberhard

[Hopf] Hopf Heinz (1894-1971) 

[Hopkins] Hopkins William (1793-1866) 

[Hopkinson] Hopkinson John (1849-1898) 

[Hopper] Hopper Grace (1906-1992) 

[Horner] Horner William (1786-1837) 

[Householder] Householder Alston (1904-1993) 

[Hsu] Hsu Pao-Lu (1910-1970) 

[Hubble] Hubble Edwin (1889-1953) 

[Hudde] Hudde Johann (1628-1704) 

Hopf 

Hopkins 

Hopkinson 

Hopper 

Horner 

Householder 

Hsu 

Hubble 

Hudde

[HumbertPierre] Humbert Pierre (1891-1953) 

[HumbertGeorges] Humbert Georges (1859-1921) 

[Huntington] Huntington Edward (1874-1952) 

[Hurewicz] Hurewicz Witold (1904-1956) 

[Hurwitz] Hurwitz Adolf (1859-1919) 

[Hutton] Hutton Charles (1737-1823) 

[Huygens] Huygens Christiaan (1629-1695) 

[Hypatia] Hypatia of Alexandria (370-415) 

[Hypsicles] Hypsicles of Alexandria (190BC-120BC) 

HumbertPierre 

HumbertGeorges 

Huntington 

Hurewicz 

Hurwitz 

Hutton 

Huygens 

Hypatia 

Hypsicles

[Ibrahim] Ibrahim ibn Sinan (908-946) 

[Ingham] Ingham Albert (1900-1967) 

[Ito] Ito Kiyosi 

[Ivory] Ivory James (1765-1842) 

[Iwasawa] Iwasawa Kenkichi (1917-1998) 

[Iyanaga] Iyanaga Skokichi 

[JabiribnAflah] Jabir ibn Aflah (1100-1160) 

[Jacobi] Jacobi Carl (1804-1851) 

[Jacobson] Jacobson Nathan (1910-1999) 

Ibrahim 

Ingham 

Ito 

Ivory 

Iwasawa 

Iyanaga 

JabiribnAflah 

Jacobi 

Jacobson

[Jagannatha] Jagannatha Samrat 

[James] James Ioan 

[Janiszewski] Janiszewski Zygmunt (1888-1920) 

[Janovskaja] Janovskaja Sof’ja (1896-1966) 

[Jarnik] Jarnik Vojtech (1897-1970) 

[Al-Jawhari] Jawhari al-Abbas al (800-860) 

[Al-Jayyani] Jayyani Abu al 

[Jeans] Jeans Sir James (1877-1946) 

[Jeffrey] Jeffrey George (1891-1957) 

Jagannatha 

James 

Janiszewski 

Janovskaja 

Jarnik 

Al-Jawhari 

Al-Jayyani 

Jeans 

Jeffrey

[Jeffreys] Jeffreys Sir Harold (1891-1989) 

[Jensen] Jensen Johan (1859-1925) 

[Jerrard] Jerrard George (1804-1863) 

[Jevons] Jevons William (1835-1882) 

[Joachimsthal] Joachimsthal Ferdinand (1818-1861) 

[John] John Fritz 

[JohnsonBarry] Johnson Barry 

[Johnson] Johnson William (1858-1931) 

[JonesBurton] Jones F (1910-1999) 

Jeffreys 

Jensen 

Jerrard 

Jevons 

Joachimsthal 

John 

JohnsonBarry 

Johnson 

JonesBurton

[Jones] Jones William (1675-1749) 

[JonesVaughan] Jones Vaughan 

[Jonquieres] Jonquières Ernest de (1820-1901) 

[Jordan] Jordan Camille (1838-1922) 

[Jordanus] Jordanus Nemorarius (1225-1260) 

[Jourdain] Jourdain Philip (1879-1921) 

[Juel] Juel Christian (1855-1935) 

[Julia] Julia Gaston (1893-1978) 

[Jungius] Jungius Joachim (1587-1657) 

Jones 

JonesVaughan 

Jonquieres 

Jordan 

Jordanus 

Jourdain 

Juel 

Julia 

Jungius

[Jyesthadeva] Jyesthadeva (1500-1575) 

[KonigJulius] König Julius (1849-1913) 

[KonigSamuel] König Samuel (1712-1757) 

[Konigsberger] Königsberger Leo (1837-1921) 

[Kurschak] Kürschäk József (1864-1933) 

[Kac] Kac Mark (1914-1984) 

[Kaestner] Kaestner Abraham (1719-1800) 

[Kagan] Kagan Benjamin (1869-1953) 

[Kakutani] Kakutani Shizuo 

Jyesthadeva 

KonigJulius 

KonigSamuel 

Konigsberger 

Kurschak 

Kac 

Kaestner 

Kagan 

Kakutani

[Kalmar] Kalmär Läszló (1905-1976) 

[Kaluza] Kaluza Theodor (1885-1945) 

[Kaluznin] Kaluznin Lev (1914-1990) 

[Al-Farisi] Kamal al-Farisi (1260-1320) 

[Kamalakara] Kamalakara (1616-1700) 

[AbuKamil] Kamil Abu Shuja (850-930) 

[Kantorovich] Kantorovich Leonid (1912-1986) 

[Kaplansky] Kaplansky Irving 

[Al-Karaji] Karkhi al 

Kalmar 

Kaluza 

Kaluznin 

Al-Farisi 

Kamalakara 

AbuKamil 

Kantorovich 

Kaplansky 

Al-Karaji

[Karp] Karp Carol (1926-1972) 

[Al-Kashi] Kashi Ghiyath al (1390-1450) 

[Katyayana] Katyayana (200BC-140BC) 

[Keill] Keill John (1671-1721) 

[Kelland] Kelland Philip (1808-1879) 

[Kellogg] Kellogg Oliver (1878-1957) 

[Kemeny] Kemeny John (1926-1992) 

[Kempe] Kempe Alfred (1849-1922) 

[KendallMaurice] Kendall Maurice (1907-1983) 

Karp 

Al-Kashi 

Katyayana 

Keill 

Kelland 

Kellogg 

Kemeny 

Kempe 

KendallMaurice

[Kendall] Kendall David 

[Kepler] Kepler Johannes (1571-1630) 

[Kerekjarto] Kerékjärtó Béla (1898-1946) 

[Keynes] Keynes John Maynard (1883-1946) 

[Al-Khalili] Khalili Shams al (1320-1380) 

[Al-Khazin] Khazin Abu Jafar al (900-971) 

[Khinchin] Khinchin Aleksandr (1894-1959) 

[Al-Khujandi] Khujandi Abu al 

[Al-Khwarizmi] Khwarizmi Abu al- (790-850) 

Kendall 

Kepler 

Kerekjarto 

Keynes 

Al-Khalili 

Al-Khazin 

Khinchin 

Al-Khujandi 

Al-Khwarizmi

[Killing] Killing Wilhelm (1847-1923) 

[Al-Kindi] Kindi Abu al (805-873) 

[Kingman] Kingman John 

[Kirchhoff] Kirchhoff Gustav (1824-1887) 

[Kirkman] Kirkman Thomas (1806-1895) 

[Klugel] Klügel Georg (1739-1812) 

[Kleene] Kleene Stephen (1909-1994) 

[KleinOskar] Klein Oskar (1894-1977) 

[Klein] Klein Felix (1849-1925) 

Killing 

Al-Kindi 

Kingman 

Kirchhoff 

Kirkman 

Klugel 

Kleene 

KleinOskar 

Klein

[Klingenberg] Klingenberg Wilhelm 

[Kloosterman] Kloosterman Hendrik (1900-1968) 

[Kneser] Kneser Adolf (1862-1930) 

[KneserHellmuth] Kneser Hellmuth (1898-1973) 

[Knopp] Knopp Konrad (1882-1957) 

[Kober] Kober Hermann (1888-1973) 

[Kochin] Kochin Nikolai (1901-1944) 

[Kodaira] Kodaira Kunihiko (1915-1997) 

[Koebe] Koebe Paul (1882-1945) 

Klingenberg 

Kloosterman 

Kneser 

KneserHellmuth 

Knopp 

Kober 

Kochin 

Kodaira 

Koebe

[Koenigs] Koenigs Gabriel (1858-1931) 

Koenigs 

Kolmogorov 

[Kolmogorov] Kolmogorov Andrey (1903-1987) Russian probabilist who established in 1933 the mathematical 

foundation of probability theory and did important work also in other fields like Hamiltonian dynamics (KAM 

theorem) or turbulence Kolmogorov scaling. 

[Kolosov] Kolosov Gury (1867-1936) 

[KonigDenes] Konig Denes (1884-1944) 

[Korteweg] Korteweg Diederik (1848-1941) 

[Kotelnikov] Kotelnikov Aleksandr (1865-1944) 

[Kovalevskaya] Kovalevskaya Sofia (1850-1891) 

[Kramp] Kramp Christian (1760-1826) 

[Krawtchouk] Krawtchouk Mikhail (1892-1942) 

Kolosov 

KonigDenes 

Korteweg 

Kotelnikov 

Kovalevskaya 

Kramp 

Krawtchouk

[Krein] Krein Mark (1907-1989) 

[Kreisel] Kreisel Georg 

[Kronecker] Kronecker Leopold (1823-1891) 

[Krull] Krull Wolfgang (1899-1971) 

[KrylovAleksei] Krylov Aleksei (1863-1945) 

[KrylovNikolai] Krylov Nikolai (1879-1955) 

[Kulik] Kulik Yakov (1783-1863) 

[Kumano-Go] Kumano-Go Hitoshi (1935-1982) 

[Kummer] Kummer Eduard (1810-1893) 

Krein 

Kreisel 

Kronecker 

Krull 

KrylovAleksei 

KrylovNikolai 

Kulik 

Kumano-Go 

Kummer

[Kuratowski] Kuratowski Kazimierz (1896-1980) 

[Kurosh] Kurosh Aleksandr (1908-1971) 

[Kutta] Kutta Martin (1867-1944) 

[Kuttner] Kuttner Brian (1908-1992) 

[Leger] Léger Emile (1795-1838) 

[LevyPaul] Lévy Paul (1886-1971) 

[Lowenheim] Löwenheim Leopold (1878-1957) 

[Loewner] Löwner Karl (1893-1968) 

[DeL’Hopital] L’Hopital Guillaume de (1661-1704) 

Kuratowski 

Kurosh 

Kutta 

Kuttner 

Leger 

LevyPaul 

Lowenheim 

Loewner 

DeL’Hopital

[LaHire] La Hire Philippe de (1640-1718) 

[LaFaille] La Faille Charles de (1597-1652) 

[LaCondamine] La Condamine Charles de (1701-1774) 

[Lacroix] Lacroix Sylvestre (1765-1843) 

[Lagny] Lagny Thomas de (1660-1734) 

[Lagrange] Lagrange Joseph-Louis (1736-1813) 

[Laguerre] Laguerre Edmond (1834-1886) 

[Lakatos] Lakatos Imre (1922-1974) 

[Lalla] Lalla (720-790) 

LaHire 

LaFaille 

LaCondamine 

Lacroix 

Lagny 

Lagrange 

Laguerre 

Lakatos 

Lalla

[Lame] Lamé Gabriel (1795-1870) 

[Lamb] Lamb Horace (1849-1934) 

[Lambert] Lambert Johann (1728-1777) 

[HermannofReichenau] Lame Hermann the (1013-1054) 

[Lamy] Lamy Bernard (1640-1715) 

[Lanczos] Lanczos Cornelius (1893-1974) 

[Landau] Landau Edmund (1877-1938) 

[LandauLev] Landau Lev (1908-1968) 

[Landen] Landen John (1719-1790) 

Lame 

Lamb 

Lambert 

HermannofReichenau 

Lamy 

Lanczos 

Landau 

LandauLev 

Landen

[Landsberg] Landsberg Georg (1865-1912) 

[Langlands] Langlands Robert 

[Laplace] Laplace Pierre-Simon (1749-1827) 

[Larmor] Larmor Sir Joseph (1857-1942) 

[Lasker] Lasker Emanuel (1868-1941) 

[Kramer] Lassar Edna Kramer (1902-1984) 

[LaurentHermann] Laurent Hermann (1841-1908) 

[LaurentPierre] Laurent Pierre (1813-1854) 

[Lavanha] Lavanha Joao Baptista (1550-1624) 

Landsberg 

Langlands 

Laplace 

Larmor 

Lasker 

Kramer 

LaurentHermann 

LaurentPierre 

Lavanha

[Lavrentev] Lavrentev Mikhail (1900-1980) 

[Lax] Lax Gaspar (1487-1560) 

[LeFevre] Le Fèvre Jean (1652-1706) 

[Lebesgue] Lebesgue Henri (1875-1941) 

[Ledermann] Ledermann Walter 

[Leech] Leech John (1926-1992) 

[Lefschetz] Lefschetz Solomon (1884-1972) 

[Legendre] Legendre Adrien-Marie (1752-1833) 

[Lemoine] Lemoine Emile (1840-1912) 

Lavrentev 

Lax 

LeFevre 

Lebesgue 

Ledermann 

Leech 

Lefschetz 

Legendre 

Lemoine

[Leray] Leray Jean (1906-1998) 

[Lerch] Lerch Mathias (1860-1922) 

[Leshniewski] Leshniewski Stanislaw (1886-1939) 

[Leslie] Leslie John (1766-1832) 

[Leucippus] Leucippus (480BC-420BC) 

[Levi] Levi ben Gerson (1288-1344) 

[Levi-Civita] Levi-Civita Tullio (1873-1941) 

[Levinson] Levinson Norman (1912-1975) 

[LevyHyman] Levy Hyman (1889-1975) 

Leray 

Lerch 

Leshniewski 

Leslie 

Leucippus 

Levi 

Levi-Civita 

Levinson 

LevyHyman

[Levytsky] Levytsky Volodymyr (1872-1956) 

[Lexell] Lexell Anders (1740-1784) 

[Lexis] Lexis Wilhelm (1837-1914) 

[Lhuilier] Lhuilier Simon (1750-1840) 

[Libri] Libri Guglielmo (1803-1869) 

[Lie] Lie Sophus (1842-1899) 

[Lifshitz] Lifshitz Evgenii (1915-1985) 

[Lighthill] Lighthill Sir James (1924-1998) 

[Lindelof] Lindelöf Ernst (1870-1946) 

Levytsky 

Lexell 

Lexis 

Lhuilier 

Libri 

Lie 

Lifshitz 

Lighthill 

Lindelof

[Linnik] Linnik Yuri (1915-1972) 

[Lions] Lions Pierre-Louis 

[Liouville] Liouville Joseph (1809-1882) 

[Lipschitz] Lipschitz Rudolf (1832-1903) 

[Lissajous] Lissajous Jules (1822-1880) 

[Listing] Listing Johann (1808-1882) 

[Littlewood] Littlewood John E (1885-1977) 

[LittlewoodDudley] Littlewood Dudley (1903-1979) 

[Livsic] Livsic Moshe 

Linnik 

Lions 

Liouville 

Lipschitz 

Lissajous 

Listing 

Littlewood 

LittlewoodDudley 

Livsic

[Llull] Llull Ramon (1235-1316) 

[Lobachevsky] Lobachevsky Nikolai (1792-1856) 

[Loewy] Loewy Alfred (1873-1935) 

[Lopatynsky] Lopatynsky Yaroslav (1906-1981) 

[Lorentz] Lorentz Hendrik (1853-1928) 

[Love] Love Augustus (1863-1940) 

[Lovelace] Lovelace Augusta Ada (1815-1852) 

[Loyd] Loyd Samuel (1841-1911) 

[Lucas] Lucas F Edouard (1842-1891) 

Llull 

Lobachevsky 

Loewy 

Lopatynsky 

Lorentz 

Love 

Lovelace 

Loyd 

Lucas

[Lueroth] Lueroth Jacob (1844-1910) 

[Lukacs] Lukacs Eugene (1906-1987) 

[Lukasiewicz] Lukasiewicz Jan (1878-1956) 

[Luke] Luke Yudell (1918-1983) 

[Luzin] Luzin Nikolai (1883-1950) 

[Lyapunov] Lyapunov Aleksandr (1857-1918) 

[Lyndon] Lyndon Roger (1917-1988) 

[Meray] Méray Charles (1835-1911) 

[Mobius] Möbius August (1790-1868) 

Lueroth 

Lukacs 

Lukasiewicz 

Luke 

Luzin 

Lyapunov 

Lyndon 

Meray 

Mobius

[MacCullagh] MacCullagh James (1809-1896) 

[MacLane] MacLane Saunders 

[MacMahon] MacMahon Percy (1854-1929) 

[Macaulay] Macaulay Francis (1862-1937) 

[Macdonald] Macdonald Hector (1865-1935) 

[Maclaurin] Maclaurin Colin (1698-1746) 

[Madhava] Madhava Sangamagramma (1350-1425) 

[Al-Maghribi] Maghribi Muhyi al (1220-1280) 

[Magnitsky] Magnitsky Leonty (1669-1739) 

MacCullagh 

MacLane 

MacMahon 

Macaulay 

Macdonald 

Maclaurin 

Madhava 

Al-Maghribi 

Magnitsky

[Magnus] Magnus Wilhelm (1907-1990) 

[Al-Mahani] Mahani Abu al (820-880) 

[Mahavira] Mahavira Mahavira (800-870) 

[MahendraSuri] Mahendra Suri (1340-1410) 

[Mahler] Mahler Kurt (1903-1988) 

[Maior] Maior John (1469-1550) 

[Malcev] Malcev Anatoly (1909-1967) 

[Malebranche] Malebranche Nicolas (1638-1715) 

[Malfatti] Malfatti Francesco (1731-1807) 

Magnus 

Al-Mahani 

Mahavira 

MahendraSuri 

Mahler 

Maior 

Malcev 

Malebranche 

Malfatti

[Malus] Malus Etienne Louis (1775-1812) 

[Manava] Manava (750BC-690BC) 

[Mandelbrot] Mandelbrot Benoit 

[Mannheim] Mannheim Amédée (1831-1906) 

[Mansion] Mansion Paul (1844-1919) 

[Mansur] Mansur ibn Ali Abu 

[Marchenko] Marchenko Vladimir 

[Marcinkiewicz] Marcinkiewicz Jozef (1910-1940) 

[Marczewski] Marczewski Edward (1907-1976) 

Malus 

Manava 

Mandelbrot 

Mannheim 

Mansion 

Mansur 

Marchenko 

Marcinkiewicz 

Marczewski

[Margulis] Margulis Gregori 

[Marinus] Marinus of Neapolis (450-500) 

[Markov] Markov Andrei (1856-1922) 

[Al-Banna] Marrakushi al (1256-1321) 

[Mascheroni] Mascheroni Lorenzo (1750-1800) 

[Maschke] Maschke Heinrich (1853-1908) 

[Maseres] Maseres Francis (1731-1824) 

[Maskelyne] Maskelyne Nevil (1732-1811) 

[Mason] Mason Max (1877-1961) 

Margulis 

Marinus 

Markov 

Al-Banna 

Mascheroni 

Maschke 

Maseres 

Maskelyne 

Mason

[Mathews] Mathews George (1861-1922) 

[MathieuClaude] Mathieu Claude-Louis (1783-1875) 

[MathieuEmile] Mathieu Emile (1835-1890) 

[Matsushima] Matsushima Yozo (1921-1983) 

[Mauchly] Mauchly John (1907-1980) 

[Maupertuis] Maupertuis Pierre de (1698-1759) 

[Maurolico] Maurolico Francisco (1494-1575) 

[Maxwell] Maxwell James Clerk (1831-1879) 

[MayerAdolph] Mayer Adolph (1839-1903) 

Mathews 

MathieuClaude 

MathieuEmile 

Matsushima 

Mauchly 

Maupertuis 

Maurolico 

Maxwell 

MayerAdolph

[MayerTobias] Mayer Tobias (1723-1762) 

[Mazur] Mazur Stanislaw (1905-1981) 

[Mazurkiewicz] Mazurkiewicz Stefan (1888-1945) 

[McClintock] McClintock John (1840-1916) 

[McDuff] McDuff Margaret 

[McShane] McShane Edward (1904-1989) 

[Meissel] Meissel Ernst (1826-1895) 

[Mellin] Mellin Hjalmar (1854-1933) 

[Menabrea] Menabrea Luigi (1809-1896) 

MayerTobias 

Mazur 

Mazurkiewicz 

McClintock 

McDuff 

McShane 

Meissel 

Mellin 

Menabrea

[Menaechmus] Menaechmus (380BC-320BC) 

[Menelaus] Menelaus of Alexandria (70AD-130) 

[Menger] Menger Karl 

[Mengoli] Mengoli Pietro (1626-1686) 

[Menshov] Menshov Dmitrii (1892-1988) 

[MercatorGerardus] Mercator Gerardus (1512-1592) 

[MercatorNicolaus] Mercator Nicolaus (1620-1687) 

[Mercer] Mercer James (1883-1932) 

[Merrifield] Merrifield Charles (1827-1884) 

Menaechmus 

Menelaus 

Menger 

Mengoli 

Menshov 

MercatorGerardus 

MercatorNicolaus 

Mercer 

Merrifield

[Merrill] Merrill Winifred (1862-1951) 

[Mersenne] Mersenne Marin (1588-1648) 

[Mertens] Mertens Franz (1840-1927) 

[Meshchersky] Meshchersky Ivan (1859-1935) 

[Meyer] Meyer Wilhelm (1856-1934) 

[Miller] Miller George (1863-1951) 

[Milne] Milne Edward (1896-1950) 

[Milnor] Milnor John 

[Minding] Minding Ferdinand (1806-1885) 

Merrill 

Mersenne 

Mertens 

Meshchersky 

Meyer 

Miller 

Milne 

Milnor 

Minding

[Mineur] Mineur Henri (1899-1954) 

[Minkowski] Minkowski Hermann (1864-1909) 

[Mirsky] Mirsky Leon (1918-1983) 

[Mittag-Leffler] Mittag-Leffler Gösta (1846-1927) 

[Mohr] Mohr Georg (1640-1697) 

[DeMoivre] Moivre Abraham de (1667-1754) 

[Molin] Molin Fedor (1861-1941) 

[Monge] Monge Gaspard (1746-1818) 

[Monte] Monte Guidobaldo del (1545-1607) 

Mineur 

Minkowski 

Mirsky 

Mittag-Leffler 

Mohr 

DeMoivre 

Molin 

Monge 

Monte

[Montel] Montel Paul (1876-1975) 

[Montmort] Montmort Pierre Rémond de (1678-1719) 

[Montucla] Montucla Jean (1725-1799) 

[MooreJonas] Moore Jonas (1627-1679) 

[MooreRobert] Moore Robert (1882-1974) 

[MooreEliakim] Moore Eliakim (1862-1932) 

[Morawetz] Morawetz Cathleen 

[Mordell] Mordell Louis (1888-1972) 

[DeMorgan] Morgan Augustus De (1806-1871) 

Montel 

Montmort 

Montucla 

MooreJonas 

MooreRobert 

MooreEliakim 

Morawetz 

Mordell 

DeMorgan

[Mori] Mori Shigefumi 

[Morin] Morin Arthur (1795-1880) 

[MorinJean-Baptiste] Morin Jean-Baptiste 

[Morley] Morley Frank (1860-1937) 

[Morse] Morse Harald Marston (1892-1977) 

[Mostowski] Mostowski Andrzej (1913-1975) 

[Motzkin] Motzkin Theodore (1908-1970) 

[Moufang] Moufang Ruth (1905-1977) 

[Mouton] Mouton Gabriel (1618-1694) 

Mori 

Morin 

MorinJean-Baptiste 

Morley 

Morse 

Mostowski 

Motzkin 

Moufang 

Mouton

[Muir] Muir Thomas (1844-1934) 

[Mumford] Mumford David 

[Mydorge] Mydorge Claude (1585-1647) 

[Mytropolshy] Mytropolshy Yurii 

[Naimark] Naimark Mark (1909-1978) 

[Napier] Napier John (1550-1617) 

[Narayana] Narayana Pandit (1340-1400) 

[Al-Nasawi] Nasawi Abu al (1010-1075) 

[Nash] Nash John 

Muir 

Mumford 

Mydorge 

Mytropolshy 

Naimark 

Napier 

Narayana 

Al-Nasawi 

Nash

[Navier] Navier Claude (1785-1836) 

[Al-Nayrizi] Nayrizi Abu’l al (875-940) 

[Neile] Neile William (1637-1670) 

[Nekrasov] Nekrasov Aleksandr (1883-1957) 

[Netto] Netto Eugen (1848-1919) 

[Neuberg] Neuberg Joseph (1840-1926) 

[Neugebauer] Neugebauer Otto (1899-1990) 

[NeumannHanna] Neumann Hanna (1914-1971) 

[NeumannCarl] Neumann Carl Gottfried (1832-1925) 

Navier 

Al-Nayrizi 

Neile 

Nekrasov 

Netto 

Neuberg 

Neugebauer 

NeumannHanna 

NeumannCarl

[NeumannFranz] Neumann Franz Ernst (1798-1895) 

[NeumannBernhard] Neumann Bernhard 

[Nevanlinna] Nevanlinna Rolf (1895-1980) 

[Newcomb] Newcomb Simon (1835-1909) 

[Newman] Newman Maxwell (1897-1984) 

[Newton] Newton Sir Isaac (1643-1727) 

[Neyman] Neyman Jerzy (1894-1981) 

[Nicolson] Nicolson Phyllis (1917-1968) 

[Nicomachus] Nicomachus of Gerasa (60AD-120) 

NeumannFranz 

NeumannBernhard 

Nevanlinna 

Newcomb 

Newman 

Newton 

Neyman 

Nicolson 

Nicomachus

[Nicomedes] Nicomedes (280BC-210BC) 

[Nielsen] Nielsen Niels (1865-1931) 

[NielsenJakob] Nielsen Jacob 

[Nightingale] Nightingale Florence (1820-1910) 

[Nilakantha] Nilakantha Somayaji (1444-1544) 

[Niven] Niven William (1843-1917) 

[NoetherMax] Noether Max (1844-1921) 

[NoetherEmmy] Noether Emmy (1882-1935) 

[Novikov] Novikov Petr (1901-1975) 

Nicomedes 

Nielsen 

NielsenJakob 

Nightingale 

Nilakantha 

Niven 

NoetherMax 

NoetherEmmy 

Novikov

[NovikovSergi] Novikov Sergi 

[Oenopides] Oenopides of Chios (490BC-420BC) 

[Ohm] Ohm Georg Simon (1789-1854) 

[Oka] Oka Kiyoshi (1901-1978) 

[Olivier] Olivier Théodore (1793-1853) 

[Khayyam] Omar Khayyam (1048-1122) 

[Oresme] Oresme Nicole d’ (1323-1382) 

[Orlicz] Orlicz Wladyslaw (1903-1990) 

[Ortega] Ortega Juan de (1480-1568) 

NovikovSergi 

Oenopides 

Ohm 

Oka 

Olivier 

Khayyam 

Oresme 

Orlicz 

Ortega

[Osgood] Osgood William (1864-1943) 

[Osipovsky] Osipovsky Timofei (1765-1832) 

[Ostrogradski] Ostrogradski Mikhail (1801-1862) 

[Ostrowski] Ostrowski Alexander (1893-1986) 

[Oughtred] Oughtred William (1574-1660) 

[D’Ovidio] Ovidio Enrico D’ (1842-1933) 

[Ozanam] Ozanam Jacques (1640-1717) 

[Peres] Pérès Joseph (1890-1962) 

[Peter] Péter Rózsa (1905-1977) 

Osgood 

Osipovsky 

Ostrogradski 

Ostrowski 

Oughtred 

D’Ovidio 

Ozanam 

Peres 

Peter

[Polya] Pólya George (1887-1985) 

[Pacioli] Pacioli Luca (1445-1517) 

[Pade] Padé Henri (1863-1953) 

[Padoa] Padoa Alessandro (1868-1937) 

[LePaige] Paige Constantin Le (1852-1929) 

[Painleve] Painlevé Paul (1863-1933) 

[Paley] Paley Raymond (1907-1933) 

[Paman] Paman Roger (1710-1748) 

[Panini] Panini (520BC-460BC) 

Polya 

Pacioli 

Pade 

Padoa 

LePaige 

Painleve 

Paley 

Paman 

Panini

[Papin] Papin Denis (1647-1712) 

[Pappus] Pappus of Alexandria (290-350) 

[Pars] Pars Leopold (1896-1985) 

[Parseval] Parseval des Chees M-A (1755-1836) 

[Pascal] Pascal Blaise (1623-1662) 

[PascalEtienne] Pascal Etienne (1588-1640) 

[Pasch] Pasch Moritz (1843-1930) 

[Patodi] Patodi Vijay (1945-1976) 

[Pauli] Pauli Wolfgang (1900-1958) 

Papin 

Pappus 

Pars 

Parseval 

Pascal 

PascalEtienne 

Pasch 

Patodi 

Pauli

[Peacock] Peacock George (1791-1858) 

[Peano] Peano Giuseppe (1858-1932) 

[Pearson] Pearson Karl (1857-1936) 

[PearsonEgon] Pearson Egon (1895-1980) 

[PeirceBenjamin] Peirce Benjamin (1809-1880) 

[PeirceCharles] Peirce Charles (1839-1914) 

[Pell] Pell John (1611-1685) 

[Penney] Penney Bill (1909-1991) 

[Perron] Perron Oskar (1880-1975) 

Peacock 

Peano 

Pearson 

PearsonEgon 

PeirceBenjamin 

PeirceCharles 

Pell 

Penney 

Perron

[Perseus] Perseus (180BC-120BC) 

[Petersen] Petersen Julius (1839-1910) 

[Peterson] Peterson Karl (1828-1881) 

[Petit] Petit Aléxis (1791-1820) 

[Petrovsky] Petrovsky Ivan (1901-1973) 

[Petryshyn] Petryshyn Volodymyr 

[Petzval] Petzval Józeph (1807-1891) 

[Peurbach] Peurbach Georg (1423-1461) 

[Pfaff] Pfaff Johann (1765-1825) 

Perseus 

Petersen 

Peterson 

Petit 

Petrovsky 

Petryshyn 

Petzval 

Peurbach 

Pfaff

[Pfeiffer] Pfeiffer Georgii (1872-1946) 

[Philon] Philon of Byzantium (280BC-220BC) 

[PicardEmile] Picard Emile (1856-1941) 

[PicardJean] Picard Jean (1620-1682) 

[Pieri] Pieri Mario (1860-1913) 

[Francesca] Piero della Francesca (1412-1492) 

[Pillai] Pillai K C Sreedharan (1920-1980) 

[Pincherle] Pincherle Salvatore (1853-1936) 

[Fibonacci] Pisano Leonardo Fibonacci (1170-1250) 

Pfeiffer 

Philon 

PicardEmile 

PicardJean 

Pieri 

Francesca 

Pillai 

Pincherle 

Fibonacci

[Pitiscus] Pitiscus Bartholomeo (1561-1613) 

[Plucker] Plücker Julius (1801-1868) 

[Plana] Plana Giovanni (1781-1864) 

[Planck] Planck Max (1858-1947) 

[Plateau] Plateau Joseph (1801-1883) 

[Plato] Plato (428BC-347BC) 

[Playfair] Playfair John (1748-1819) 

[Plessner] Plessner Abraham 

[Poincare] Poincaré J Henri (1854-1912) 

Pitiscus 

Plucker 

Plana 

Planck 

Plateau 

Plato 

Playfair 

Plessner 

Poincare

[Poinsot] Poinsot Louis (1777-1859) 

[Poisson] Poisson Siméon (1781-1840) 

[Poleni] Poleni Giovanni (1683-1761) 

[Polozii] Polozii Georgii (1914-1968) 

[Poncelet] Poncelet Jean-Victor (1788-1867) 

[Pontryagin] Pontryagin Lev (1908-1988) 

[Poretsky] Poretsky Platon (1846-1907) 

[Porphyry] Porphyry of Malchus (233-309) 

[Porta] Porta Giambattista Della (1535-1615) 

Poinsot 

Poisson 

Poleni 

Polozii 

Poncelet 

Pontryagin 

Poretsky 

Porphyry 

Porta

[Posidonius] Posidonius of Rhodes (135BC-51BC) 

[Post] Post Emil (1897-1954) 

[Potapov] Potapov Vladimir (1914-1980) 

[Prufer] Prüfer Heinz (1896-1934) 

[Pratt] Pratt John (1809-1871) 

[Pringsheim] Pringsheim Alfred (1850-1941) 

[Privalov] Privalov Ivan (1891-1941) 

Posidonius 

Post 

Potapov 

Prufer 

Pratt 

Pringsheim 

Privalov 

PrivatdeMolieres 

[PrivatdeMolieres] Privat de Molières Joseph (1677-1742) 

[Proclus] Proclus Diadochus (411-485) 

Proclus

[DeProny] Prony Gaspard de (1755-1839) 

[Prthudakasvami] Prthudakasvami (830-890) 

[Ptolemy] Ptolemy (85AD-165) 

[Puiseux] Puiseux Victor (1820-1883) 

[Puissant] Puissant Louis (1769-1943) 

[Pythagoras] Pythagoras of Samos (580BC-520BC) 

[Al-Qalasadi] Qalasadi Abu’l al (1412-1486) 

[Quetelet] Quetelet Adolphe (1796-1874) 

[Al-Quhi] Quhi Abu al 

DeProny 

Prthudakasvami 

Ptolemy 

Puiseux 

Puissant 

Pythagoras 

Al-Qalasadi 

Quetelet 

Al-Quhi

[Quillen] Quillen Daniel 

[Quine] Quine Willard Van (1908-2000) 

[Renyi] Rényi Alfréd (1921-1970) 

[Rado] Radó Tibor (1895-1965) 

[Rademacher] Rademacher Hans (1892-1969) 

[RadoRichard] Rado Richard (1906-1989) 

[Radon] Radon Johann (1887-1956) 

[Rahn] Rahn Johann (1622-1676) 

[Rajagopal] Rajagopal Cadambathur (1903-1978) 

Quillen 

Quine 

Renyi 

Rado 

Rademacher 

RadoRichard 

Radon 

Rahn 

Rajagopal

[Ramanujam] Ramanujam Chidambaram (1938-1974) 

[Ramanujan] Ramanujan Srinivasa (1887-1920) 

[Ramsden] Ramsden Jesse (1735-1800) 

[Ramsey] Ramsey Frank (1903-1930) 

[Ramus] Ramus Peter (1515-1572) 

[Rankin] Rankin Robert (1915-2001) 

[Rankine] Rankine William (1820-1872) 

[Raphson] Raphson Joseph (1648-1715) 

[Rasiowa] Rasiowa Helena (1917-1994) 

Ramanujam 

Ramanujan 

Ramsden 

Ramsey 

Ramus 

Rankin 

Rankine 

Raphson 

Rasiowa

[Razmadze] Razmadze Andrei (1889-1929) 

[Recorde] Recorde Robert (1510-1558) 

[Rees] Rees Mina (1902-1997) 

[Regiomontanus] Regiomontanus Johann (1436-1476) 

[Reichenbach] Reichenbach Hans (1891-1953) 

[Reidemeister] Reidemeister Kurt (1893-1971) 

[Reiner] Reiner Irving (1924-1986) 

[Remak] Remak Robert (1888-1942) 

[Remez] Remez Evgeny (1896-1975) 

Razmadze 

Recorde 

Rees 

Regiomontanus 

Reichenbach 

Reidemeister 

Reiner 

Remak 

Remez

[ReyPastor] Rey Pastor Julio (1888-1962) 

[Reye] Reye Theodor (1838-1919) 

ReyPastor 

Reye 

DuBois-Reymond 

[DuBois-Reymond] Reymond Paul du Bois- (1831-1889) 

[Reynaud] Reynaud Antoine-André (1771-1844) 

[Reyneau] Reyneau Charles (1656-1728) 

[Reynolds] Reynolds Osborne (1842-1912) 

[DeRham] Rham Georges de (1903-1990) 

[Rheticus] Rheticus Georg Joachim (1514-1574) 

[Riccati] Riccati Jacopo (1676-1754) 

Reynaud 

Reyneau 

Reynolds 

DeRham 

Rheticus 

Riccati

[RiccatiVincenzo] Riccati Vincenzo (1707-1775) 

[RicciMatteo] Ricci Matteo (1552-1610) 

[Ricci] Ricci Michelangelo (1619-1682) 

RiccatiVincenzo 

RicciMatteo 

Ricci 

Ricci-Curbastro 

[Ricci-Curbastro] Ricci-Curbastro Georgorio (1853-1925) 

[RichardLouis] Richard Louis (1795-1849) 

[RichardJules] Richard Jules (1862-1956) 

[Richardson] Richardson Lewis (1881-1953) 

[Richer] Richer Jean (1630-1696) 

[Richmond] Richmond Herbert (1863-1948) 

RichardLouis 

RichardJules 

Richardson 

Richer 

Richmond

[Riemann] Riemann G F Bernhard (1826-1866) 

[Ries] Ries Adam (1492-1559) 

[RieszMarcel] Riesz Marcel (1886-1969) 

[Riesz] Riesz Frigyes (1880-1956) 

[Ringrose] Ringrose John 

[Roberts] Roberts Samuel (1827-1913) 

[Roberval] Roberval Gilles de (1602-1675) 

[Robins] Robins Benjamin (1707-1751) 

[RobinsonJulia] Robinson Julia Bowman (1919-1985) 

Riemann 

Ries 

RieszMarcel 

Riesz 

Ringrose 

Roberts 

Roberval 

Robins 

RobinsonJulia

[Robinson] Robinson Abraham (1918-1974) 

[Rocard] Rocard Yves-André (1903-1992) 

[LaRoche] Roche Estienne de La (1470-1530) 

[Rogers] Rogers Ambrose 

[Rohn] Rohn Karl (1855-1920) 

[Rolle] Rolle Michel (1652-1719) 

[Rosanes] Rosanes Jakob (1842-1922) 

[Rosenhain] Rosenhain Johann (1816-1887) 

[Rota] Rota Gian-Carlo (1932-1999) 

Robinson 

Rocard 

LaRoche 

Rogers 

Rohn 

Rolle 

Rosanes 

Rosenhain 

Rota

[Roth] Roth Leonard (1904-1968) 

[RothKlaus] Roth Klaus 

[Routh] Routh Edward (1831-1907) 

[Rudio] Rudio Ferdinand (1856-1929) 

[Rudolff] Rudolff Christoff (1499-1545) 

[Ruffini] Ruffini Paolo (1765-1822) 

[Runge] Runge Carle (1856-1927) 

[RussellScott] Russell John (1808-1882) 

[Russell] Russell Bertrand (1872-1970) 

Roth 

RothKlaus 

Routh 

Rudio 

Rudolff 

Ruffini 

Runge 

RussellScott 

Russell

[Rutherford] Rutherford Daniel E (1906-1966) 

[Rydberg] Rydberg Johannes (1854-1919) 

[Saccheri] Saccheri Giovanni (1667-1733) 

[Sacrobosco] Sacrobosco Johannes de (1195-1256) 

[Saks] Saks Stanislaw (1897-1942) 

[Nunez] Salaciense Pedro Nunez (1502-1587) 

[Salem] Salem Raphaël (1898-1963) 

[Salmon] Salmon George (1819-1904) 

[Al-Samarqandi] Samarqandi Shams al (1250-1310) 

Rutherford 

Rydberg 

Saccheri 

Sacrobosco 

Saks 

Nunez 

Salem 

Salmon 

Al-Samarqandi

[Al-Samawal] Samawal Ibn al (1130-1180) 

[Samoilenko] Samoilenko Anatoly 

[Sang] Sang Edward (1805-1890) 

[Sankara] Sankara Narayana (840-900) 

[Sasaki] Sasaki Shigeo 

[Saurin] Saurin Joseph (1659-1737) 

[Savage] Savage Leonard (1917-1971) 

[Savart] Savart Felix (1791-1841) 

[Savary] Savary Félix (1797-1841) 

Al-Samawal 

Samoilenko 

Sang 

Sankara 

Sasaki 

Saurin 

Savage 

Savart 

Savary

[Abraham] Savasorda (1070-1130) 

[Savile] Savile Sir Henry (1549-1622) 

[Schonflies] Schönflies Arthur (1853-1928) 

[Schatten] Schatten Robert (1911-1977) 

[Schauder] Schauder Juliusz (1899-1943) 

[Scheffe] Scheffé Henry (1907-1977) 

[Scheffers] Scheffers Georg (1866-1945) 

[Schickard] Schickard Wilhelm (1592-1635) 

[Schlafli] Schläfli Ludwig (1814-1895) 

Abraham 

Savile 

Schonflies 

Schatten 

Schauder 

Scheffe 

Scheffers 

Schickard 

Schlafli

[Schlomilch] Schlömilch Oscar (1823-1901) 

[Schmidt] Schmidt Erhard (1876-1959) 

[Schoenberg] Schoenberg Isaac (1903-1990) 

[Schottky] Schottky Friedrich (1851-1935) 

[Schoute] Schoute Pieter (1846-1923) 

[Schouten] Schouten Jan (1883-1971) 

[Schroder] Schröder Ernst (1841-1902) 

[Schrodinger] Schrödinger Erwin (1887-1961) 

[Schreier] Schreier Otto (1901-1929) 

Schlomilch 

Schmidt 

Schoenberg 

Schottky 

Schoute 

Schouten 

Schroder 

Schrodinger 

Schreier

[Schroeter] Schroeter Heinrich (1829-1892) 

[Schubert] Schubert Hermann (1848-1911) 

[Schur] Schur Issai (1875-1941) 

[Schwartz] Schwartz Laurent 

[SchwarzStefan] Schwarz Stefan (1914-1996) 

[Schwarz] Schwarz Herman (1843-1921) 

[Schwarzschild] Schwarzschild Karl (1873-1916) 

[Schwinger] Schwinger Julian (1918-1994) 

[Scott] Scott Charlotte (1858-1931) 

Schroeter 

Schubert 

Schur 

Schwartz 

SchwarzStefan 

Schwarz 

Schwarzschild 

Schwinger 

Scott

[Macintyre] Scott Sheila (1910-1960) 

[SegreBeniamino] Segre Beniamino (1903-1977) 

[SegreCorrado] Segre Corrado (1863-1924) 

[Seifert] Seifert Karl (1907-1996) 

[Selberg] Selberg Atle 

[Selten] Selten Reinhard 

[Semple] Semple Jack (1904-1985) 

[Serenus] Serenus (300-360) 

[Serre] Serre Jean-Pierre 

Macintyre 

SegreBeniamino 

SegreCorrado 

Seifert 

Selberg 

Selten 

Semple 

Serenus 

Serre

[Serret] Serret Joseph (1819-1885) 

[Servois] Servois Francois (1768-1847) 

[Severi] Severi Francesco (1879-1961) 

[Shanks] Shanks William (1812-1882) 

[Shannon] Shannon Claude (1916-2001) 

[Sharkovsky] Sharkovsky Oleksandr 

[Shatunovsky] Shatunovsky Samuil (1859-1929) 

[Shen] Shen Kua (1031-1095) 

[Shewhart] Shewhart Walter (1891-1967) 

Serret 

Servois 

Severi 

Shanks 

Shannon 

Sharkovsky 

Shatunovsky 

Shen 

Shewhart

[Shields] Shields Allen (1927-1989) 

[Shnirelman] Shnirelman Lev (1905-1938) 

[Shoda] Shoda Kenjiro (1902-1977) 

[Shtokalo] Shtokalo Josif (1897-1987) 

[Siacci] Siacci Francesco (1839-1907) 

[Siegel] Siegel Carl (1896-1981) 

[Sierpinski] Sierpinski Waclaw (1882-1969) 

[Siguenza] Siguenza y Gongora (1645-1700) 

[Al-Sijzi] Sijzi Abu al 

Shields 

Shnirelman 

Shoda 

Shtokalo 

Siacci 

Siegel 

Sierpinski 

Siguenza 

Al-Sijzi

[Simplicius] Simplicius Simplicius (490-560) 

[Simpson] Simpson Thomas (1710-1761) 

[Simson] Simson Robert (1687-1768) 

[Avicenna] Sina ibn 

[Sinan] Sinan ibn Thabit (880-943) 

[Sintsov] Sintsov Dmitrii (1867-1946) 

[Sitter] Sitter Willem de (1872-1934) 

[Skolem] Skolem Thoralf (1887-1963) 

[Slaught] Slaught Herbert (1861-1937) 

Simplicius 

Simpson 

Simson 

Avicenna 

Sinan 

Sintsov 

Sitter 

Skolem 

Slaught

[Sleszynski] Sleszynski Ivan (1854-1931) 

[Slutsky] Slutsky Evgeny (1880-1948) 

[Sluze] Sluze René de (1622-1685) 

[Smale] Smale Stephen 

[Smirnov] Smirnov Vladimir (1887-1974) 

[Smith] Smith Henry (1826-1883) 

[Sneddon] Sneddon Ian (1919-2000) 

[Snell] Snell Willebrord (1580-1626) 

[Snyder] Snyder Virgil (1869-1950) 

Sleszynski 

Slutsky 

Sluze 

Smale 

Smirnov 

Smith 

Sneddon 

Snell 

Snyder

[Sobolev] Sobolev Sergei (1908-1989) 

[Sokhotsky] Sokhotsky Yulian-Karl (1842-1927) 

[Sokolov] Sokolov Yurii (1896-1971) 

[Somerville] Somerville Mary (1780-1872) 

[Sommerfeld] Sommerfeld Arnold (1868-1951) 

[Sommerville] Sommerville Duncan (1879-1934) 

[Somov] Somov Osip (1815-1876) 

[Sonin] Sonin Nikolay (1849-1915) 

[Spanier] Spanier Edwin (1921-1996) 

Sobolev 

Sokhotsky 

Sokolov 

Somerville 

Sommerfeld 

Sommerville 

Somov 

Sonin 

Spanier

[Spence] Spence William (1777-1815) 

[Sporus] Sporus of Nicaea (240-300) 

[Spottiswoode] Spottiswoode William (1825-1883) 

[Sridhara] Sridhara Sridhara (870-930) 

[Sripati] Sripati (1019-1066) 

[Stackel] Stäckel Paul (1862-1919) 

[Stampioen] Stampioen Jan (1610-1690) 

[Steenrod] Steenrod Norman (1910-1971) 

[StefanJosef] Stefan Josef (1835-1893) 

Spence 

Sporus 

Spottiswoode 

Sridhara 

Sripati 

Stackel 

Stampioen 

Steenrod 

StefanJosef

[StefanPeter] Stefan Peter (1941-1978) 

[Steiner] Steiner Jakob (1796-1863) 

[Steinhaus] Steinhaus Hugo (1887-1972) 

[Steinitz] Steinitz Ernst (1871-1928) 

[Steklov] Steklov Vladimir A (1864-1926) 

[Stepanov] Stepanov Vyacheslaw V (1889-1950) 

[Stevin] Stevin Simon (1548-1620) 

[Stewart] Stewart Matthew (1717-1785) 

[Stewartson] Stewartson Keith (1925-1983) 

StefanPeter 

Steiner 

Steinhaus 

Steinitz 

Steklov 

Stepanov 

Stevin 

Stewart 

Stewartson

[Stieltjes] Stieltjes Thomas Jan (1856-1894) 

[Stifel] Stifel Michael (1487-1567) 

[Stirling] Stirling James (1692-1770) 

[Stokes] Stokes George Gabriel (1819-1903) 

[Stolz] Stolz Otto (1842-1905) 

[Stone] Stone Marshall (1903-1989) 

[Stott] Stott Alicia Boole (1860-1940) 

[Struik] Struik Dirk (1894-2000) 

[Rayleigh] Strutt (1842-1919) 

Stieltjes 

Stifel 

Stirling 

Stokes 

Stolz 

Stone 

Stott 

Struik 

Rayleigh

[Study] Study Eduard (1862-1930) 

[Sturm] Sturm J Charles-Francois (1803-1855) 

[SturmRudolf] Sturm Rudolf (1841-1919) 

[Subbotin] Subbotin Mikhail (1893-1966) 

[Suetuna] Suetuna Zyoiti (1898-1970) 

[Suter] Suter Heinrich (1848-1922) 

[Suvorov] Suvorov Georgii (1919-1984) 

[Swain] Swain Lorna (1891-1936) 

[Sylow] Sylow Ludwig (1832-1918) 

Study 

Sturm 

SturmRudolf 

Subbotin 

Suetuna 

Suter 

Suvorov 

Swain 

Sylow

[Sylvester] Sylvester James Joseph (1814-1897) 

[Synge] Synge John (1897-1995) 

[Szasz] Szäsz Otto (1884-1952) 

[Szego] Szegö Gäbor (1895-1985) 

[Tacquet] Tacquet Andrea (1612-1660) 

[Al-Baghdadi] Tahir ibn 

[Tait] Tait Peter Guthrie (1831-1901) 

[Takagi] Takagi Teiji (1875-1960) 

[Seki] Takakazu (1642-1708) 

Sylvester 

Synge 

Szasz 

Szego 

Tacquet 

Al-Baghdadi 

Tait 

Takagi 

Seki

[Talbot] Talbot Henry Fox (1800-1877) 

[Taniyama] Taniyama Yutaka (1927-1958) 

[TanneryPaul] Tannery Paul (1843-1904) 

[TanneryJules] Tannery Jules (1848-1910) 

[Tarry] Tarry Gaston (1843-1913) 

[Tarski] Tarski Alfred (1902-1983) 

[Tartaglia] Tartaglia Niccolo Fontana (1500-1557) 

[Tauber] Tauber Alfred (1866-1942) 

[Taurinus] Taurinus Franz (1794-1874) 

Talbot 

Taniyama 

TanneryPaul 

TanneryJules 

Tarry 

Tarski 

Tartaglia 

Tauber 

Taurinus

[Taussky-Todd] Taussky-Todd Olga 

[TaylorGeoffrey] Taylor Geoffrey (1886-1975) 

[Taylor] Taylor Brook (1685-1731) 

[Teichmuller] Teichmüller Oswald (1913-1943) 

[Temple] Temple George (1901-1992) 

[LeTenneur] Tenneur Jacques (1610-1660) 

[Tetens] Tetens Johannes (1736-1807) 

[Thabit] Thabit ibn Qurra Abu’l (826-901) 

[Thales] Thales of Miletus (624BC-546BC) 

Taussky-Todd 

TaylorGeoffrey 

Taylor 

Teichmuller 

Temple 

LeTenneur 

Tetens 

Thabit 

Thales

[Theaetetus] Theaetetus of Athens (415BC-369BC) 

[Theodorus] Theodorus of Cyrene (465BC-398BC) 

[Theodosius] Theodosius of Bithynia (160BC-90BC) 

[TheonofSmyrna] Theon of Smyrna (70AD-135) 

[Theon] Theon of Alexandria (335-395) 

[Thiele] Thiele Thorvald (1838-1910) 

[Thom] Thom René 

[Thomae] Thomae Johannes (1840-1921) 

[Thomason] Thomason Bob (1952-1995) 

Theaetetus 

Theodorus 

Theodosius 

TheonofSmyrna 

Theon 

Thiele 

Thom 

Thomae 

Thomason

[ThompsonJohn] Thompson John 

[ThompsonD’Arcy] Thompson D’Arcy W (1860-1948) 

[Thomson] Thomson W (1824-1907) 

[Thue] Thue Axel (1863-1922) 

[Thurston] Thurston Bill 

[Thymaridas] Thymaridas (400BC-350BC) 

[Tibbon] Tibbon Jacob ben (1236-1312) 

[Tietze] Tietze Heinrich (1880-1964) 

[Tilly] Tilly Joseph de (1837-1906) 

ThompsonJohn 

ThompsonD’Arcy 

Thomson 

Thue 

Thurston 

Thymaridas 

Tibbon 

Tietze 

Tilly

[Tinbergen] Tinbergen Jan (1903-1994) 

[Tinseau] Tinseau Charles (1748-1822) 

[Tisserand] Tisserand Félix (1845-1896) 

[Titchmarsh] Titchmarsh Edward (1899-1963) 

[Todd] Todd John (1908-1994) 

[Todhunter] Todhunter Isaac (1820-1884) 

[Toeplitz] Toeplitz Otto (1881-1940) 

[Torricelli] Torricelli Evangelista (1608-1647) 

[Trail] Trail William (1746-1831) 

Tinbergen 

Tinseau 

Tisserand 

Titchmarsh 

Todd 

Todhunter 

Toeplitz 

Torricelli 

Trail

[Tricomi] Tricomi Francesco (1897-1978) 

[Troughton] Troughton Edward (1753-1836) 

[Tsu] Tsu Ch’ung Chi (430-501) 

[Tukey] Tukey John (1915-2000) 

[Tunstall] Tunstall Cuthbert (1474-1559) 

[Turan] Turän Paul (1910-1976) 

[Turing] Turing Alan (1912-1954) 

[Turnbull] Turnbull Herbert (1885-1961) 

[Turner] Turner Peter (1586-1652) 

Tricomi 

Troughton 

Tsu 

Tukey 

Tunstall 

Turan 

Turing 

Turnbull 

Turner

[Al-TusiSharaf] Tusi Sharaf al (1135-1213) 

[Al-TusiNasir] Tusi Nasir al (1201-1274) 

[Tikhonov] Tychonoff Andrey (1906-1993) 

[UhlenbeckKaren] Uhlenbeck Karen 

[Uhlenbeck] Uhlenbeck George (1900-1988) 

[Ulam] Ulam Stanislaw (1909-1984) 

[UlughBeg] Ulugh Beg (1393-1449) 

[Al-Umawi] Umawi Abu al (1400-1489) 

[Upton] Upton Francis (1852-1921) 

Al-TusiSharaf 

Al-TusiNasir 

Tikhonov 

UhlenbeckKaren 

Uhlenbeck 

Ulam 

UlughBeg 

Al-Umawi 

Upton

[Al-Uqlidisi] Uqlidisi Abu’l al (920-980) 

[Urysohn] Urysohn Pavel (1898-1924) 

[Vacca] Vacca Giovanni (1872-1953) 

[Vailati] Vailati Giovanni (1863-1909) 

[DuVal] Val Patrick du (1903-1987) 

[Valerio] Valerio Luca (1552-1618) 

[ValleePoussin] Vallée Poussin C de (1866-1962) 

[Vandermonde] Vandermonde Alexandre (1735-1796) 

[Vandiver] Vandiver Harry (1882-1973) 

Al-Uqlidisi 

Urysohn 

Vacca 

Vailati 

DuVal 

Valerio 

ValleePoussin 

Vandermonde 

Vandiver

[Varahamihira] Varahamihira Varahamihira (505-587) 

[Varignon] Varignon Pierre (1654-1722) 

[Veblen] Veblen Oswald (1880-1960) 

[Saint-Venant] Venant Adhémar de St- (1797-1886) 

[Venn] Venn John (1834-1923) 

[Verhulst] Verhulst Pierre (1804-1849) 

[Vernier] Vernier Pierre (1584-1637) 

[Veronese] Veronese Giuseppe (1854-1917) 

[LeVerrier] Verrier Urbain Le (1811-1877) 

Varahamihira 

Varignon 

Veblen 

Saint-Venant 

Venn 

Verhulst 

Vernier 

Veronese 

LeVerrier

[Vessiot] Vessiot Ernest (1865-1952) 

[Viete] Viète Francois (1540-1603) 

[Vijayanandi] Vijayanandi 

[Saint-Vincent] Vincent Gregorius Saint- (1584-1667) 

[Leonardo] Vinci Leonardo da (1452-1519) 

[Vinogradov] Vinogradov Ivan (1891-1983) 

[Vitali] Vitali Giuseppe (1875-1932) 

[Viviani] Viviani Vincenzo (1622-1703) 

[Vlacq] Vlacq Adriaan (1600-1667) 

Vessiot 

Viete 

Vijayanandi 

Saint-Vincent 

Leonardo 

Vinogradov 

Vitali 

Viviani 

Vlacq

[VanVleck] Vleck Edward van (1863-1943) 

[Volterra] Volterra Vito (1860-1940) 

[Voronoy] Voronoy Georgy (1868-1908) 

[Vranceanu] Vranceanu Gheorghe (1900-1979) 

[VanderWaerden] Waerden Bartel van der (1903-1996) 

[Abu’l-Wafa] Wafa al-Buzjani Abu’l (940-998) 

[Wald] Wald Abraham (1902-1950) 

[WalkerJohn] Walker John (1825-1900) 

[WalkerArthur] Walker Geoffrey (1909-2001) 

VanVleck 

Volterra 

Voronoy 

Vranceanu 

VanderWaerden 

Abu’l-Wafa 

Wald 

WalkerJohn 

WalkerArthur

[Wall] Wall C Terence 

[Wallace] Wallace William (1768-1843) 

[Wallis] Wallis John (1616-1703) 

[Wang] Wang Hsien Chung (1918-1978) 

[Wangerin] Wangerin Albert (1844-1933) 

[Wantzel] Wantzel Pierre (1814-1894) 

[Waring] Waring Edward (1734-1798) 

[Watson] Watson G N (1886-1965) 

[WatsonHenry] Watson Henry (1827-1903) 

Wall 

Wallace 

Wallis 

Wang 

Wangerin 

Wantzel 

Waring 

Watson 

WatsonHenry

[Wazewski] Wazewski Tadeusz (1896-1972) 

[Weatherburn] Weatherburn Charles (1884-1974) 

[Weber] Weber Wilhelm (1804-1891) 

[WeberHeinrich] Weber Heinrich Martin (1842-1913) 

[Wedderburn] Wedderburn Joseph (1882-1948) 

[Weierstrass] Weierstrass Karl (1815-1897) 

[Weil] Weil André (1906-1998) 

[Weingarten] Weingarten Julius (1836-1910) 

[Weinstein] Weinstein Alexander (1897-1979) 

Wazewski 

Weatherburn 

Weber 

WeberHeinrich 

Wedderburn 

Weierstrass 

Weil 

Weingarten 

Weinstein

[Weisbach] Weisbach Julius (1806-1871) 

[Weldon] Weldon Raphael (1860-1906) 

[Werner] Werner Johann (1468-1522) 

[Wessel] Wessel Caspar (1745-1818) 

[West] West John (1756-1817) 

[Weyl] Weyl Hermann (1885-1955) 

[Weyr] Weyr Emil (1848-1894) 

[Wheeler] Wheeler Anna J Pell (1883-1966) 

[Whiston] Whiston William (1667-1752) 

Weisbach 

Weldon 

Werner 

Wessel 

West 

Weyl 

Weyr 

Wheeler 

Whiston

[White] White Henry (1861-1943) 

[Whitehead] Whitehead Alfred N (1861-1947) 

[WhiteheadHenry] Whitehead J Henry C (1904-1960) 

[Whitney] Whitney Hassler (1907-1989) 

[Whittaker] Whittaker Edmund (1873-1956) 

[WhittakerJohn] Whittaker John (1905-1984) 

[Whyburn] Whyburn Gordon (1904-1969) 

[Widman] Widman Johannes (1462-1498) 

[Wielandt] Wielandt Helmut (1910-2001) 

White 

Whitehead 

WhiteheadHenry 

Whitney 

Whittaker 

WhittakerJohn 

Whyburn 

Widman 

Wielandt

[Wien] Wien Wilhelm (1864-1928) 

[WienerNorbert] Wiener Norbert (1894-1964) 

[WienerChristian] Wiener Christian (1826-1896) 

[Wigner] Wigner Eugene (1902-1995) 

[Wilczynski] Wilczynski Ernest (1876-1932) 

[Wiles] Wiles Andrew 

[Wilkins] Wilkins John (1614-1672) 

[Wilkinson] Wilkinson Jim (1919-1986) 

[Wilks] Wilks Samuel (1906-1964) 

Wien 

WienerNorbert 

WienerChristian 

Wigner 

Wilczynski 

Wiles 

Wilkins 

Wilkinson 

Wilks

[Ockham] William of Ockham (1285-1349) 

[WilsonEdwin] Wilson Edwin (1879-1964) 

[WilsonJohn] Wilson John (1741-1793) 

[WilsonAlexander] Wilson Alexander (1714-1786) 

[Winkler] Winkler Wilhelm (1884-1984) 

[Wintner] Wintner Aurel (1903-1958) 

[Wirtinger] Wirtinger Wilhelm (1865-1945) 

[Wishart] Wishart John (1898-1956) 

[DeWitt] Witt Johan de (1625-1672) 

Ockham 

WilsonEdwin 

WilsonJohn 

WilsonAlexander 

Winkler 

Wintner 

Wirtinger 

Wishart 

DeWitt

[Witt] Witt Ernst (1911-1991) 

[Witten] Witten Edward 

[Wittgenstein] Wittgenstein Ludwig (1889-1951) 

[Wolf] Wolf Rudolph (1816-1893) 

[Wolfowitz] Wolfowitz Jacob (1910-1981) 

[Wolstenholme] Wolstenholme Joseph (1829-1891) 

[Woodhouse] Woodhouse Robert (1773-1827) 

[Woodward] Woodward Robert (1849-1924) 

[Wren] Wren Sir Christopher (1632-1723) 

Witt 

Witten 

Wittgenstein 

Wolf 

Wolfowitz 

Wolstenholme 

Woodhouse 

Woodward 

Wren

[Wronski] Wronski Hoëné (1778-1853) 

[Xenocrates] Xenocrates of Chalcedon (396BC-314BC) 

[Yang] Yang Hui (1238-1298) 

[Yates] Yates Frank (1902-1994) 

[Yativrsabha] Yativrsabha (500-570) 

[Yau] Yau Shing-Tung 

[Yavanesvara] Yavanesvara (120-180) 

[Yoccoz] Yoccoz Jean-Christophe 

[Youden] Youden William (1900-1971) 

Wronski 

Xenocrates 

Yang 

Yates 

Yativrsabha 

Yau 

Yavanesvara 

Yoccoz 

Youden

[Young] Young William (1863-1942) 

[YoungAlfred] Young Alfred (1873-1940) 

[ChisholmYoung] Young Grace Chisholm (1868-1944) 

[Yule] Yule George (1871-1951) 

[Yunus] Yunus Abu’l-Hasan ibn 

[Yushkevich] Yushkevich Adolph P (1906-1993) 

[Ahmed] Yusuf Ahmed ibn (835-912) 

[QadiZada] Zada al-Rumi Qadi (1364-1436) 

[Vashchenko] Zakharchenko M V- (1825-1912) 

Young 

YoungAlfred 

ChisholmYoung 

Yule 

Yunus 

Yushkevich 

Ahmed 

QadiZada 

Vashchenko

[Zarankiewicz] Zarankiewicz Kazimierz (1902-1959) 

[Zaremba] Zaremba Stanislaw (1863-1942) 

[Zariski] Zariski Oscar (1899-1986) 

[Zassenhaus] Zassenhaus Hans (1912-1991) 

[Zeckendorf] Zeckendorf Edouard (1901-1983) 

[Zeeman] Zeeman Chris 

[Zelmanov] Zelmanov Efim 

[ZenoofSidon] Zeno of Sidon (150BC-70BC) 

[ZenoofElea] Zeno of Elea (490BC-430BC) 

Zarankiewicz 

Zaremba 

Zariski 

Zassenhaus 

Zeckendorf 

Zeeman 

Zelmanov 

ZenoofSidon 

ZenoofElea

[Zenodorus] Zenodorus (200BC-140BC) 

[Zermelo] Zermelo Ernst (1871-1951) 

[Zeuthen] Zeuthen Hieronymous (1839-1920) 

[Chu] Zhu Shie-jie (1270-1330) 

[Zhukovsky] Zhukovsky Nikolay (1847-1921) 

[Zolotarev] Zolotarev Egor (1847-1878) 

[Zorn] Zorn Max (1906-1993) 

[Zuse] Zuse Konrad (1910-1995) 

[Zygmund] Zygmund Antoni (1900-1992) 

Zenodorus 

Zermelo 

Zeuthen 

Chu 

Zhukovsky 

Zolotarev 

Zorn 

Zuse 

Zygmund

[D’Alembert] d’Alembert Jean (1717-1783) 

D’Alembert 

BudandeBoislaurent 

[BudandeBoislaurent] de Boislaurent Budan (1761-1840) 

[Coulomb] de Coulomb Charles (1736-1806) 

[DeBeaune] de Beaune Florimond (1601-1652) 

[Carcavi] de Carcavi Pierre (1600-1684) 

[Broglie] de Broglie Louis duc (1892-1987) 

[Bougainville] de Bougainville Louis (1729-1811) 

[Billy] de Billy Jacques (1602-1679) 

[Coriolis] de Coriolis Gustave (1792-1843) 

Coulomb 

DeBeaune 

Carcavi 

Broglie 

Bougainville 

Billy 

Coriolis

[Hunayn] ibn Ishaq Hunayn (808-873) 

[Lansberge] van Lansberge Philip (1561-1632) 

[Roomen] van Roomen Adriaan (1561-1615) 

[Schooten] van Schooten Frans (1615-1660) 

[Dantzig] van Dantzig David (1900-1959) 

[Heuraet] van Heuraet Hendrik (1633-1660) 

[VanCeulen] van Ceulen Ludolph (1540-1610) 

[Amringe] van Amringe Howard (1835-1915) 

[Geiringer] von Mises Hilda Geiringer (1893-1973) 

Hunayn 

Lansberge 

Roomen 

Schooten 

Dantzig 

Heuraet 

VanCeulen 

Amringe 

Geiringer

[Helmholtz] von Helmholtz Hermann (1821-1894) 

[Lindemann] von Lindemann Carl (1852-1939) 

[Eotvos] von Eötvös Roland (1848-1919) 

[Segner] von Segner Johann (1704-1777) 

[Seidel] von Seidel Philipp (1821-1896) 

[VonNeumann] von Neumann John (1903-1957) 

[Vega] von Vega Georg (1754-1802) 

[Mises] von Mises Richard (1883-1953) 

[VonDyck] von Dyck Walther (1856-1934) 

Helmholtz 

Lindemann 

Eotvos 

Segner 

Seidel 

VonNeumann 

Vega 

Mises 

VonDyck

[Koch] von Koch Helge (1870-1924) 

[Tschirnhaus] von Tschirnhaus E (1651-1708) 

[Karman] von Kärmän Theodore (1881-1963) 

[Leibniz] von Leibniz Gottfried (1646-1719) 

[VonStaudt] von Staudt Karl (1798-1867) 

[VonBrill] von Brill Alexander (1842-1935) 

Koch 

Tschirnhaus 

Karman 

Leibniz 

VonStaudt 

VonBrill 



Index 

Abbe, 1 

Abel, 1 

Abraham, 129 

AbrahamMax, 1 

Abu’l-Wafa, 153 

AbuKamil, 76 

Ackermann, 1 

Adams, 1 

AdamsFrank, 1 

Adelard, 1 

Adler, 1 

Adrain, 2 

Aepinus, 2 

Agnesi, 2 

Ahlfors, 2 

Ahmed, 162 

Ahmes, 2 

Aida, 2 

Aiken, 2 

Airy, 2 

Aitken, 2 

Ajima, 3 

Akhiezer, 3 

Al-Baghdadi, 142 

Al-Banna, 96 

Al-Battani, 11 

Al-Biruni, 15 

Al-Farisi, 76 

Al-Haytham, 63 

Al-Jawhari, 72 

Al-Jayyani, 72 

Al-Karaji, 76 

Al-Kashi, 77 

Al-Khalili, 78 

Al-Khazin, 78 

Al-Khujandi, 78 

Al-Khwarizmi, 78 

Al-Kindi, 79 

Al-Maghribi, 93 

Al-Mahani, 94 

Al-Nasawi, 104 

Al-Nayrizi, 105 

Al-Qalasadi, 118 

Al-Quhi, 118 

Al-Samarqandi, 127 

Al-Samawal, 128 

Al-Sijzi, 134 

Al-TusiNasir, 149 

Al-TusiSharaf, 149 

Al-Umawi, 149 

Al-Uqlidisi, 150 

Albanese, 3 

Albert, 3 

AlbertAbraham, 3 

Alberti, 3 

Albertus, 3 

Alcuin, 3 

Aleksandrov, 4 

AleksandrovAleksandr, 4 

Alexander, 3 

169 

AlexanderArchie, 4 

Ampere, 4 

Amringe, 166 

Amsler, 4 

Anaxagoras, 4 

Anderson, 4 

Andreev, 4 

Angeli, 4 

Anstice, 5 

Anthemius, 5 

Antiphon, 5 

Apastamba, 5 

Apollonius, 5 

Appell, 5 

Arago, 5 

Arbogast, 5 

Arbuthnot, 5 

Archimedes, 6 

Archytas, 6 

Arf, 6 

Argand, 6 

Aristaeus, 6 

Aristarchus, 6 

Aristotle, 6 

Arnauld, 6 

Aronhold, 6 

Artin, 7 

AryabhataI, 7 

AryabhataII, 7 

Atiyah, 7 

Atwood, 7 

Auslander, 7 

Autolycus, 7 

Avicenna, 135 

Babbage, 8 

Bachet, 8 

Bachmann, 8 

Backus, 8 

Bacon, 8 

Baer, 8 

Baire, 8 

Baker, 9 

BakerAlan, 8 

Ball, 9 

Balmer, 9 

Banach, 9 

Banneker, 9 

BanuMusa, 9 

BanuMusaAhmad, 9 

BanuMusaal-Hasan, 9 

BanuMusaMuhammad, 9 

Barbier, 10 

Bari, 10 

Barlow, 10 

Barnes, 10 

Barocius, 10 

Barrow, 10 

Bartholin, 10 

Batchelor, 10

Bateman, 10 

Battaglini, 11 

Baudhayana, 11 

Bayes, 11 

Beaugrand, 11 

Bell, 11 

Bellavitis, 11 

Beltrami, 11 

Bendixson, 11 

Benedetti, 12 

Bergman, 12 

Berkeley, 12 

Bernays, 12 

BernoulliDaniel, 12 

BernoulliJacob, 12 

BernoulliJohann, 12 

BernoulliNicolaus, 12 

Bernstein, 12 

BernsteinFelix, 13 

Bers, 13 

Bertini, 13 

Bertrand, 13 

Berwald, 13 

Berwick, 13 

Besicovitch, 13 

Bessel, 13 

Betti, 13 

Beurling, 14 

Bezout, 7 

BhaskaraI, 14 

BhaskaraII, 14 

Bianchi, 14 

Bieberbach, 14 

Bienayme, 14 

Billy, 165 

Binet, 14 

Bing, 14 

Biot, 14 

Birkhoff, 15 

BirkhoffGarrett, 15 

BjerknesCarl, 15 

BjerknesVilhelm, 15 

Black, 15 

Blaschke, 15 

Blichfeldt, 15 

Bliss, 15 

Bloch, 16 

Bobillier, 16 

Bocher, 7 

Bochner, 16 

Boethius, 16 

Boggio, 16 

Bohl, 16 

BohrHarald, 16 

BohrNiels, 16 

Boltzmann, 16 

Bolyai, 17 

BolyaiFarkas, 17 

Bolza, 17 

Bolzano, 17 

Bombelli, 17 

Bombieri, 17 

Bonferroni, 17 

Bonnet, 17 

Boole, 17 

Boone, 18 

Borchardt, 18 

Borda, 18 

Borel, 18 

Borgi, 18 

Born, 18 

Borsuk, 18 

Bortkiewicz, 18 

Bortolotti, 18 

Bosanquet, 19 

Boscovich, 19 

Bose, 19 

Bossut, 19 

Bougainville, 165 

Bouguer, 19 

Boulliau, 19 

Bouquet, 19 

Bour, 19 

Bourbaki, 19 

Bourgain, 20 

Boutroux, 20 

Bowditch, 20 

Bowen, 20 

Boyle, 20 

Boys, 20 

Bradwardine, 20 

Brahe, 20 

Brahmadeva, 20 

Brahmagupta, 21 

Braikenridge, 21 

Bramer, 21 

Brashman, 21 

Brauer, 21 

BrauerAlfred, 21 

Brianchon, 21 

Briggs, 21 

Brillouin, 21 

Bring, 22 

Brioschi, 22 

Briot, 22 

Brisson, 22 

Britton, 22 

Brocard, 22 

Brodetsky, 22 

Broglie, 165 

Bromwich, 22 

Bronowski, 22 

Brouncker, 23 

Brouwer, 23 

Brown, 23 

Browne, 23 

Bruno, 23 

Bruns, 23 

Bryson, 23 

BudandeBoislaurent, 165 

Buffon, 23 

Bugaev, 23 

Bukreev, 24 

Bunyakovsky, 24

Burali-Forti, 50 

Burchnall, 24 

Burgi, 8 

Burkhardt, 24 

Burkill, 24 

Burnside, 24 

Caccioppoli, 24 

Cajori, 24 

Calderon, 24 

Callippus, 25 

Campanus, 25 

Campbell, 25 

Camus, 25 

Cannell, 25 

Cantelli, 25 

Cantor, 25 

CantorMoritz, 25 

Caramuel, 25 

Caratheodory, 26 

Carcavi, 165 

Cardan, 26 

Carlitz, 26 

Carlyle, 26 

Carnot, 26 

CarnotSadi, 26 

Carslaw, 26 

Cartan, 26 

CartanHenri, 26 

Cartwright, 27 

Casorati, 27 

Cassels, 27 

Cassini, 27 

Castel, 27 

Castelnuovo, 27 

Castigliano, 27 

Castillon, 27 

Catalan, 27 

Cataldi, 28 

Cauchy, 28 

Cavalieri, 28 

Cayley, 28 

Cech, 28 

Cesaro, 28 

CevaGiovanni, 28 

CevaTommaso, 28 

Ch’in, 30 

Chandrasekhar, 29 

Chang, 29 

Chaplygin, 29 

Chapman, 29 

Chasles, 29 

Chatelet, 28 

Chebotaryov, 29 

Chebyshev, 29 

Chern, 29 

Chernikov, 29 

Chevalley, 30 

ChisholmYoung, 162 

Chowla, 30 

Christoffel, 30 

Chrysippus, 30 

Chrystal, 30 

Chu, 164 

Chuquet, 30 

Church, 30 

Clairaut, 30 

Clapeyron, 31 

Clarke, 31 

Clausen, 31 

Clausius, 31 

Clavius, 31 

Clebsch, 31 

Cleomedes, 31 

Clifford, 31 

Coates, 31 

Coble, 32 

Cochran, 32 

Cocker, 32 

Codazzi, 32 

Cohen, 32 

Cole, 32 

Collingwood, 32 

Collins, 32 

Condorcet, 32 

Connes, 33 

Conon, 33 

Conway, 33 

ConwayArthur, 33 

Coolidge, 33 

Cooper, 33 

Copernicus, 33 

Copson, 33 

Coriolis, 165 

Cosserat, 33 

Cotes, 34 

Coulomb, 165 

Courant, 34 

Cournot, 34 

Couturat, 34 

Cox, 34 

Coxeter, 34 

Craig, 34 

Cramer, 34 

CramerHarald, 34 

Crank, 35 

Crelle, 35 

Cremona, 35 

Crighton, 35 

Cunha, 35 

Cunningham, 35 

Curry, 35 

Cusa, 35 

D’Alembert, 165 

D’Ovidio, 109 

Dandelin, 36 

Danti, 36 

Dantzig, 166 

DantzigGeorge, 36 

Darboux, 36 

Darwin, 36 

Dase, 36 

Davenport, 36 

Davidov, 36 

Davies, 36

DeBeaune, 165 

Dechales, 37 

Dedekind, 37 

Dee, 37 

DeGroot, 59 

Dehn, 37 

DeL’Hopital, 83 

Delamain, 37 

Delambre, 37 

Delaunay, 37 

Deligne, 37 

Delone, 37 

Delsarte, 38 

Democritus, 38 

DeMoivre, 101 

DeMorgan, 102 

Denjoy, 38 

Deparcieux, 38 

DeProny, 118 

DeRham, 122 

Desargues, 38 

Descartes, 38 

DeWitt, 159 

Dickson, 38 

Dickstein, 38 

Dieudonne, 38 

Digges, 39 

Dilworth, 39 

Dinghas, 39 

Dini, 39 

Dinostratus, 39 

Diocles, 39 

Dionis, 39 

Dionysodorus, 39 

Diophantus, 39 

Dirac, 40 

Dirichlet, 40 

Dixon, 40 

DixonArthur, 40 

Dodgson, 40 

Doeblin, 40 

Domninus, 40 

Donaldson, 40 

Doob, 40 

Doppelmayr, 41 

Doppler, 41 

Douglas, 41 

Dowker, 41 

Drach, 41 

Drinfeld, 41 

DuBois-Reymond, 122 

Dubreil, 41 

Dudeney, 41 

Duhamel, 41 

Duhem, 42 

Dupin, 42 

Dupre, 42 

Durer, 35 

DuVal, 150 

Dynkin, 42 

EckertJohn, 42 

EckertWallace, 42 

Eckmann, 42 

Eddington, 42 

Edge, 42 

Edgeworth, 43 

Egorov, 43 

Ehrenfest, 43 

Ehresmann, 43 

Eilenberg, 43 

Einstein, 43 

Eisenhart, 43 

Eisenstein, 43 

Elliott, 43 

Empedocles, 44 

Engel, 44 

Enriques, 44 

Enskog, 44 

Eotvos, 167 

Epstein, 44 

Eratosthenes, 44 

Erdelyi, 44 

Erdos, 44 

Erlang, 44 

Escher, 45 

Esclangon, 45 

Euclid, 45 

Eudemus, 45 

Eudoxus, 45 

Euler, 45 

Eutocius, 45 

Evans, 45 

Ezra, 45 

FaadiBruno, 46 

Faber, 46 

Fabri, 46 

FagnanoGiovanni, 46 

FagnanoGiulio, 46 

Faltings, 46 

Fano, 46 

Faraday, 46 

Farey, 46 

Fatou, 47 

Faulhaber, 47 

Fefferman, 47 

Feigenbaum, 47 

Feigl, 47 

Fejer, 47 

Feller, 47 

Fermat, 47 

Ferrar, 47 

Ferrari, 48 

Ferrel, 48 

Ferro, 48 

Feuerbach, 48 

Feynman, 48 

Fibonacci, 114 

Fields, 48 

Finck, 48 

Fincke, 48 

Fine, 49 

FineHenry, 48 

Finsler, 49 

Fischer, 49

Fisher, 49 

Fiske, 49 

FitzGerald, 49 

Flamsteed, 49 

Flugge-Lotz, 49 

Fomin, 49 

FontainedesBertins, 50 

Fontenelle, 50 

Forsyth, 50 

Fourier, 50 

Fowler, 50 

Fox, 50 

Fraenkel, 50 

FrancaisFrancois, 51 

FrancaisJacques, 51 

Francesca, 114 

Francoeur, 51 

Frank, 51 

Franklin, 51 

FranklinBenjamin, 51 

Frattini, 51 

Frechet, 50 

Fredholm, 51 

Freedman, 51 

Frege, 52 

Freitag, 52 

Frenet, 52 

FrenicledeBessy, 52 

Frenkel, 52 

Fresnel, 52 

Freudenthal, 52 

Freundlich, 52 

Friedmann, 52 

Friedrichs, 53 

Frisi, 53 

Frobenius, 53 

Fubini, 53 

Fuchs, 53 

Fueter, 53 

Fuller, 53 

Fuss, 53 

Galerkin, 54 

Galileo, 54 

Gallarati, 54 

Galois, 54 

Galton, 54 

Gassendi, 54 

Gauss, 54 

Gegenbauer, 54 

Geiringer, 166 

Geiser, 55 

Gelfand, 55 

Gelfond, 55 

Gellibrand, 55 

Geminus, 55 

GemmaFrisius, 55 

Genocchi, 55 

Gentzen, 55 

Gergonne, 55 

Germain, 56 

Gherard, 56 

Ghetaldi, 56 

Gibbs, 56 

GirardAlbert, 56 

GirardPierre, 56 

Glaisher, 56 

Glenie, 56 

Gnedenko, 67 

Godel, 53 

Gohberg, 56 

Goldbach, 57 

Goldstein, 57 

Gompertz, 57 

Goodstein, 57 

Gopel, 54 

Gordan, 57 

Gorenstein, 57 

Gosset, 57 

Goursat, 57 

Govindasvami, 57 

Graffe, 58 

Gram, 58 

Grandi, 58 

Granville, 58 

Grassmann, 58 

Grave, 58 

Green, 58 

Greenhill, 58 

Gregory, 58 

GregoryDavid, 59 

GregoryDuncan, 59 

Grosseteste, 59 

Grossmann, 59 

Grothendieck, 59 

Grunsky, 59 

Guarini, 59 

Guccia, 59 

Gudermann, 60 

Guenther, 60 

Guinand, 60 

Guldin, 60 

Gunter, 60 

Haar, 61 

Hachette, 61 

Hadamard, 61 

Hadley, 61 

Hahn, 61 

Hajek, 60 

Hall, 61 

Halley, 61 

HallMarshall, 61 

Halmos, 61 

Halphen, 62 

Halsted, 62 

Hamill, 62 

Hamilton, 62 

HamiltonWilliam, 62 

Hamming, 62 

Hankel, 62 

Hardy, 62 

HardyClaude, 62 

Harish-Chandra, 63 

Harriot, 63 

Hartley, 63

Hartree, 63 

Hasse, 63 

Hausdorff, 63 

Hawking, 63 

Heath, 63 

Heaviside, 64 

Heawood, 64 

Hecht, 64 

Hecke, 64 

Hedrick, 64 

Heegaard, 64 

Heilbronn, 64 

Heine, 64 

Heisenberg, 64 

Hellinger, 65 

Helly, 65 

Helmholtz, 167 

Heng, 65 

Henrici, 65 

Hensel, 65 

Heraclides, 65 

Herbrand, 65 

Herigone, 60 

Hermann, 65 

HermannofReichenau, 85 

Hermite, 65 

Heron, 66 

Herschel, 66 

HerschelCaroline, 66 

Herstein, 66 

Hesse, 66 

Heuraet, 166 

Heyting, 66 

Higman, 66 

Hilbert, 66 

Hill, 66 

Hille, 67 

Hindenburg, 67 

Hipparchus, 67 

Hippias, 67 

Hippocrates, 67 

Hironaka, 67 

Hirsch, 67 

Hirst, 67 

Hobbes, 68 

Hobson, 68 

Hodge, 68 

Holder, 60 

Hollerith, 68 

Holmboe, 68 

Honda, 68 

Hooke, 68 

Hopf, 69 

HopfEberhard, 68 

Hopkins, 69 

Hopkinson, 69 

Hopper, 69 

Hormander, 60 

Horner, 69 

Houel, 68 

Householder, 69 

Hsu, 69 

Hubble, 69 

Hudde, 69 

HumbertGeorges, 70 

HumbertPierre, 70 

Hunayn, 166 

Huntington, 70 

Hurewicz, 70 

Hurwitz, 70 

Hutton, 70 

Huygens, 70 

Hypatia, 70 

Hypsicles, 70 

Ibrahim, 71 

Ingham, 71 

Ito, 71 

Ivory, 71 

Iwasawa, 71 

Iyanaga, 71 

JabiribnAflah, 71 

Jacobi, 71 

Jacobson, 71 

Jagannatha, 72 

James, 72 

Janiszewski, 72 

Janovskaja, 72 

Jarnik, 72 

Jeans, 72 

Jeffrey, 72 

Jeffreys, 73 

Jensen, 73 

Jerrard, 73 

Jevons, 73 

Joachimsthal, 73 

John, 73 

Johnson, 73 

JohnsonBarry, 73 

Jones, 74 

JonesBurton, 73 

JonesVaughan, 74 

Jonquieres, 74 

Jordan, 74 

Jordanus, 74 

Jourdain, 74 

Juel, 74 

Julia, 74 

Jungius, 74 

Jyesthadeva, 75 

Kac, 75 

Kaestner, 75 

Kagan, 75 

Kakutani, 75 

Kalmar, 76 

Kaluza, 76 

Kaluznin, 76 

Kamalakara, 76 

Kantorovich, 76 

Kaplansky, 76 

Karman, 168 

Karp, 77 

Katyayana, 77

Keill, 77 

Kelland, 77 

Kellogg, 77 

Kemeny, 77 

Kempe, 77 

Kendall, 78 

KendallMaurice, 77 

Kepler, 78 

Kerekjarto, 78 

Keynes, 78 

Khayyam, 108 

Khinchin, 78 

Killing, 79 

Kingman, 79 

Kirchhoff, 79 

Kirkman, 79 

Kleene, 79 

Klein, 79 

KleinOskar, 79 

Klingenberg, 80 

Kloosterman, 80 

Klugel, 79 

Kneser, 80 

KneserHellmuth, 80 

Knopp, 80 

Kober, 80 

Koch, 168 

Kochin, 80 

Kodaira, 80 

Koebe, 80 

Koenigs, 81 

Kolmogorov, 81 

Kolosov, 81 

KonigDenes, 81 

KonigJulius, 75 

KonigSamuel, 75 

Konigsberger, 75 

Korteweg, 81 

Kotelnikov, 81 

Kovalevskaya, 81 

Kramer, 86 

Kramp, 81 

Krawtchouk, 81 

Krein, 82 

Kreisel, 82 

Kronecker, 82 

Krull, 82 

KrylovAleksei, 82 

KrylovNikolai, 82 

Kulik, 82 

Kumano-Go, 82 

Kummer, 82 

Kuratowski, 83 

Kurosh, 83 

Kurschak, 75 

Kutta, 83 

Kuttner, 83 

LaCondamine, 84 

Lacroix, 84 

LaFaille, 84 

Lagny, 84 

Lagrange, 84 

Laguerre, 84 

LaHire, 84 

Lakatos, 84 

Lalla, 84 

Lamb, 85 

Lambert, 85 

Lame, 85 

Lamy, 85 

Lanczos, 85 

Landau, 85 

LandauLev, 85 

Landen, 85 

Landsberg, 86 

Langlands, 86 

Lansberge, 166 

Laplace, 86 

Larmor, 86 

LaRoche, 125 

Lasker, 86 

LaurentHermann, 86 

LaurentPierre, 86 

Lavanha, 86 

Lavrentev, 87 

Lax, 87 

Lebesgue, 87 

Ledermann, 87 

Leech, 87 

LeFevre, 87 

Lefschetz, 87 

Legendre, 87 

Leger, 83 

Leibniz, 168 

Lemoine, 87 

Leonardo, 152 

LePaige, 110 

Leray, 88 

Lerch, 88 

Leshniewski, 88 

Leslie, 88 

LeTenneur, 144 

Leucippus, 88 

LeVerrier, 151 

Levi, 88 

Levi-Civita, 88 

Levinson, 88 

LevyHyman, 88 

LevyPaul, 83 

Levytsky, 89 

Lexell, 89 

Lexis, 89 

Lhuilier, 89 

Libri, 89 

Lie, 89 

Lifshitz, 89 

Lighthill, 89 

Lindelof, 89 

Lindemann, 167 

Linnik, 90 

Lions, 90 

Liouville, 90 

Lipschitz, 90 

Lissajous, 90

Listing, 90 

Littlewood, 90 

LittlewoodDudley, 90 

Livsic, 90 

Llull, 91 

Lobachevsky, 91 

Loewner, 83 

Loewy, 91 

Lopatynsky, 91 

Lorentz, 91 

Love, 91 

Lovelace, 91 

Lowenheim, 83 

Loyd, 91 

Lucas, 91 

Lueroth, 92 

Lukacs, 92 

Lukasiewicz, 92 

Luke, 92 

Luzin, 92 

Lyapunov, 92 

Lyndon, 92 

Macaulay, 93 

MacCullagh, 93 

Macdonald, 93 

Macintyre, 132 

MacLane, 93 

Maclaurin, 93 

MacMahon, 93 

Madhava, 93 

Magnitsky, 93 

Magnus, 94 

Mahavira, 94 

MahendraSuri, 94 

Mahler, 94 

Maior, 94 

Malcev, 94 

Malebranche, 94 

Malfatti, 94 

Malus, 95 

Manava, 95 

Mandelbrot, 95 

Mannheim, 95 

Mansion, 95 

Mansur, 95 

Marchenko, 95 

Marcinkiewicz, 95 

Marczewski, 95 

Margulis, 96 

Marinus, 96 

Markov, 96 

Mascheroni, 96 

Maschke, 96 

Maseres, 96 

Maskelyne, 96 

Mason, 96 

Mathews, 97 

MathieuClaude, 97 

MathieuEmile, 97 

Matsushima, 97 

Mauchly, 97 

Maupertuis, 97 

Maurolico, 97 

Maxwell, 97 

MayerAdolph, 97 

MayerTobias, 98 

Mazur, 98 

Mazurkiewicz, 98 

McClintock, 98 

McDuff, 98 

McShane, 98 

Meissel, 98 

Mellin, 98 

Menabrea, 98 

Menaechmus, 99 

Menelaus, 99 

Menger, 99 

Mengoli, 99 

Menshov, 99 

Meray, 92 

MercatorGerardus, 99 

MercatorNicolaus, 99 

Mercer, 99 

Merrifield, 99 

Merrill, 100 

Mersenne, 100 

Mertens, 100 

Meshchersky, 100 

Meyer, 100 

Miller, 100 

Milne, 100 

Milnor, 100 

Minding, 100 

Mineur, 101 

Minkowski, 101 

Mirsky, 101 

Mises, 167 

Mittag-Leffler, 101 

Mobius, 92 

Mohr, 101 

Molin, 101 

Monge, 101 

Monte, 101 

Montel, 102 

Montmort, 102 

Montucla, 102 

MooreEliakim, 102 

MooreJonas, 102 

MooreRobert, 102 

Morawetz, 102 

Mordell, 102 

Mori, 103 

Morin, 103 

MorinJean-Baptiste, 103 

Morley, 103 

Morse, 103 

Mostowski, 103 

Motzkin, 103 

Moufang, 103 

Mouton, 103 

Muir, 104 

Mumford, 104 

Mydorge, 104 

Mytropolshy, 104

Naimark, 104 

Napier, 104 

Narayana, 104 

Nash, 104 

Navier, 105 

Neile, 105 

Nekrasov, 105 

Netto, 105 

Neuberg, 105 

Neugebauer, 105 

NeumannBernhard, 106 

NeumannCarl, 105 

NeumannFranz, 106 

NeumannHanna, 105 

Nevanlinna, 106 

Newcomb, 106 

Newman, 106 

Newton, 106 

Neyman, 106 

Nicolson, 106 

Nicomachus, 106 

Nicomedes, 107 

Nielsen, 107 

NielsenJakob, 107 

Nightingale, 107 

Nilakantha, 107 

Niven, 107 

NoetherEmmy, 107 

NoetherMax, 107 

Novikov, 107 

NovikovSergi, 108 

Nunez, 127 

Ockham, 159 

Oenopides, 108 

Ohm, 108 

Oka, 108 

Olivier, 108 

Oresme, 108 

Orlicz, 108 

Ortega, 108 

Osgood, 109 

Osipovsky, 109 

Ostrogradski, 109 

Ostrowski, 109 

Oughtred, 109 

Ozanam, 109 

Pacioli, 110 

Pade, 110 

Padoa, 110 

Painleve, 110 

Paley, 110 

Paman, 110 

Panini, 110 

Papin, 111 

Pappus, 111 

Pars, 111 

Parseval, 111 

Pascal, 111 

PascalEtienne, 111 

Pasch, 111 

Patodi, 111 

Pauli, 111 

Peacock, 112 

Peano, 112 

Pearson, 112 

PearsonEgon, 112 

PeirceBenjamin, 112 

PeirceCharles, 112 

Pell, 112 

Penney, 112 

Peres, 109 

Perron, 112 

Perseus, 113 

Peter, 109 

Petersen, 113 

Peterson, 113 

Petit, 113 

Petrovsky, 113 

Petryshyn, 113 

Petzval, 113 

Peurbach, 113 

Pfaff, 113 

Pfeiffer, 114 

Philon, 114 

PicardEmile, 114 

PicardJean, 114 

Pieri, 114 

Pillai, 114 

Pincherle, 114 

Pitiscus, 115 

Plana, 115 

Planck, 115 

Plateau, 115 

Plato, 115 

Playfair, 115 

Plessner, 115 

Plucker, 115 

Poincare, 115 

Poinsot, 116 

Poisson, 116 

Poleni, 116 

Polozii, 116 

Polya, 110 

Poncelet, 116 

Pontryagin, 116 

Poretsky, 116 

Porphyry, 116 

Porta, 116 

Posidonius, 117 

Post, 117 

Potapov, 117 

Pratt, 117 

Pringsheim, 117 

Privalov, 117 

PrivatdeMolieres, 117 

Proclus, 117 

Prthudakasvami, 118 

Prufer, 117 

Ptolemy, 118 

Puiseux, 118 

Puissant, 118 

Pythagoras, 118 

QadiZada, 162

Quetelet, 118 

Quillen, 119 

Quine, 119 

Rademacher, 119 

Rado, 119 

Radon, 119 

RadoRichard, 119 

Rahn, 119 

Rajagopal, 119 

Ramanujam, 120 

Ramanujan, 120 

Ramsden, 120 

Ramsey, 120 

Ramus, 120 

Rankin, 120 

Rankine, 120 

Raphson, 120 

Rasiowa, 120 

Rayleigh, 140 

Razmadze, 121 

Recorde, 121 

Rees, 121 

Regiomontanus, 121 

Reichenbach, 121 

Reidemeister, 121 

Reiner, 121 

Remak, 121 

Remez, 121 

Renyi, 119 

Reye, 122 

Reynaud, 122 

Reyneau, 122 

Reynolds, 122 

ReyPastor, 122 

Rheticus, 122 

Riccati, 122 

RiccatiVincenzo, 123 

Ricci, 123 

Ricci-Curbastro, 123 

RicciMatteo, 123 

RichardJules, 123 

RichardLouis, 123 

Richardson, 123 

Richer, 123 

Richmond, 123 

Riemann, 124 

Ries, 124 

Riesz, 124 

RieszMarcel, 124 

Ringrose, 124 

Roberts, 124 

Roberval, 124 

Robins, 124 

Robinson, 125 

RobinsonJulia, 124 

Rocard, 125 

Rogers, 125 

Rohn, 125 

Rolle, 125 

Roomen, 166 

Rosanes, 125 

Rosenhain, 125 

Rota, 125 

Roth, 126 

RothKlaus, 126 

Routh, 126 

Rudio, 126 

Rudolff, 126 

Ruffini, 126 

Runge, 126 

Russell, 126 

RussellScott, 126 

Rutherford, 127 

Rydberg, 127 

Saccheri, 127 

Sacrobosco, 127 

Saint-Venant, 151 

Saint-Vincent, 152 

Saks, 127 

Salem, 127 

Salmon, 127 

Samoilenko, 128 

Sang, 128 

Sankara, 128 

Sasaki, 128 

Saurin, 128 

Savage, 128 

Savart, 128 

Savary, 128 

Savile, 129 

Schatten, 129 

Schauder, 129 

Scheffe, 129 

Scheffers, 129 

Schickard, 129 

Schlafli, 129 

Schlomilch, 130 

Schmidt, 130 

Schoenberg, 130 

Schonflies, 129 

Schooten, 166 

Schottky, 130 

Schoute, 130 

Schouten, 130 

Schreier, 130 

Schroder, 130 

Schrodinger, 130 

Schroeter, 131 

Schubert, 131 

Schur, 131 

Schwartz, 131 

Schwarz, 131 

Schwarzschild, 131 

SchwarzStefan, 131 

Schwinger, 131 

Scott, 131 

Segner, 167 

SegreBeniamino, 132 

SegreCorrado, 132 

Seidel, 167 

Seifert, 132 

Seki, 142 

Selberg, 132 

Selten, 132

Semple, 132 

Serenus, 132 

Serre, 132 

Serret, 133 

Servois, 133 

Severi, 133 

Shanks, 133 

Shannon, 133 

Sharkovsky, 133 

Shatunovsky, 133 

Shen, 133 

Shewhart, 133 

Shields, 134 

Shnirelman, 134 

Shoda, 134 

Shtokalo, 134 

Siacci, 134 

Siegel, 134 

Sierpinski, 134 

Siguenza, 134 

Simplicius, 135 

Simpson, 135 

Simson, 135 

Sinan, 135 

Sintsov, 135 

Sitter, 135 

Skolem, 135 

Slaught, 135 

Sleszynski, 136 

Slutsky, 136 

Sluze, 136 

Smale, 136 

Smirnov, 136 

Smith, 136 

Sneddon, 136 

Snell, 136 

Snyder, 136 

Sobolev, 137 

Sokhotsky, 137 

Sokolov, 137 

Somerville, 137 

Sommerfeld, 137 

Sommerville, 137 

Somov, 137 

Sonin, 137 

Spanier, 137 

Spence, 138 

Sporus, 138 

Spottiswoode, 138 

Sridhara, 138 

Sripati, 138 

Stackel, 138 

Stampioen, 138 

Steenrod, 138 

StefanJosef, 138 

StefanPeter, 139 

Steiner, 139 

Steinhaus, 139 

Steinitz, 139 

Steklov, 139 

Stepanov, 139 

Stevin, 139 

Stewart, 139 

Stewartson, 139 

Stieltjes, 140 

Stifel, 140 

Stirling, 140 

Stokes, 140 

Stolz, 140 

Stone, 140 

Stott, 140 

Struik, 140 

Study, 141 

Sturm, 141 

SturmRudolf, 141 

Subbotin, 141 

Suetuna, 141 

Suter, 141 

Suvorov, 141 

Swain, 141 

Sylow, 141 

Sylvester, 142 

Synge, 142 

Szasz, 142 

Szego, 142 

Tacquet, 142 

Tait, 142 

Takagi, 142 

Talbot, 143 

Taniyama, 143 

TanneryJules, 143 

TanneryPaul, 143 

Tarry, 143 

Tarski, 143 

Tartaglia, 143 

Tauber, 143 

Taurinus, 143 

Taussky-Todd, 144 

Taylor, 144 

TaylorGeoffrey, 144 

Teichmuller, 144 

Temple, 144 

Tetens, 144 

Thabit, 144 

Thales, 144 

Theaetetus, 145 

Theodorus, 145 

Theodosius, 145 

Theon, 145 

TheonofSmyrna, 145 

Thiele, 145 

Thom, 145 

Thomae, 145 

Thomason, 145 

ThompsonD’Arcy, 146 

ThompsonJohn, 146 

Thomson, 146 

Thue, 146 

Thurston, 146 

Thymaridas, 146 

Tibbon, 146 

Tietze, 146 

Tikhonov, 149 

Tilly, 146

Tinbergen, 147 

Tinseau, 147 

Tisserand, 147 

Titchmarsh, 147 

Todd, 147 

Todhunter, 147 

Toeplitz, 147 

Torricelli, 147 

Trail, 147 

Tricomi, 148 

Troughton, 148 

Tschirnhaus, 168 

Tsu, 148 

Tukey, 148 

Tunstall, 148 

Turan, 148 

Turing, 148 

Turnbull, 148 

Turner, 148 

Uhlenbeck, 149 

UhlenbeckKaren, 149 

Ulam, 149 

UlughBeg, 149 

Upton, 149 

Urysohn, 150 

Vacca, 150 

Vailati, 150 

Valerio, 150 

ValleePoussin, 150 

VanCeulen, 166 

Vandermonde, 150 

VanderWaerden, 153 

Vandiver, 150 

VanVleck, 153 

Varahamihira, 151 

Varignon, 151 

Vashchenko, 162 

Veblen, 151 

Vega, 167 

Venn, 151 

Verhulst, 151 

Vernier, 151 

Veronese, 151 

Vessiot, 152 

Viete, 152 

Vijayanandi, 152 

Vinogradov, 152 

Vitali, 152 

Viviani, 152 

Vlacq, 152 

Volterra, 153 

VonBrill, 168 

VonDyck, 167 

VonNeumann, 167 

VonStaudt, 168 

Voronoy, 153 

Vranceanu, 153 

Wald, 153 

WalkerArthur, 153 

WalkerJohn, 153 

Wall, 154 

Wallace, 154 

Wallis, 154 

Wang, 154 

Wangerin, 154 

Wantzel, 154 

Waring, 154 

Watson, 154 

WatsonHenry, 154 

Wazewski, 155 

Weatherburn, 155 

Weber, 155 

WeberHeinrich, 155 

Wedderburn, 155 

Weierstrass, 155 

Weil, 155 

Weingarten, 155 

Weinstein, 155 

Weisbach, 156 

Weldon, 156 

Werner, 156 

Wessel, 156 

West, 156 

Weyl, 156 

Weyr, 156 

Wheeler, 156 

Whiston, 156 

White, 157 

Whitehead, 157 

WhiteheadHenry, 157 

Whitney, 157 

Whittaker, 157 

WhittakerJohn, 157 

Whyburn, 157 

Widman, 157 

Wielandt, 157 

Wien, 158 

WienerChristian, 158 

WienerNorbert, 158 

Wigner, 158 

Wilczynski, 158 

Wiles, 158 

Wilkins, 158 

Wilkinson, 158 

Wilks, 158 

WilsonAlexander, 159 

WilsonEdwin, 159 

WilsonJohn, 159 

Winkler, 159 

Wintner, 159 

Wirtinger, 159 

Wishart, 159 

Witt, 160 

Witten, 160 

Wittgenstein, 160 

Wolf, 160 

Wolfowitz, 160 

Wolstenholme, 160 

Woodhouse, 160 

Woodward, 160 

Wren, 160 

Wronski, 161

Xenocrates, 161 

Yang, 161 

Yates, 161 

Yativrsabha, 161 

Yau, 161 

Yavanesvara, 161 

Yoccoz, 161 

Youden, 161 

Young, 162 

YoungAlfred, 162 

Yule, 162 

Yunus, 162 

Yushkevich, 162 

Zarankiewicz, 163 

Zaremba, 163 

Zariski, 163 

Zassenhaus, 163 

Zeckendorf, 163 

Zeeman, 163 

Zelmanov, 163 

Zenodorus, 164 

ZenoofElea, 163 

ZenoofSidon, 163 

Zermelo, 164 

Zeuthen, 164 

Zhukovsky, 164 

Zolotarev, 164 

Zorn, 164 

Zuse, 164 

Zygmund, 164

ENTRY MATH MOVIES 

[ENTRY MATH MOVIES] Author: Oliver Knill: March 2000 -March 2004 Literature: actual DVD’s and 

corresponding movie websites 

Enigma 

[Enigma] is an espionage thriller set during WW II. Most of the story is fictional. The main character Tom 

Jericho who serves with the British ”Government Communication Headquarters” at Bletchley Park and played 

a significant role in breaking the German ”Enigma” codes using a machine called ”Collossus” to decipher the 

Enigma codes. The story is inspired by the lif of the mathematician Alan Turing who indeed contributed to 

the deciphering of Enigma during WW II. 

A beautiful mind 

The movie [A beautiful mind] describes the life of the Mathematician John Nash. Nash is introduced while 

entering Princeton as a young graduate student. The movie shows how Nash was struggling writing his PhD 

with the title ”Non-cooperative games”, a work which later would give him the Nobel prize. Nash is described 

as an impossible college teacher. In a calculus class, he introduces the following problem: 

Find a subset X of three dimensional space which has the property that if V 

is the set of vector fields F on the complement of X, which satisfy curl(F ) = 0 

and W is the set of vector fields F which are conservative F = grad(f). Then, 

the space V/W should be 8 dimensional. 

Good will hunting 

The movie [Good will hunting] shows a math prodigy Will Hunting who grew up in a succession of orphanages in 

South Boston. Working as a janitor at MIT, he has taught himself mathematics. He would anonymously solve 

complex math problems which were left overnight on blackboards. From an AMS review: ”The mathematics 

referred to later on ranges from basic linear algebra, through simple graph theory, to Parseval’s theorem, Fourier 

analysis, and on to what seem to be some deeper graph theoretical results. Mathematics is referred to constantly, 

but in no scene is it presented coherently.” 

Cube 

[Cube] Six strangers wake up in a maze of cubes equiped with movie traps and have to find their way out. Each 

room is equipped with a triple of numbers and colored. If all numbers are simultaneously not prime, then the 

room is trapped and entering it would kill the person entering it. 

Hypercube 

[Hypercube] In this horror movie, eight strangers wake up in a bizare cube-shaped room not knowing how they 

got there or how to escape. They soon learn that their ”hypercube” operates in the fourth dimension and shifts 

into an endless maze of danger and in the end everyone dies. The movie is the sequel to the 1999 cult hit cube. 

Cube 2 was directed by Andrzej Sekula.

Sneakers 

[Sneakers] An espionage thriller with Robert Redford. A hunt for a futuristic device which allows to decrypt 

secret messages. The device was built by a ”genious Mathematician” who appears in the movie giving a 

pompeous lecture on factorization algorithms. The movie which appeared in 1992 is not totally unrealistic from 

the mathematical point of view. Shortly after the movie was released, in the year 1993, mathematicians have 

shown that in principle, a quantum computer could break the factorization difficulty which is the fundament 

for many modern encryption algorithms. An other interpretation for the device would be that a new algorithm 

for factoring large integers would be found secretly and be hardwired into a chip. 

Pi 

[Pi] In the movie Pi, the pursuit of the infinite takes on a deeper meaning. Max Cohen is a number theorist 

living in New York obsessed with a potentially unsolvable problem. Yet, what the story and the age-old problem 

uncovers is the deeper link between the mysteries of life and other topics of consciousness as seemingly disparate 

as the stock market, the Kaballah, technology, the DNA and the stars in the sky. 

”11:15 Restate my assumptions: 

• Mathematics is the language of nature. 

• Everything around us can be represented and understood through numbers. 

• If you graph these numbers, patterns emerge. Therefore: There are patterns 

everywhere in nature.” 

Max Cohen in Pi 

Old School 

[Old School] In the college comedy ”old school”, three men, disenchanted with their life try to recapture their 

college life and wild youth by opening a frat house. In the movie, some aereal shots of Harvard appear evenso 

the movie seems have no scenes at all taken in Cambridge. At one point, the fraternaty members have to take 

a test in which they are asked about Hariotts method to solve cubics. 



Index 

A beautiful mind, 1 

Cube, 1 

Enigma, 1 

Good will hunting, 1 

Hypercube, 1 

Old School, 2 

Pi, 2 

Sneakers, 2 

3

ENTRY MEASURE THEORY 

[ENTRY MEASURE THEORY] Authors: Oliver Knill: 2003 Literature: measure theory 

analytic set 

An [analytic set] in a complete seperable metric space is the continuous image of a Borel set. Also called A-set. 

Any A-set is Lebesgue measurable. Any uncountable A-set topologically contains a perfect Cantor set. Suslins 

criterium tells that an analytic set is a Borel set if and only if its complement is also an analytic set. 

atom 

An [atom] is a measurable set Y of positive measure in a measure space such that every subset Z of Y has 

either zero or the same measure. Often an atom consists of only one point. More generally, an atom is minimal, 

non-zero element in a Boolean algebra. 

atom 

A property which holds up to a set of measure zero is said to hold [almost everywhere] (= almost surely). 

Banach-Tarski theorem 

The [Banach-Tarski theorem]: a ball in Euclidean space of dimension 3 can be decomposed into finitely many 

sets and rearranged by rigid motion to obtain two balls. The 

barycentre 

The [barycentre] of a Lebesgue measurable set S in an Euclidean space is the point � 

Boolean algebra 

S xdx. 

A [Boolean algebra] is a set S with two binary operations + and * which are commutative monoids (S, +, 0), 

(S, ∗, 1) and satisfy the two distributive laws (x ∗ (y + z) = x ∗ y + y ∗ z, x + (y ∗ z) = (x + y) ∗ (x + z) as well 

as the complementary laws x ∗ x = 1, y + y = 0. A Boolean algebra is especially an algebra. Examples are the 

algebra of classes, where + is the union and ∗ is the intersection or the algebra of propositions, for which + is 

and and ∗ is or. 

Boolean ring 

A [Boolean ring] is a ring in which every member is idempotent.

Borel-Cantelli lemma 

The [Borel-Cantelli lemma]: if Yn are events in a probability space and the sum of their probabilities is finite, 

then the probability that infinitely many events occur is zero. If the events are independent and the sum of 

their probabilities is infinite, then the probability that infinitely many events occur is one. 

Borel measure 

A [Borel measure] is a measure on the sigma-algebra of Borel sets. 

Borel set 

A [Borel set] (=Borel measurable set) in a topological space is an element in the smallest sigma-algebra which 

contains all compact sets. Borel sets are also called B-sets. One can say that a B-set is a set which can be 

obtained of not more than a countable number of operations of union and intersection of closed open sets in a 

topological space. Borel sets are special cases of analytic sets. 

Borel set 

The smallest sigma-algebra A of subsets of a topological space (X,O) containing O is called a Borel sigmaalgebra. 

absolutely continuous 

A measure µ is [absolutely continuous] to a measure ν if ν(Y ) = 0 implies µ(Y ) = 0. 

centre of mass 

The [centre of mass] (=barycentre) of a Borel measure µ in a Euclidean space X is the point x = � 

X xµ(x). 

For example, if µ is supported on finitely many points xi and mi = µ(xi) then x = � 

i mixi. If µ is the mass 

distribution of a body, then its centre of mass is called the centre of gravity. 

abstract integral 

[abstract integral]. Denote by L, L + the set of measureable maps from a measure space (X, A, µ) to the real 

line (R, B), where B is the Borel sigma-algebra on R, R + . For f ∈ S = {f = �n i=1 αi · 1Ai αi � � 

∈ R }, define 

f dµ := a∈f(X) a · µ{X = a}. For f ∈ L+ define � 

� 

g dµ . For f ∈ L finally define 

X X f dµ = sup � � � g∈S 

+ − + − + f = f − f , where f (x) = max(f(x), 0) and f (x) = −(−f) (x). 

X

abstract integral 

A sigma-additive function µ : A → [0, ∞] on a measurable space (X,A) is called a [measure]. It is called a finite 

measure if µ(X) < ∞. 

measure 

A map f : X → Y where (X, A), (Y, B) are measurable spaces is called [measurable] if f −1 (B) ∈ A for all 

B ∈ B. 

The pair (X, A) is called a [measurable space] if 

measurable space 

measure space 

(X, A, µ) is called a [measure space] if (X, A) is a measurable space and µ is a measure on (X, A). 

measure space 

[sigma-additive: a real-valued function on a set A of subsets of X is called sigma-additive if for all disjoint 

Yn ∈ A, one has µ( � 

n Yn) = � 

n µ(Yn). 

sigma-algebra 

A set A of subsets of a set X is called a [sigma-algebra] if 

• X is in A 

• Y ∈ A implies Y c ∈ A. (iii) Yn ∈ A, n = 1, 2, 3, ... implies � ∞ 

n=1 Yn ∈ A. 



Index 

absolutely continuous, 2 

abstract integral, 2, 3 

analytic set, 1 

atom, 1 

Banach-Tarski theorem, 1 

barycentre, 1 

Boolean algebra, 1 

Boolean ring, 1 

Borel measure, 2 

Borel set, 2 

Borel-Cantelli lemma, 2 

centre of mass, 2 

measurable space, 3 

measure, 3 

measure space, 3 

sigma-algebra, 3 

4

ENTRY NUMBER THEORY 

[ENTRY NUMBER THEORY] Authors: Oliver Knill: 2003 Literature: Hua, introduction to number theory 

ABC conjecture 

[ABC conjecture] If a,b,c are positive integers, let N(a,b,c) be the product of the prime divisors of a,b,c, with 

each divisor counted only once. The conjecture claims that for every ɛ > 0, there is a constant mu¿1 such that 

for all coprime a,b and c=a+b, then max(|a|, |b|, |c|) ≤ muN(a, b, c) ( 1 + epsilon). 

irreducible polynomial 

A root of an [irreducible polynomial] with integer coefficients is called an [algebraic number]. 

amicable 

Two integers are called [amicable] if each is the sum of the distinct proper factors of the other. For example: 

220 and 284 are amicable. Amicable numbers are 2-periodic orbits of the sigma function σ(n) which is the sum 

of the divisors of n. 

Apery’s theorem 

[Apery’s theorem]: the value of the zeta function at z=3 is rational. 

arithmetic function 

An [arithmetic function] is a function f(n) whose domain is the set of positive integers. An important class of 

arithmetic functions are multiplicative functions f(nm) = f(n)f(m). An example is the Möbius function µ(n) 

defined by µ(1) = 1, µ(n) = (−1) r if n is the product of r distinct primes and µ(n) = 0 otherwise. 

Artinian conjecture 

[Artinian conjecture]: a quantitative form of the conjecture that every non-square integer is a primitive root of 

infinitely many primes. [Beal’s conjecture] If a x + b y = c z , where a,b,c,x,y,z are positive integers and x, y, z > 2, 

then a, b, c must have a common factor. It is known that for every x, y, z, there are only finitely many solutions. 

The Beal conjecture is a ageneralization of Fermat’s last theorem. The conjecture was announced in December 

1997. The prize is now 100’000 Dollars for either a proof or a counterexample. The conjecture was discovered 

by the Texan number theory enthusiast and banker Andrew Beal.

Bertrands postulate 

[Bertrands postulate] tells that for any integer n greater than 3, there is a prime between n and 2n − 2. The 

postulate is a theorem, proven by Tchebychef in 1850. 

Bezout’s lemma 

[Bezout’s lemma] tells that if f and g are polynomials over a field K and d is the greatest common divisor of f 

and g, then d = af + bg, where a,b are two other polynomials. This generalizes Euclid’s theorem for integers. 

Brun’s constant 

The [Brun’s constant] is the sum of the reciprocals of all the prime twins. It is estimated to be about 

1.9021605824. While one does not know, whether infinitely many prime twins exist, the sum of their reciprocals 

is known to be finite. This has been proven by the Norwegian Mathematician Viggo Brun (1885-1978) 

in 1919. 

Carmichael numbers 

[Carmichael numbers] are natural numbers which are Fermat pseudoprime to any base. Named after R.D. 

Carmichael who discoved them in 1909. It is known that there are infinitely many Carmichael numbers. The 

Carmichael numbers under 100’000 are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 

46657, 52633, 62745, 63973, and 75361. 

Chineese reminder problem 

The [Chineese reminder problem] tells that if n1, . . . , nk are natural numbers which are pairwise relatively prime 

and if a1, . . . , ak are any integers, then there exists an integer x which solves simultaneously the congruences 

x == a1(modn1), . . . , x == ak(modnk). All solutions are congruent to a given solution modulo �k j=1 nj. The 

theorem was established by Qin Jiushao in 1247. 

coprime 

Two numbers a,b are called [coprime] if their greatest common divisor is 1.

ElGamal 

[ElGamal] The ElGamal cryptosystem is based on the difficulty to solve the discrete logarithm problem modulo 

a large number n = p r , where p is a prime and and r is a positive integer: solving g x = bmodn for x is 

computationally hard. Suppose you want to send a message encoded as an integer m to Alice. A large integer n 

and a base g are chosen and public. Alice who has a secrete integer a has published the integer c = g k mod(n) as 

her public key. Everybody knows n, g, c. To send Alice a message m, we chose an integer k at random and send 

Alice the pair (A, B) = (g m , kg am ) modulo n. (We can compute g am = (g a ) m = c m using the publically availble 

information only.) Alice can recover from this the secret message m = A a /B. However, somebody intercepting 

the message is not able to recover m without knowing a. He would have to find the discrete logarithm of g m 

with base g to do so but this is believed to be a computationally difficult problem. 

Farey Sequence 

The [Farey Sequence] of order n is the finite sequence of rational numbers a/b, with 0 ≤ a ≤ b ≤ n such that 

a, b have no common divisor different from 1 and which are arranged in increasing order. 

[Number]: 

F1 = (0/1, 1/1) 

F2 = (0/1, 1/2, 1) 

F3 = (0/1, 1/3, 1/2, 2/3, 1/1) 

F4 = (0/1, 1/4, 1/3, 1/2, 2/3, 3/4) 

F5 = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4/4/5) 

Number 

N: natural numbers e.g. 1, 2, 3, 4, ... 

Z: integers e.g. −1, 0, 1, 2, ... 

Q: rational numbers e.g. 5/6, 5, −8/10 

R: real numbers e.g. 1, π, e, � (2), 5/4 

C: complex numbers e.g. i, 2, e + iπ/2, 4π/e 

The numbers 1, 2, 3, ... are called the [natural numbers]. 

natural numbers 

Fermats last theorem 

[Fermats last theorem]. For any ineger n bigger than 2, the equation x n + y n = z n has no solutions in positive 

integers. This theorem was proven in 1995 by Andrew Wiles with the assistance of Richard Taylor. The theorem 

has a long history: in an annotation of his copy ”Diophantus”, Fermat wrote a note: ”On the other hand, it 

is impossible to seperate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power 

except a squre into two powers with the same exponent. I have discovered a truely marvelous proof of this 

which however the margin is not large enough to contain.”

Fermat-Catalan Conjecture 

[Fermat-Catalan Conjecture] There are only finitely many triples of coprime integer powers x q , y q , z r for which 

x p + y q = z r with 1/p + 1/q + 1/r < 1. 

Fermats little theorem 

[Fermats little theorem] If p is prime and a is an integer which is not a multiple of p, then a ( p − 1) = 1modp. 

Example: 2 4 = 16 = 1 (mod 5). Fermats little theorem is a consequence of the Lagrange theorem in algebra, 

which says that for finite groups, the order of a subgroup divides the order of the group. Fermats theorem is the 

special case, when the finite group is the cyclic group with p − 1 elements. Fermats little theorem is somtimes 

also stated in the form: for every integer a and prime number p, the number a p − a is a multiple of p. 

Fermat numbers 

Numbers Fn = 2 ( 2 n ) + 1 are called [Fermat numbers]. Examples are 

F0 = 3 prime 

F1 = 5 prime 

F2 = 17 prime 

F3 = 257 prime 

F4 = 65537 prime 

F5 = 641 · 6700417 composite 

No other prime Fermat number beside the first 5 had been found so far. 

fundamental theorem of arithmetic 

The [fundamental theorem of arithmetic] assures that every natural number n has a unique prime factorization. 

In other words, there is only one way in which one can write a number as a product of prime numbers if the 

order of the product does not matter. For example, 84 = 2237 is the prime factorization of 84. 

Goldbach’s conjecture 

[Goldbach’s conjecture]: Every even integer n greater than two is the sum of two primes. For example: 8 = 5+3 

or 20 = 13 + 7. The conjecture has not been proven yet. 

greatest common divisor 

The [greatest common divisor] of two integers n and m is the largest integer d such that d divides n and d divides 

m. One writes d = gcd(n, m). For example, gcd(6, 9) = 3. There are few recursive algorithms for gcd: one of 

them is the Euclidean algorithm: gcd(m, n) = {k = m mod n; if (k == 0) return(n); else return(gcd(n, k))}.

prime number or prime 

A positive integer n is called a [prime number] or [prime], if it is divisible by 1 and n only. The first prime 

numbers are 2, 3, 5, 7, 11, 13, 17. An example of a non prime number is 12 because it is divisible by 3. There are 

infinitely many primes. Every natural number n can be factorized uniquely into primes: for example 42 = 237. 

prime factorization 

The [prime factorization] of a positive integer n is a sequence of primes whose product is n. For example: 

18 = 3 · 3 · 2 or 100 = 2 · 2 · 5 · 5 or 17 = 17. Every integer has a unique prime factorization. 

Pells equation 

Fermat claimed first that [Pells equation] dy 2 + 1 = x 2 , where d is an integer has always integer solutions x 

and y. The name ”Pell equation” was given by Euler evenso Pell seems nothing have to do with the equation. 

Lagrange was the first to prove the existence of solutions. One can find solutions by performing the Continued 

fraction expansion of the square root of d. 

Fermat number 

A [Fermat number] is an integer of the form Fk = 2 ( 2 k ) + 1. The first Fermat numbers are F0 = 3, F1 = 5, F2 = 

17, F3 = 257, F4 = 65537. They are all primes and called Fermat primes. Fermat had claimed that all Fk are 

primes. Euler disproved that showing that 641 divides F5. The Fermat numbers F5 until F9 are known to be 

not prime and also have been factored. Fermat numbers play a role in constructing regular polygons with ruler 

and compass. The factorization of Fermat numbers serves as a challenge to factorization algorithms. 

A [Fermat prime] is a Fermat number which is prime. 

Fermat prime 

Mersenne number 

An integer 2 n − 1 is called a [Mersenne number]. If a Mersenne number is prime, it is called a Mersenne prime. 

Mersenne number 

If a Mersenne number 2 n − 1 is prime, it is called a [Mersenne prime]. In that case, n must be prime. Known 

examples are n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 

9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 

2976221, 3021377. It is not known whether there are infinitely many Mersenne primes.

perfect number 

An integer n is called a [perfect number] if it is equal to the sum of its proper divisors. For example 6 = 1+2+3 

or 28 = 1 + 2 + 4 + 7 + 14 are perfect numbers. Also, if 2 n − 1 is prime then 2 n−1 (2 n − 1) is perfect because 

1 + 2 + 4 + ... + 2 ( n − 1) = 2 n − 1. This was known already to Euclid. Every even perfect number is of the form 

p(p + 1)/2, where p = 2 n − 1 is a Mersenne prime. It is not known whether there is an odd perfect number, nor 

whether there are infinitely many Mersenne primes. 

partition 

The [partition] of a number n is a decomposition of n into a sum of integers. Examples are 5 = 1 + 2 + 2. The 

number of partitions of a number n is denoted by p(n) and plays a role in the theory of representations of finite 

groups. For example: p(4) = 5 because of the following partitions 4 = 3+1 = 2+2 = 1+1+2 = 1+1+1+1 Euler 

introduced the Power series f(x) = � p(n)x n which is (1−x) −1 (1−x 2 ) −1 .... The algebra of formal power series 

leads to powerful identites like (1+x) −1 (1+x 2 ) −1 (1+x 3 ) −1 ... = (1+x)(1−x)(1+x 2 )(1−x 2 ).../(1−x)(1−x 2 )... = 

(1−x 2 )(1−x 4 ).../(1−x)(1−x 2 )... = (1−x) −1 (1−x 3 ) −1 (1−x 5 ) −1 .... The left hand side is the generating function 

for a(n), the number of partitions of n into distinct numbers. The right hand side is the generating function of 

b(n), the number of partitions of n into an odd number of summands. The algebraic identity has shown that 

a(n) = b(n). For example, for n = 5, one has a(5) = 3 decomposition 5 = 5 = 4 + 1 = 3 + 2 into different 

summands and also b(5) = 3 decompositions into an odd number of summands 5 = 5 = 2+2+1 = 1+1+1+1+1. 

prime number 

A [prime number] is an positive integer which is divisible only by 1 or itself. For example, 12 is not a prime 

number because it is divisible by 3 but the integer 13 is a prime number. The first prime numbers are 

2, 3, 5, 7, 11, 13, 17, 23.... There are infinitely many prime numbers because if there were only finitely many, 

their product p1p2...pk = n has the property that n + 1 is not divisible by any pi. Therefore, n + 1 would either 

be a new prime number of be divisible by a new prime number. This contradicts the assumption that there are 

only finitely many. 

prime twin 

[prime twin] Two positive integers p, p + 2 are called prime twins if both p and p + 2 are prime numbers. For 

example (3, 5), (11, 13) and 17, 19 are prime twins. It is unknown, whether there are infinitely many prime 

twins. One knows that � 

i 1/pi, where (pi, pi + 2) are prime times is finite. 

Pythagorean triple 

Three integers x, y, z form a [Pythagorean triple] if x 2 + y 2 = z 2 . An example is 3 2 + 4 2 = 5 2 . Pythagorean 

triples define triangles with a right angle and integer side lengths x, y, z. They were known and useful already 

by the Babylonians and used to triangulate rectangular regions. The Pythagorean triples with even x can be 

parameterized with p > q and x = 2pq, y = p 2 − q 2 , z = p 2 + q 2 . Each Pythagrean triple corresponds to rational 

points on the unit circle: X 2 + Y 2 = 1, where X = x/z, Y = y/z.

elatively prime 

Two integers n and m are [relatively prime] if their greatest common divisor gcd(n, m) is 1. In other words, one 

does not find a common factor of n and m other than 1. 

Sieve of Eratosthenes 

The [Sieve of Eratosthenes] allows to construct prime numbers. By sieving away all multiplies of 2, 3, ..., N and 

listing what is left, one obtains a list of all the prime numbers smaller than N 2 . For example: 

multiples of 2: 4,6,8,10,12,14,16,18,20,22,24,26,... 

multiplis of 3: 6,9,12,15,18,21,24,... 

multiples of 5: 10,15,20,25,... 

The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 do not appear in this list and are all the prime numbers smaller or equal 

than 5 2 . To list all the prime number up to N 2 , one would have to list all the multiples of k for k ≤ N. 

quadratic residue 

A square modulo m, then n is called a A [quadratic residue] modulo m is an integer n which is a square modulo 

m. That is one can find an integer x such that n = x 2 (modm). If m is not a quadratic residue, it is called a 

quadratic non-residue modulo n. Examples: 

• 2 is a quadratic residue modulo 7 because 3 2 = 2 modulo 7. 

• If p is an odd prime, then there are (p − 1)/2 quadratic residues and (p − 1)/2 quadratic nonresidues 

modulo p. 

Legendre symbol 

The [Legendre symbol] encodes, whether n is a quadratic residue modulo a prime number p or not: 

Legendre(n, p) = 1 if n is a quadratic residue and Legendre(n, p) = −1 if n is not a quadratic residue. If 

p is a prime number, then Legendre(−1, p) = (−1) ( (p − 1)/2) and Legendre(2, p) = (−1) ( (p 2 − 1)/8). 

law of quadratic reciprocity 

The [law of quadratic reciprocity] tells that if p, q are distinct odd prime numbers, then Legendre(p, q) · 

Legendre(q, p) = (−1) n , where n = (p − 1)(q − 1)/4. Gauss called this result the ”queen of number theory”. 

The theorem implies that if p = 3 (mod4) and q = 3 (mod4), then exactly one of the two congruences 

x 2 = p (modq) or x 2 = q (modp) is solvable. Otherwise, either both or none is solvable. 

Jacobi symbol 

[Jacobi symbol] If m = p1...pk is the prime factorization of m, define Jacobi(n, m) as the product of 

Legendre(n, pi), where Legendre(n, p) denotes the Legendre symbol of n and p and m = p1 · · · pk is the prime 

factorization of m.

Jacobi symbol 

A positive integer a which generates the multiplicative group modulo a prime number p is called a [primitive 

root of p]. Examples: 

• a = 2 is a primitive root modulo p = 5, because 1 = a 0 , 2 = a 1 , 4 = a 2 , 3 = a 3 is already the list of 

elements in the multiplicative group of 5. 

• a = 4 is not a primitive root modulo p − 5 because 4 2 = 1 mod5. 

Liouville 

An irrational number a is called [Liouville] if there exists for every integer m a sequence pn/qn of irreducible 

fractions such that limn→∞ q m n |a − pn/qn| = 0. Liouville numbers form a class of irrational numbers which can 

be approximated well by rational numbers. An example of a Liouville number is 0.10100100001000000001... 

,where the number of zeros between the 1’s grows exponentially. 

Möbius function 

The [Möbius function] µ is an example of multiplicative arithmetic function. It is defined as 

⎧ 

⎨ 1 n = 1 

µ(n) = 

⎩ 

(−1) r 0 

n = p1 · .. · pr, pi < pi+1 

otherwise 

. 

sigma function 

Orbits of the [sigma function] σ(n) giving the sum of the divisors of n is called an [aliquot sequence]. One starts 

with a number n and forms σ(n), σ(σ(n)) etc. Example: 12, 16, 15, 9, 4, 3, 1. Perfect numbers are fixed points, 

amicable numbers are periodic orbits. Higher periodic orbits are called sociable chains. The Catalan conjecture 

states that every aliquot sequence 



Index 

ABC conjecture, 1 

amicable, 1 

Apery’s theorem, 1 

arithmetic function, 1 

Artinian conjecture, 1 

Bertrands postulate, 2 

Bezout’s lemma, 2 

Brun’s constant, 2 

Carmichael numbers, 2 

Chineese reminder problem, 2 

coprime, 2 

ElGamal, 3 

Farey Sequence, 3 

Fermat number, 5 

Fermat numbers, 4 

Fermat prime, 5 

Fermat-Catalan Conjecture, 4 

Fermats last theorem, 3 

Fermats little theorem, 4 

fundamental theorem of arithmetic, 4 

Goldbach’s conjecture, 4 

greatest common divisor, 4 

irreducible polynomial, 1 

Jacobi symbol, 7, 8 

law of quadratic reciprocity, 7 

Legendre symbol, 7 

Liouville, 8 

Möbius function, 8 

Mersenne number, 5 

natural numbers, 3 

Number, 3 

partition, 6 

Pells equation, 5 

perfect number, 6 

prime factorization, 5 

prime number, 6 

prime number or prime, 5 

prime twin, 6 

Pythagorean triple, 6 

quadratic residue, 7 

relatively prime, 7 

Sieve of Eratosthenes, 7 

sigma function, 8 

9

ENTRY PHYSICS 

[ENTRY PHYSICS] Authors: Oliver Knill: 2002 Literature: Fundamental formulas in physics (edited by D.H. 

Menzel) Kneubuehl, Repetitorium in physics 

angular momentum 

[angular momentum] If r(t) is the position of a mass point of mass m, then the vector L = mrxr ′ is called the 

angular momentum. 

C14 method 

[C14 method] Used to estimate the age of substances containing carbon. It is useful in the range between 500 

and 50’000 years. The C14 method is based on the asumption that the atmosphere has a constant C14 isotope 

conentration. The decay of C14 is compensated by creation of C14 in the stratosphere through cosmic radiation. 

A living plant has the same C14 concentration as the atmosphere. When it dies, the exchange of air stops and 

the C14 concentration in the plant will decay. 

Gravity 

[Gravity] a fundamental force which is responsible for the attraction of different masses like for example the 

Sun and the earth. 

Keplers laws 

[Keplers laws] 

1. Law: (1609) Planets move on ellipses around the sun. The sun is in a focal point of the ellipse. 

2. Law: (1609) The radius vector from the sun to the planet covers equal area in equal time. 

3. Law: (1619) The squares of the periods of the planets are proportional to the cubes of the semiaxes of the 

planets. 

Newton laws 

[Newton laws] (Newton 1686) established four axioms 

1. law) Bodies not subject to forces move along straight lines: r ′′ = 0. 

2. law) Force is mass times acceleration F = mr ′′ . 

3. law) To every action there is a reation: F12 = −F21. 

4. law) Forces add like vectors: two forces F1, F2 acting on a body can be replaced by F1 + F2.

Mass 

[Mass] measures amount of material in a body. The SI unit is 1 kilogram = 1kg. One liter of water at 

temperature 4 ◦ C has the mass of one kilogram. Typical masses are 

• Electron 0.9 · 10 −30 kg 

• Hydrogen atom 210 −27 kg 

• Virus 610 −19 kg 

• Earth 6 · 10 24 kg 

• Sun 2 · 10 30 kg 

• Milkyway 10 41 kg 

• Universe 10 52 kg 

Length 

[Length] measures the position in space. The SI unit is 1 meter = 1m. The meter was originally defined as 1/40 

millionth of the meridian of the earth but is sine 1960 defined spectroscopically. Typical lengths are 

• Diameter of an atomic nucleus 3 · 10 −15 m. 

• Wave length of the visible light 5 · 10 −7 m. 

• Diameter of the earth 1.3 · 10 7 m 

• Diameter of the sun 1.4 · 10 9 m 

• Distance to alpha centauri 4 · 10 16 m 

• Diameter of the Milkyway 7 · 10 20 m 

• Diameter of universe 10 26 m. 

Maser 

[Maser] Maser stands for Microwave Amplification by Stiumlated Emision of Radiation. Masers are oscillators 

whose frequency are determined by quantized states of atoms or molecules. The frequencies of a Maser are in 

the microwave range. 

[Power] is work per time or force times space. The SI unit is 1 Watt 1W = 1kgm 2 /s 3 .

Time 

[Time] measures the position on the time axes. The SI unit is 1 second = 1s. The second was originally defined 

as a fraction of one tropical year. Since 1967 it is defined as 1s=9’192’631 177 periods of a Cesium 133 Maser 

oscillation. Typicl times: 

• Light passing through kernel 10 −24 s. 

• Light passes through atom 10 −19 s. 

• Period of light 10 −15 s. 

• Period of sound 10 −3 s. 

• One day 10 5 s. 

• Life of a human 210 9 s. 

• Age of earth 1.3 · 10 17 s 

• Age of universe 5 · 10 17 s 

Relativitistic addition of velocities 

[Relativitistic addition of velocities] As Poincare has realized first, the Maxwell equations are not invariant 

under Galilei transformations but under Lorentz transformations. The Michelson-Morely measurements of 

the speed of lights showed that light has a constant speed. The addition of velocities has therefore to be 

modified to a relativistic addition v = (v1 + v2)/(1 + v1v2/c 2 ). Using a mass which depends on the velocity 

m = m0/sqrt1 − v 2 /c 2 the Newton laws hold unmodified. 



Index 

angular momentum, 1 

C14 method, 1 

Gravity, 1 

Keplers laws, 1 

Length, 2 

Maser, 2 

Mass, 2 

Newton laws, 1 

Relativitistic addition of velocities, 3 

Time, 3 

4

ENTRY POLYHEDRA 

[ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data 

given in http://netlib.bell-labs.com/netlib 

tetrahedron 

The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. 

cube 

The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. 

hexahedron 

The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. 

octahedron 

The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. 

dodecahedron 

The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. 

icosahedron 

The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. 

small stellated dodecahedron 

The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called 

great dodecahedron. 

great dodecahedron 

The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small 

stellated dodecahedron.

great stellated dodecahedron 

The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called 

great icosahedron. 

great icosahedron 

The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great 

stellated dodecahedron. 

truncated tetrahedron 

The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis 

tetrahedron. 

cuboctahedron 

The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic 

dodecahedron. 

truncated cube 

The [truncated cube] is a polyhedron with 24 vertices and 14 faces. The dual polyhedron is called triakis 

octahedron. 

truncated octahedron 

The [truncated octahedron] is a polyhedron with 24 vertices and 14 faces. The dual polyhedron is called tetrakis 

hexahedron. 

rhombicuboctahedron 

The [rhombicuboctahedron] is a polyhedron with 24 vertices and 26 faces. The dual polyhedron is called 

trapezoidal icositetrahedron. 

great rhombicuboctahedron 

The [great rhombicuboctahedron] is a polyhedron with 48 vertices and 26 faces. The dual polyhedron is called 

hexakis octahedron.

snub cube 

The [snub cube] is a polyhedron with 24 vertices and 38 faces. The dual polyhedron is called pentagonal 

icositetrahedron. 

icosidodecahedron 

The [icosidodecahedron] is a polyhedron with 30 vertices and 32 faces. The dual polyhedron is called rhombic 

triacontahedron. 

truncated dodecahedron 

The [truncated dodecahedron] is a polyhedron with 60 vertices and 42 faces. The dual polyhedron is called 

triakis icosahedron. 

truncated icosahedron 

The [truncated icosahedron] is a polyhedron with 60 vertices and 32 faces. The dual polyhedron is called 

pentakis dodecahedron. 

rhombicosidodecahedron 

The [rhombicosidodecahedron] is a polyhedron with 60 vertices and 62 faces. The dual polyhedron is called 

trapezoidal hexecontahedron. 

great rhombicosidodecahedron 

The [great rhombicosidodecahedron] is a polyhedron with 120 vertices and 62 faces. The dual polyhedron is 

called hexakis icosahedron. 

snub dodecahedron 

The [snub dodecahedron] is a polyhedron with 60 vertices and 92 faces. The dual polyhedron is called pentagonal 

hexacontahedron. 

triangular prism 

The [triangular prism] is a polyhedron with 6 vertices and 5 faces.

pentagonal prism 

The [pentagonal prism] is a polyhedron with 10 vertices and 7 faces. 

hexagonal prism 

The [hexagonal prism] is a polyhedron with 12 vertices and 8 faces. 

octagonal prism 

The [octagonal prism] is a polyhedron with 16 vertices and 10 faces. 

decagonal prism 

The [decagonal prism] is a polyhedron with 20 vertices and 12 faces. 

square antiprism 

The [square antiprism] is a polyhedron with 8 vertices and 10 faces. 

pentagonal antiprism 

The [pentagonal antiprism] is a polyhedron with 10 vertices and 12 faces. 

hexagonal antiprism 

The [hexagonal antiprism] is a polyhedron with 12 vertices and 13 faces. 

octagonal antiprism 

The [octagonal antiprism] is a polyhedron with 16 vertices and 18 faces. 

decagonal antiprism 

The [decagonal antiprism] is a polyhedron with 20 vertices and 22 faces.

triakis tetrahedron 

The [triakis tetrahedron] is a polyhedron with 8 vertices and 12 faces. The dual polyhedron is called truncated 

tetrahedron. 

rhombic dodecahedron 

The [rhombic dodecahedron] is a polyhedron with 14 vertices and 12 faces. The dual polyhedron is called 

cuboctahedron. 

triakis octahedron 

The [triakis octahedron] is a polyhedron with 14 vertices and 24 faces. The dual polyhedron is called truncated 

cube. 

tetrakis hexahedron 

The [tetrakis hexahedron] is a polyhedron with 14 vertices and 24 faces. The dual polyhedron is called truncated 

octahedron. 

trapezoidal icositetrahedron 

The [trapezoidal icositetrahedron] is a polyhedron with 26 vertices and 24 faces. The dual polyhedron is called 

rhombicuboctahedron. 

hexakis octahedron 

The [hexakis octahedron] is a polyhedron with 26 vertices and 48 faces. The dual polyhedron is called great 

rhombicuboctahedron. 

pentagonal icositetrahedron 

The [pentagonal icositetrahedron] is a polyhedron with 38 vertices and 24 faces. The dual polyhedron is called 

snub cube. 

rhombic triacontahedron 

The [rhombic triacontahedron] is a polyhedron with 32 vertices and 30 faces. The dual polyhedron is called 

icosidodecahedron.

triakis icosahedron 

The [triakis icosahedron] is a polyhedron with 32 vertices and 60 faces. The dual polyhedron is called truncated 

dodecahedron. 

pentakis dodecahedron 

The [pentakis dodecahedron] is a polyhedron with 32 vertices and 60 faces. The dual polyhedron is called 

truncated icosahedron. 

trapezoidal hexecontahedron 

The [trapezoidal hexecontahedron] is a polyhedron with 62 vertices and 60 faces. The dual polyhedron is called 

rhombicosidodecahedron. 

hexakis icosahedron 

The [hexakis icosahedron] is a polyhedron with 62 vertices and 120 faces. The dual polyhedron is called great 

rhombicosidodecahedron. 

pentagonal hexecontahedron 

The [pentagonal hexecontahedron] is a polyhedron with 92 vertices and 60 faces. The dual polyhedron is called 

snub dodecahedron. 

square pyramid 

The [square pyramid] is a polyhedron with 5 vertices and 5 faces. 

pentagonal pyramid 

The [pentagonal pyramid] is a polyhedron with 6 vertices and 6 faces. 

triangular cupola 

The [triangular cupola] is a polyhedron with 9 vertices and 8 faces. 

square cupola 

The [square cupola] is a polyhedron with 12 vertices and 10 faces.

pentagonal cupola 

The [pentagonal cupola] is a polyhedron with 15 vertices and 12 faces. 

pentagonal rotunda 

The [pentagonal rotunda] is a polyhedron with 20 vertices and 17 faces. 

elongated triangular pyramid 

The [elongated triangular pyramid] is a polyhedron with 7 vertices and 7 faces. 

elongated square pyramid 

The [elongated square pyramid] is a polyhedron with 9 vertices and 9 faces. 

elongated pentagonal pyramid 

The [elongated pentagonal pyramid] is a polyhedron with 11 vertices and 11 faces. 

gyroelongated square pyramid 

The [gyroelongated square pyramid] is a polyhedron with 9 vertices and 13 faces. 

gyroelongated pentagonal pyramid 

The [gyroelongated pentagonal pyramid] is a polyhedron with 11 vertices and 16 faces. 

triangular dipyramid 

The [triangular dipyramid] is a polyhedron with 5 vertices and 6 faces. 

pentagonal dipyramid 

The [pentagonal dipyramid] is a polyhedron with 7 vertices and 10 faces.

elongated triangular dipyramid 

The [elongated triangular dipyramid] is a polyhedron with 8 vertices and 9 faces. 

elongated square dipyramid 

The [elongated square dipyramid] is a polyhedron with 10 vertices and 12 faces. 

elongated pentagonal dipyramid 

The [elongated pentagonal dipyramid] is a polyhedron with 12 vertices and 15 faces. 

gyroelongated square dipyramid 

The [gyroelongated square dipyramid] is a polyhedron with 10 vertices and 16 faces. 

elongated triangular cupola 

The [elongated triangular cupola] is a polyhedron with 15 vertices and 14 faces. 

elongated square cupola 

The [elongated square cupola] is a polyhedron with 20 vertices and 18 faces. 

elongated pentagonal cupola 

The [elongated pentagonal cupola] is a polyhedron with 25 vertices and 22 faces. 

elongated pentagonal rotunds 

The [elongated pentagonal rotunds] is a polyhedron with 30 vertices and 27 faces. 

gyroelongated triangular cupola 

The [gyroelongated triangular cupola] is a polyhedron with 15 vertices and 20 faces.

gyroelongated square cupola 

The [gyroelongated square cupola] is a polyhedron with 20 vertices and 26 faces. 

gyroelongated pentagonal cupola 

The [gyroelongated pentagonal cupola] is a polyhedron with 25 vertices and 32 faces. 

gyroelongated pentagonal rotunda 

The [gyroelongated pentagonal rotunda] is a polyhedron with 30 vertices and 37 faces. 

gyrobifastigium 

The [gyrobifastigium] is a polyhedron with 8 vertices and 8 faces. 

triangular orthobicupola 

The [triangular orthobicupola] is a polyhedron with 12 vertices and 14 faces. 

square orthobicupola 

The [square orthobicupola] is a polyhedron with 16 vertices and 18 faces. 

square gyrobicupola 

The [square gyrobicupola] is a polyhedron with 16 vertices and 18 faces. 

pentagonal orthobicupola 

The [pentagonal orthobicupola] is a polyhedron with 20 vertices and 22 faces. 

pentagonal gyrobicupola 

The [pentagonal gyrobicupola] is a polyhedron with 20 vertices and 22 faces.

pentagonal orthocupolarontunda 

The [pentagonal orthocupolarontunda] is a polyhedron with 25 vertices and 27 faces. 

pentagonal gyrocupolarotunda 

The [pentagonal gyrocupolarotunda] is a polyhedron with 25 vertices and 27 faces. 

pentagonal orthobirotunda 

The [pentagonal orthobirotunda] is a polyhedron with 30 vertices and 32 faces. 

elongated triangular orthobicupola 

The [elongated triangular orthobicupola] is a polyhedron with 18 vertices and 20 faces. 

elongated triangular gyrobicupola 

The [elongated triangular gyrobicupola] is a polyhedron with 18 vertices and 20 faces. 

elongated square gyrobicupola 

The [elongated square gyrobicupola] is a polyhedron with 24 vertices and 26 faces. 

elongated pentagonal orthobicupola 

The [elongated pentagonal orthobicupola] is a polyhedron with 30 vertices and 32 faces. 

elongated pentagonal gyrobicupola 

The [elongated pentagonal gyrobicupola] is a polyhedron with 30 vertices and 32 faces. 

elongated pentagonal orthocupolarotunda 

The [elongated pentagonal orthocupolarotunda] is a polyhedron with 35 vertices and 37 faces.

elongated pentagonal gyrocupolarotunda 

The [elongated pentagonal gyrocupolarotunda] is a polyhedron with 35 vertices and 37 faces. 

elongated pentagonal orthobirotunda 

The [elongated pentagonal orthobirotunda] is a polyhedron with 40 vertices and 42 faces. 

elongated pentagonal gyrobirotunda 

The [elongated pentagonal gyrobirotunda] is a polyhedron with 40 vertices and 42 faces. 

gyroelongated triangular bicupola 

The [gyroelongated triangular bicupola] is a polyhedron with 18 vertices and 26 faces. 

gyroelongated square bicupola 

The [gyroelongated square bicupola] is a polyhedron with 24 vertices and 34 faces. 

gyroelongated pentagonal bicupola 

The [gyroelongated pentagonal bicupola] is a polyhedron with 30 vertices and 42 faces. 

gyroelongated pentagonal cupolarotunda 

The [gyroelongated pentagonal cupolarotunda] is a polyhedron with 35 vertices and 47 faces. 

gyroelongated pentagonal birotunda 

The [gyroelongated pentagonal birotunda] is a polyhedron with 40 vertices and 52 faces. 

augmented triangular prism 

The [augmented triangular prism] is a polyhedron with 7 vertices and 8 faces.

iaugmented triangular prism 

The [biaugmented triangular prism] is a polyhedron with 8 vertices and 11 faces. 

triaugmented triangular prism 

The [triaugmented triangular prism] is a polyhedron with 9 vertices and 14 faces. 

augmented pentagonal prism 

The [augmented pentagonal prism] is a polyhedron with 11 vertices and 10 faces. 

biaugmented pentagonal prism 

The [biaugmented pentagonal prism] is a polyhedron with 12 vertices and 13 faces. 

augmented hexagonal prism 

The [augmented hexagonal prism] is a polyhedron with 13 vertices and 11 faces. 

parabiaugmented hexagonal prism 

The [parabiaugmented hexagonal prism] is a polyhedron with 14 vertices and 14 faces. 

metabiaugmented hexagonal prism 

The [metabiaugmented hexagonal prism] is a polyhedron with 14 vertices and 14 faces. 

triaugmented hexagonal prism 

The [triaugmented hexagonal prism] is a polyhedron with 15 vertices and 17 faces. 

augmented dodecahedron 

The [augmented dodecahedron] is a polyhedron with 21 vertices and 16 faces.

parabiaugmented dodecahedron 

The [parabiaugmented dodecahedron] is a polyhedron with 22 vertices and 20 faces. 

metabiaugmented dodecahedron 

The [metabiaugmented dodecahedron] is a polyhedron with 22 vertices and 20 faces. 

triaugmented dodecahedron 

The [triaugmented dodecahedron] is a polyhedron with 23 vertices and 24 faces. 

metabidiminished icosahedron 

The [metabidiminished icosahedron] is a polyhedron with 10 vertices and 12 faces. 

tridiminished icosahedron 

The [tridiminished icosahedron] is a polyhedron with 9 vertices and 8 faces. 

augmented tridiminished icosahedron 

The [augmented tridiminished icosahedron] is a polyhedron with 10 vertices and 10 faces. 

augmented truncated tetrahedron 

The [augmented truncated tetrahedron] is a polyhedron with 15 vertices and 14 faces. 

augmented truncated cube 

The [augmented truncated cube] is a polyhedron with 28 vertices and 22 faces. 

biaugmented truncated cube 

The [biaugmented truncated cube] is a polyhedron with 32 vertices and 30 faces.

augmented truncated dodecahedron 

The [augmented truncated dodecahedron] is a polyhedron with 65 vertices and 42 faces. 

parabiaugmented truncated dodecahedron 

The [parabiaugmented truncated dodecahedron] is a polyhedron with 70 vertices and 52 faces. 

metabiaugmented truncated dodecahedron 

The [metabiaugmented truncated dodecahedron] is a polyhedron with 70 vertices and 52 faces. 

triaugmented truncated dodecahedron 

The [triaugmented truncated dodecahedron] is a polyhedron with 75 vertices and 62 faces. 

gyrate rhombicosidodecahedron 

The [gyrate rhombicosidodecahedron] is a polyhedron with 60 vertices and 62 faces. 

parabigyrate rhombicosidodecahedron 

The [parabigyrate rhombicosidodecahedron] is a polyhedron with 60 vertices and 62 faces. 

metabigyrate rhombicosidodecahedron 

The [metabigyrate rhombicosidodecahedron] is a polyhedron with 60 vertices and 62 faces. 

trigyrate rhombicosidodecahedron 

The [trigyrate rhombicosidodecahedron] is a polyhedron with 60 vertices and 62 faces. 

diminished rhombicosidodecahedron 

The [diminished rhombicosidodecahedron] is a polyhedron with 55 vertices and 52 faces.

paragyrate diminished rhombicosidodecahedron 

The [paragyrate diminished rhombicosidodecahedron] is a polyhedron with 55 vertices and 52 faces. 

metagyrate diminished rhombicosidodecahedron 

The [metagyrate diminished rhombicosidodecahedron] is a polyhedron with 55 vertices and 52 faces. 

bigyrate diminished rhombicosidodecahedron 

The [bigyrate diminished rhombicosidodecahedron] is a polyhedron with 55 vertices and 52 faces. 

parabidiminished rhombicosidodecahedron 

The [parabidiminished rhombicosidodecahedron] is a polyhedron with 50 vertices and 42 faces. 

metabidiminished rhombicosidodecahedron 

The [metabidiminished rhombicosidodecahedron] is a polyhedron with 50 vertices and 42 faces. 

gyrate bidiminished rhombicosidodecahedron 

The [gyrate bidiminished rhombicosidodecahedron] is a polyhedron with 50 vertices and 42 faces. 

tridiminished rhombicosidodecahedron 

The [tridiminished rhombicosidodecahedron] is a polyhedron with 45 vertices and 32 faces. 

snub disphenoid 

The [snub disphenoid] is a polyhedron with 8 vertices and 12 faces. 

snub square antiprism 

The [snub square antiprism] is a polyhedron with 16 vertices and 26 faces.

sphenocorona 

The [sphenocorona] is a polyhedron with 10 vertices and 14 faces. 

augmented sphenocorona 

The [augmented sphenocorona] is a polyhedron with 11 vertices and 17 faces. 

sphenomegacorona 

The [sphenomegacorona] is a polyhedron with 12 vertices and 18 faces. 

hebesphenomegacorona 

The [hebesphenomegacorona] is a polyhedron with 14 vertices and 21 faces. 

disphenocingulum 

The [disphenocingulum] is a polyhedron with 16 vertices and 24 faces. 

bilunabirotunda 

The [bilunabirotunda] is a polyhedron with 14 vertices and 14 faces. 

triangular hebesphenorotunda 

The [triangular hebesphenorotunda] is a polyhedron with 18 vertices and 20 faces. 

tetrahemihexahedron 

The [tetrahemihexahedron] is a polyhedron with 7 vertices and 16 faces. 

octahemioctahedron 

The [octahemioctahedron] is a polyhedron with 13 vertices and 32 faces.

small ditrigonal icosidodecahedron 

The [small ditrigonal icosidodecahedron] is a polyhedron with 80 vertices and 72 faces. 

dodecadodecahedron 

The [dodecadodecahedron] is a polyhedron with 110 vertices and 72 faces. 

echidnahedron 

The [echidnahedron] is a polyhedron with 92 vertices and 180 faces. 



Index 

augmented dodecahedron, 12 

augmented hexagonal prism, 12 

augmented pentagonal prism, 12 

augmented sphenocorona, 16 

augmented triangular prism, 11 

augmented tridiminished icosahedron, 13 

augmented truncated cube, 13 

augmented truncated dodecahedron, 14 

augmented truncated tetrahedron, 13 

biaugmented pentagonal prism, 12 

biaugmented triangular prism, 12 

biaugmented truncated cube, 13 

bigyrate diminished rhombicosidodecahedron, 15 

bilunabirotunda, 16 

cube, 1 

cuboctahedron, 2 

decagonal antiprism, 4 

decagonal prism, 4 

diminished rhombicosidodecahedron, 14 

disphenocingulum, 16 

dodecadodecahedron, 17 

dodecahedron, 1 

echidnahedron, 17 

elongated pentagonal cupola, 8 

elongated pentagonal dipyramid, 8 

elongated pentagonal gyrobicupola, 10 

elongated pentagonal gyrobirotunda, 11 

elongated pentagonal gyrocupolarotunda, 11 

elongated pentagonal orthobicupola, 10 

elongated pentagonal orthobirotunda, 11 

elongated pentagonal orthocupolarotunda, 10 

elongated pentagonal pyramid, 7 

elongated pentagonal rotunds, 8 

elongated square cupola, 8 

elongated square dipyramid, 8 

elongated square gyrobicupola, 10 

elongated square pyramid, 7 

elongated triangular cupola, 8 

elongated triangular dipyramid, 8 

elongated triangular gyrobicupola, 10 

elongated triangular orthobicupola, 10 

elongated triangular pyramid, 7 

great dodecahedron, 1 

great icosahedron, 2 

great rhombicosidodecahedron, 3 

great rhombicuboctahedron, 2 

great stellated dodecahedron, 2 

gyrate bidiminished rhombicosidodecahedron, 15 

gyrate rhombicosidodecahedron, 14 

gyrobifastigium, 9 

gyroelongated pentagonal bicupola, 11 

gyroelongated pentagonal birotunda, 11 

gyroelongated pentagonal cupola, 9 

gyroelongated pentagonal cupolarotunda, 11 

gyroelongated pentagonal pyramid, 7 

gyroelongated pentagonal rotunda, 9 

18 

gyroelongated square bicupola, 11 

gyroelongated square cupola, 9 

gyroelongated square dipyramid, 8 

gyroelongated square pyramid, 7 

gyroelongated triangular bicupola, 11 

gyroelongated triangular cupola, 8 

hebesphenomegacorona, 16 

hexagonal antiprism, 4 

hexagonal prism, 4 

hexahedron, 1 

hexakis icosahedron, 6 

hexakis octahedron, 5 

icosahedron, 1 

icosidodecahedron, 3 

metabiaugmented dodecahedron, 13 

metabiaugmented hexagonal prism, 12 

metabiaugmented truncated dodecahedron, 14 

metabidiminished icosahedron, 13 

metabidiminished rhombicosidodecahedron, 15 

metabigyrate rhombicosidodecahedron, 14 

metagyrate diminished rhombicosidodecahedron, 15 

octagonal antiprism, 4 

octagonal prism, 4 

octahedron, 1 

octahemioctahedron, 16 

parabiaugmented dodecahedron, 13 

parabiaugmented hexagonal prism, 12 

parabiaugmented truncated dodecahedron, 14 

parabidiminished rhombicosidodecahedron, 15 

parabigyrate rhombicosidodecahedron, 14 

paragyrate diminished rhombicosidodecahedron, 15 

pentagonal antiprism, 4 

pentagonal cupola, 7 

pentagonal dipyramid, 7 

pentagonal gyrobicupola, 9 

pentagonal gyrocupolarotunda, 10 

pentagonal hexecontahedron, 6 

pentagonal icositetrahedron, 5 

pentagonal orthobicupola, 9 

pentagonal orthobirotunda, 10 

pentagonal orthocupolarontunda, 10 

pentagonal prism, 4 

pentagonal pyramid, 6 

pentagonal rotunda, 7 

pentakis dodecahedron, 6 

rhombic dodecahedron, 5 

rhombic triacontahedron, 5 

rhombicosidodecahedron, 3 

rhombicuboctahedron, 2 

small ditrigonal icosidodecahedron, 17 

small stellated dodecahedron, 1 

snub cube, 3 

snub disphenoid, 15 

snub dodecahedron, 3

snub square antiprism, 15 

sphenocorona, 16 

sphenomegacorona, 16 

square antiprism, 4 

square cupola, 6 

square gyrobicupola, 9 

square orthobicupola, 9 

square pyramid, 6 

tetrahedron, 1 

tetrahemihexahedron, 16 

tetrakis hexahedron, 5 

trapezoidal hexecontahedron, 6 

trapezoidal icositetrahedron, 5 

triakis icosahedron, 6 

triakis octahedron, 5 

triakis tetrahedron, 5 

triangular cupola, 6 

triangular dipyramid, 7 

triangular hebesphenorotunda, 16 

triangular orthobicupola, 9 

triangular prism, 3 

triaugmented dodecahedron, 13 

triaugmented hexagonal prism, 12 

triaugmented triangular prism, 12 

triaugmented truncated dodecahedron, 14 

tridiminished icosahedron, 13 

tridiminished rhombicosidodecahedron, 15 

trigyrate rhombicosidodecahedron, 14 

truncated cube, 2 

truncated dodecahedron, 3 

truncated icosahedron, 3 

truncated octahedron, 2 

truncated tetrahedron, 2

ENTRY POTENTIAL THEORY 

[ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: ”T. Ransford”,”Potential theory 

in the complex plane”. 

Analytic 

[Analytic] Let D ⊂ C be an open set. A continuous function f : D → C is called analytic in D, if for all z ∈ D 

the complex partial derivative 

∂f 1 

:= lim (f(z + h) − f(z)) 

∂z |h|→0 h 

exists and is finite. Analytic functions are also called holomorphic. Properties: the sum and the product of 

analytic functions are analytic. If fn is a sequence of analytic maps which converges uniformly on compact 

subsets of D to a function f, then f is analytic too. 

complex partial derivative 

Define the [complex partial derivative] of a complex function f(z) = f(x + iy) in the complex plane is defined 

as ∂f 

∂z 

= 1 

2 

( ∂ 

∂x 

− i ∂ 

∂y )f. 

conformal map 

A [conformal map] is a differentiable map from the complex plane to the complex plane which preserves angles. 

• every conformal map which has continuous partial derivatives is analytic. 

• An analytic function f is conformal at every point where its derivative f ′ (z) is different from 0. 

Dirichlet problem 

Solution of the [Dirichlet problem]. If D is a regular domain in the complex plane and f is a continous function 

on the boundary of D, then there exists a unique harmonic function f on D such that h(z) = f(z) for all 

boundary points of D. 

Dirichlet problem 

Let K be a compact subset of the complex plane. Let P (K) the set of all Borel probability measure on K. 

A measure ν maximizing the potential theoretical energy in P (K) is called an [equilibrium measure] of K. 

Properties: 

• every compact K has an equilibrium measure. 

• if K is not polar then the equilibrium measure is unique.

fine topology 

The [fine topology] on the complex plane is defined as the coarsest topology on the plane which makes all 

subharmonic functions continuous. 

Frostman’s theorem 

[Frostman’s theorem]: If ν is the equilibrium measure on a compact set K, then the potential pν of ν is bounded 

below by I(ν) everywhere on C. Furthermore, pν = I(ν) everywhere on K except on a Fsigma polar subset E 

of the boundary of K. 

Frostman’s theorem 

A function h on the complex plane is called harmonic in a region D if it satisfies the mean value property on 

every disc contained in D. 

harmonic measure 

A [harmonic measure] wD on a domain D is a function from D to the set of Borel probability measures on 

the boundary of D. The measure for z is defined as the functional g ↦→ HD(g)(z), where HD(g) is the Perron 

function of g on D. 

• if the boundary of D is non-polar, there exists a unique harmonic measure for D. 

• if D = Im(z) < 0, then wD(z, a, b) = arg((z − b)/(z − a))/π 

Harnack inequality 

The [Harnack inequality] assures that for any positive harmonic function h on the disc D(w, R) and for any 

r < R and 0 < t < 2P i 

h(w)(R − r)/(R + r) ≤ h(w + re it ≤ h(w)(R + r)/(R − r) 

extended Liouville theorem 

The [extended Liouville theorem]: if f is subharmonic on the complex plane C − E, where E is a closed polar 

set and f is bounded above then f is constant. 

generalized Laplacian 

The [generalized Laplacian] ∆(f) of a subharmonic function f on a domain D is the Radon measure µ on D 

defined as the linear functional g ↦→ � 

u∆g dA. The Laplacian of a subharmonic function is also called the 

D 

Riesz measure. The Laplacian is known to exist and is unique. If pµ is the potential associated to µ, then 

∆pµ = µ.

Hadamard’s three circle theorem 

[Hadamard’s three circle theorem] assures that for any subharmonic function f on the annulus {r < |z| < R} 

the function M(f, r) = sup |z|=rf(z) is an increasing convex function of log(r). 

Jensen formula 

[Jensen formula] If f is holomorphic in the disc D = B(0, R), r < R and a1, ..., an are the zeros of f in the closure 

of D counted with multiplicity, then � 2pi 

0 

log |f(rexp(it))| dt = log |f(0)| + N log(r) − � n 

j=1 log|aj|. 

Jensen formula 

If f is a subharmonic function in a neighborhood of a point z in the complex plane, then the limit 

limrto0 M(f, r)/ log(r) exists and is called the [Lelong number] of f at z. Here M(f, r) = sup |z|=r f(z). 

hyperbolic domain 

An open set in the extended complex plane is a [hyperbolic domain] if there is a subharmonic function on G 

that is bounded above and not constant on any component of G. A domain which is not hyperbolic is called a 

parabolic domain. Known facts: 

• every bounded region is a hyperbolic domain (take f(z)=Re(z)). 

• an open not connected set is hyperbolic. 

• the complex plane is not a hyperbolic domain 

Perron function 

The [Perron function] for a domain D is defined as the functional assigning to a continous function g on the 

boundary of D the value HD(g), which is the supremum of all subharmonic functions u satisfying sup z→w u(z) ≤ 

g(w). 

potential 

A subharmonic function f is called a [potential] if f = pµ, where µ = ∆f is the Laplacian of f and pmu(z) = 

− � 

log |z − w|dµ(w)/(2π) is the potential defined by µ. 

D

logarithmic capacity 

The [logarithmic capacity] of a subset E of the complex plane is defined as c(E) = sup µ exp(−I(µ)), where I(µ) 

is the potential theoretical energy of µ and the supremum is taken over all Borel probability measures mu on C 

whose support is a compact subset of E. Known facts: 

• c(E) = 0 if and only if E is polar. 

• a disc of radius r has capacity r 

• a line segment of length h has capacity h/4. 

• if K has diameter d, then c(K) ≤ d/2. 

• if K has area A, then c(A) ≥ (A/π) 1/2 . 

mean value property 

The [mean value property] tells that if h is a harmonic function in the disc D(w, R) and 0 < r < R, then 

h(w) = � 2pi 

0 h(w + reit )dt/(2π). 

polar set 

A subset S of the complex plane is called a [polar set] if the potential theoretical energy I(µ) is −∞ for every 

finite Borel measure µ with compact support supp(µ) in S. Properties of polar sets: 

• every countable union of polar sets is polar. 

• every polar set has Lebesgue measure zero. 

Poisson integral formula 

The [Poisson integral formula]: if h is harmonic on the disk D(w, R ′ ), then for all 0 < r < R < R ′ and 

h(w + Reis )(R2 − r2 )/(R2 − 2Rrcos(s − t) + r2 )ds/(2π) 

0 < t < 2π, h(w + re it ) = � 2pi 

0 

potential theoretical energy 

The [potential theoretical energy] I(µ) of a finite Borel measure µ of compact support on the complex plane is 

defined as 

� � � � 

I(µ) = 

log|z − w|dµ(w)dµ(z) . 

C 

C

potential theoretical energy 

A function f on an open subset U of the complex plane is called [subharmonic] if it is upper semicontinous and 

satisfies the local submean inequality. Examples: 

• if g is holomorphic then f = log |g| is subharmonic 

• if µ is a Borel measure of compact support, then f(z) = � log |z − w| dµ(w) is subharmonic. 

• any harmonic function is subharmonic. 

• if g is subharmonic, then exp(g) is subharmonic. 

regular 

A boundary point w of a domain D is called [regular] if there exists a barrier at w. A barrier is a subharmonic 

function f defined in a neighborhood N of w which is negative on D ∩ N and such that limz→w f(z) = 0. It is 

known that z is a regular boundary point if and only if the complement of D is non-thin at z. 

irregular 

A boundary point w of a domain D is called [irregular] if it is not regular. It is known that if z has a neighborhood 

N such that N intersected with the boundary of D is polar, then z is irregular. 

irregular 

A domain D for which every point is regular is called a [regular domain]. For example, a simply connected 

domain D such that the complement of D in the Riemann sphere contains at least two points, is regular. 

Riemann mapping Theorem 

The [Riemann mapping Theorem]: if D is a simply connected proper subdomain of the complex plane, there 

exists a conformal map of D onto the unit disc. 

Riesz decomposition theorem 

The [Riesz decomposition theorem] tells that every subharmonic function f can be written as f = pµ + h, where 

µ = ∆f is the Laplacian of f, 2πpµ is the potential of µ and where h is harmonic.

submean inequality 

The local [submean inequality] for a function in the complex plane tells that there exists R > 0 such that for 

all 0 < r < R one has 

� 2pi 

f(w) ≤ f(w + re it ) dt/(2pi) . 

0 

submean inequality 

Let f be a subharmonic function on a domain D. The [maximum principle] says that if f attains a global 

maximum in the interior of D then f is constant. 

thin set 

A subset S of the complex plane is called a [thin set] if for all w in the closure of S − w and all subharmonic 

functions f, lim sup z→w f(z) = f(w). Examples: 

• every single point in the interiour of S is thin. 

• Fσ polar sets S are thin at every point. 

• connected sets of cardinality larger than 1 are non-thin at every point of their closure 

• A domain S is thin at a point z ∈ S if and only if z is regular. 

Wiener criterion 

The [Wiener criterion] gives a necessary and sufficient condition for a set S to be thin at a point w. Let S be a 

Fσ subset of C and let w be in S. Let a < 1 and define Sn = zinS, a n < |z − w| < a ( n − 1) . The criterion says 

that S is thin at w if and only if � 

n≥1 n/log(2/c(Sn)) < infinity, where 



Index 

Analytic, 1 

complex partial derivative, 1 

conformal map, 1 

Dirichlet problem, 1 

extended Liouville theorem, 2 

fine topology, 2 

Frostman’s theorem, 2 

generalized Laplacian, 2 

Hadamard’s three circle theorem, 3 

harmonic measure, 2 

Harnack inequality, 2 

hyperbolic domain, 3 

irregular, 5 

Jensen formula, 3 

logarithmic capacity, 4 

mean value property, 4 

Perron function, 3 

Poisson integral formula, 4 

polar set, 4 

potential, 3 

potential theoretical energy, 4, 5 

regular, 5 

Riemann mapping Theorem, 5 

Riesz decomposition theorem, 5 

submean inequality, 6 

thin set, 6 

Wiener criterion, 6 

7

Estimation theory 

[Estimation theory] part of statistics with the goal of extracting parameters from noise-corrupted observations. 

Applications of estimation theory are statistical signal processing or adaptive filter theory or adaptive optics 

which allows for example image deblurring. 

Parameter estimation problem 

[Parameter estimation problem] determine from a set L of observations a parameter vector. A paremeter 

estimate is a random vector. The estimation error ɛ is the difference between the estimated parameter and the 

parameter itself. The mean-squared error is given by the mean squared error matrix E[ɛ T ɛ]. It is a correlation 

matrix. 

Biased 

[Biased] An estimate is said to be biased, if the expected value of the estimate is different than the actual value. 

Asymptotically unbiased 

[Asymptotically unbiased] An estimate in statistics is called asymptotically unbiased, if the estimate becomes 

unbiased in the limit when the number of data points goes to infinity. 

Consistent estimate 

[Consistent estimate] An estimate in statistics is called consistent if the mean squared error matrix E[ɛ T ɛ] 

converges to the 0 matrix in the limit when the number of data points goes to infinity. 

Mean squared error matrix 

The [Mean squared error matrix] is defined as E[ɛ T ɛ], where ɛ is the difference between the estimated parameter 

and the parameter itself. 

efficient 

An estimator in statistics is called [efficient] if its mean-squared error satisfies the Cramer-Rao bound. 

Cramer-Rao bound 

[Cramer-Rao bound] The mean-squared error E[ɛ T ɛ] for any estimate of a parameter has a lower bound which 

is called the Cramer-Rao bound. In the case of unbiased estimators, the Cramer-Rao bound gives for each error 

ɛi the estimate 

E[ɛ 2 i ] ≥ [F −1 ]ii .

Fisher information matrix 

The [Fisher information matrix] is defined as the expectation of the Hessian F = E[H(− log(p))] = 

E[grad(log(p))grad(log(p)) T ] of the conditional probability p(r|θ). 

maximum likelihood estimate 

The [maximum likelihood estimate] is an estimation technique in statistics to estimate nonrandom parameters. 

A maximum likelyhood estimate is a maximizer of the log likelihood function log(p(r, θ). It is known that 

the maximum likelihood estimate is asymptotically unbiased, consistent estimate. Furthermore, the maximum 

likelihood estimate is distributed as a Gaussian random variable. 

Example. If X is a normal distributed random variable with unknown mean θ and variance 1, the likelyhood 

function is p(r, θ) = 1 

√ 2π e −(r−θ)2 /2 and the log-likelyhood function is log(p(r, θ)) = −(r − θ) 2 /2 + C. The 

maximum likelyhood estimate is r. 

The maximum likelihood estimate is difficult to compute in general for non-Gaussian random variables. 



Index 

Asymptotically unbiased, 1 

Biased, 1 

Consistent estimate, 1 

Cramer-Rao bound, 1 

efficient, 1 

Fisher information matrix, 2 

maximum likelihood estimate, 2 

Mean squared error matrix, 1 

Parameter estimation problem, 1 

3

ENTRY TOPOLOGY 

[ENTRY TOPOLOGY] Authors: Oliver Knill 2003, John Carlson 2003-2004 Literature: http://at.yorku.ca/cgibin/bell/props.cgi 

Alexander compactification 

The [Alexander compactification] Y of a Hausdorff space (X, O) is the topological space (Y = X ∪ x, P ), where 

x is an additional point. The topology P consists of the elements in O and the complements of closed subsets 

as neighborhoods of that point. The new topological space Y is compact. 

Alexander’s subbase theorem 

The [Alexander’s subbase theorem]: if every open cover of a topological space X has a finite sub-cover then X 

is compact. 

arc-connected 

A topological space is called [arc-connected] if any two points can be connected by a path, a continuous image 

of an interval. Path connected is stronger than connected but not equivalent: the subset {(x, sin(1/x)), x ∈ 

R + } ∪ {(0, y), −1 ≤ y ≤ 1 } of the plane with topology induced from the plane is connected but not path 

connected. Arc-connected is also called path-connected. 

Baire category 

[Baire category] is a measure for the size of a set in a topological space. Countable unions of nowhere dense sets 

are called of the first categorie or meager, any other set of second category. Complements of meager sets are 

called residual. Baire category is used to quantify certain sets. For example it is known that ”most” numbers 

are Liouville numbers in the sense that they form a residual set among all real numbers. 

Baire space 

A [Baire space] is a topological space with the property that the intersection of countable family of open dense 

subsets is dense. 

Baire category theorem 

The [Baire category theorem]: a complete metric space is a Baire space.

all 

A [ball] in a metric space is a set of the form {y | d(x, y) < r }. The closure of an open ball is a closed ball. To 

make clear that a ball is open, one sometimes calls it also open ball. 

barrier function 

A [barrier function] for a set S in a topological space (X, O) is a nonnegative continuous function f defined on 

X which is zero in S and positive in the complement of S. A barrier function is sometimes also called a penalty 

function. 

basis 

A [basis] of a topological space (X, T ) is a subset B of T such that 

• the empty set is in B, 

• arbitrary unions of sets in B are in B, 

• the intersection of two sets in B is a union of sets in B. 

A basis B defines the topology (X, T ). Every A ∈ T is a union of elements in B. An example: if (X, d) is a 

metric space then the set of all balls {y | d(x, y) < 1/k }, where x is taken from a dense set in X and k is a 

positive integer form a basis. 

bicontinuous 

A function is [bicontinuous] if it is continuous invertible and has a continuous inverse. A bicontinuous function 

is also called a homeomorphism. 

bounded 

A subset of a metric space is [bounded] if it is contained in some ball of finite radius. 

boundary 

The [boundary] of a set A in a topological space (X, T ) is the the set C \ B, where C is the closure of A and B 

is the interior of A. Examples: 

• if A is the open unit disc in the plane, then the boundary is the unit circle. 

• in a discrete topological space, the boundary of any set is empty. 

Cantor set 

A [Cantor set] is a topological space which is homeomorphic to the Cantor middle set.

Cantor middle set 

The [Cantor middle set] is the subset of the unit interval which is the complement of � ∞ 

n=1 Yn, where Y1 = 

(1/3, 2/3), Y2 = (1/9, 2/9) ∪ (7/9, 8/9) etc. are sucessive middle sets. It is a fractal with Hausdorff dimension 

log(2)/ log(3). 

Cantor middle set 

A topological space homeomorphic to a ball in Euclidean space is called a [cell]. Examples of cells are polyhedra 

in three dimensional space. 

closure 

The [closure] of a set A in a topological space (X, T ) is the intersection of all closed sets in X, which contain 

A. One writes Y for the closure of Y . 

dense 

A set A is called [dense] in a topological space (X, T ), if every open set Y ∈ O in X contains at least one point 

in A. 

finer 

A topology (X, T ) is [finer] than a topology (X, S) if S is a subset of T . In that case, (X, S) is called a coarser 

topology than (X, T ). Examples: 

• the discrete topology on X is finer than any other topology on X. 

• A set S of subsets of X defines a topology, the coarsest topology O which contains S. 

topological space 

A [topological space] is a pair (X,T) where T is a set of subsets of X satisfying 

• ∅ ∈ T , 

• if A, B ∈ T , then A ∩ B ∈ T , 

• an arbitrary union of subsets in T is in T. 

Elements in T are called open sets. The complement of an open set is called a closed set. Examples: 

• the discrete topology on X: T is all subsets of X 

• the indiscrete topology on X: T contains only X and the empty set, 

• the cofinite topology: T is the set of all subsets A such that their complement in X is a finite set. 

• (X, d) metric space: T is the set of sets A such that for x in A, also a small ball B = {|y − x| < a} is 

contained in A.

open set 

An [open set] of a topological space (X, T ) is an element of T . 

closed set 

A [closed set] is the complement of an open set in a topological space (X, T ). A closed set contains all its limit 

points. 

continuous 

A map f between two topological spaces (X, T ) and (Y, S) is called [continuous] if the inverse image of any 

open set is open: for all A ∈ S one has f ( − 1)(A) ∈ T . Note that f does not need to be invertible: one defines 

f −1 (A) = {x ∈ X|f(x) ∈ A}. Examples of results known: 

• The composition of two continuous maps is continuous. 

• Every map on the discrete topological space is continuous. 

• A map between metric spaces is continuous if and only if it is sequential continuous: for any xn → x, one 

has f(xn) → f(x). 

• A map between topological spaces is continuous if for every net xt → x, the net f(xt) converges to f(x). 

homeomorphism 

An invertible map f between two topological spaces (X, T ) and (Y, S) is called a [homeomorphism] if f and the 

inverse of f are both continuous. 

homeomorphic 

[homeomorphic] If there exists a homeomorphism between two topological spaces, the topological spaces are 

called homeomorphic. 

connected 

A topological space (X, T ) is called [connected], if there are no two disjoint open sets U, V whose union is X. 

For a connected topological space, the empty set ∅ or X are the only sets which are both open and closed. A 

subset A of a topological space is connected if it is connected with the on A induced topology: there are no 

disjoint open sets U, V whose union contains A.

locally connected 

A topological space is [locally connected] if every point has arbitrarily small neighborhoods which are connected. 

Examples. 

• A union of disjoint open intervals on the real line is locally connected but not connected. 

• The union of the graphs of f(x) = 2 sin(1/x) and g(x) = 1 and the y-axes all intersected with the set 

{y > 1} is connected but not locally connected because small neighborhoods of the point (0, 1) are not 

connected. 

Hausdorff 

A topological space (X, T ) is called [Hausdorff] if for every two points x, y ∈ X, there are disjoint open sets 

U, V ∈ T such that x ∈ U and y ∈ V . This is refined through seperation axioms, T 0, ..., T 4. Hausdorff is also 

called T 2. Any metric space is Hausdorff: if d is the distance betwen x and y, then balls of radius d/3 around 

x and y seperate the points. The plane X with semimetric d(x, y) = |x1 − y1| is not Hausdorff: the points 

x = (0, −1) and y = (0, 1) can not be seperated by open sets. 

seperation axioms 

[seperation axioms] define classes of topological spaces with decreasing seperability properties: T 4 ⇒ T 3 ⇒ 

T 2 ⇒ T 1 ⇒ T 0. 

T0 space: 

T1 space: 

T2 space: 

T3 space: 

T4 space: 

for two different points x, y in X one of the points has an open 

neighborhood U not containing the other point. 

for two different points x, y in X there exists an open neighborhood 

U of x and an open neighborhood V of y. such that x is not 

in V and y is not in U. 

also called Hausdorff” two different points x, y can be seperated 

with disjoint neighborhoods U, V . 

T1 and regular: any point x and any closed set F not containing 

x can be seperated by two disjoint neighborhood. 

T1 and normal: any two disjoint sets F ,G can be separated by 

two disjoint open sets. 

It is known that a T4 space with a countable basis is metrizable. 

Hausdorff topology 

The [Hausdorff topology] is a metric on the set of closed bounded subsets of a complete metric space. The 

distance between two sets A and B is the infimum over all r for which A is contained in a r-neighborhood of B 

and B is contained in a r-neighborhood of A. 

Lindeloef 

A topological space is called [Lindeloef] if every open cover of X contains a countable subcover.

compact 

A topological space is called [compact] if every open cover of X contains a finite subcover. Examples of results 

known about compactnes: 

• Heine-Borel theorem: a closed interval in the real line is compact. 

• If f : X → Y is continuous and onto and X is compact, then Y is compact. As a consequence, a 

continouous function on a compact subspace has both a maximum and a minimum. 

• In a Hausdorff space, compact sets are closed. 

• In a metric space, compact sets are closed and bounded. 

• Closed subsets of compact spaces are compact. 

• Tychonof theorem: the product of a collection of compact spaces is compact. 

countably compact 

A topological space is called [countably compact] if every countable open cover of X contains a finite subcover. 

locally compact 

A topological space is called [locally compact] if every point has a neighborhood, which has a compact closure. 

Examples. 

• The real line is compact but not locally compact. 

• A compact Hausdorff space is locally compact. 

• The n-dimensional Euclidean space R n is lcally compact but not compact. 

locally compact 

A set U of open sets in a topological space (X, O) is called locally finite if every point x ∈ X has a neighborhood 

V , such that V has a nonempty intersection with only finitely many elements in U. 

paracompact 

A topological space (X,O) is called [paracompact] if every open cover has a countable, locally finite subcover. 

relatively compact 

A subset A of a topological space (X, T ) is called [relatively compact] if the closure of A is compact.

filter 

A [filter] on a nonempty set X is a set of subsets F satisfying 

• X is in F , but the empty set ∅ is not in F. 

• If A and B are in F, then their intersection is in F . 

• If A is in F and B is a subset of A, then B is in F . 

Examples: 

• Principal filter for a nonempty subset A consists of all subsets of X which contain A. 

• Frechet filter for an infinite set consists of all subsets of X such that their complement is finite. 

• Neighborhood filter of a point x in a topological space (X,T) is the set of open neighborhoods of x. 

• Elementary filter for a sequence xn in X consists of all sets A in X such that xn is in A for large enough 

n. 

converges 

A sequence xn in a topological space [converges] to a point x, if for every neighborhood U of x, there exists an 

integer m, such that for n > m one has xn ∈ U. 

Filter convergence 

[Filter convergence] A filter F converges to x in a topological space (X, T ) if F contains the neighborhood filter 

G of x, that is if F contains all neighborhoods of x. For example, an elementary filter to a sequence xn converges 

to a point x, if and only if xn converges to x. 

accumulation point 

A point y is called an [accumulation point] of a filter F , if there exists a filter G containing F such that G 

converges to x. 

directed 

A set M is called [directed] if there exists a partial order (M,

Koch curve 

The [Koch curve] is a fractal in the plane. It has Hausdorff dimension log(4)/ log(3). It is constructed by 

building an equilateral triangle on the middle third of each side of a given equilateral triangle K0 leading to a 

curve K1 and recursively build Kn+1 from Kn by replacing each middle third of a line segment in Kn with a 

triangle. The curve is the limit of Kn, when n goes to infinity. 

metrizable 

A topological space (X, T ) is called [metrizable] if there exists a metric on X such that the topology generated 

by the metric is T . 

metric space 

A [metric space] (X, d) is a set X with a function d from X × X → [0, ∞) satisfying d(x, y) = d(y, x), d(x, y) = 

0 ⇔ x = y and d(x, z) ≤ d(x, y) + d(y, z)). The set T = {U ⊂ X | ∀x ∈ X, ∃r > 0, Br(x) = {y | d(x, y) < r} ⊂ 

U} defines a topological space (X, T ). 

metric space 

A [metric space] (X, d) is a set X with a nonnegative function d from X×X satisfying d(x, y) = d(y, x),d(x, y) = 0 

if and only if x = y and d(x, z) ≤ d(x, y)+d(y, z). A metric space defines a topological space (X,T): the topology 

T is the set of subsets A of X such that for all points x ∈ A, there is a small ball d(y, x) < r which is also 

contained in A. 

net 

A [net] with values in a topological space X is a function f: D -¿ X, where D is a directed set. For example: if 

D is the set of natural numbers, then a net is a sequence. A net defines a filter F : it is the set of all sets A such 

that xt is eventually in A. A net xt converges to a point x if and only if the associated filter converges to x. 

open cover 

[open cover] A subset U of O, where (X, O) is a topological space is called an open cover of X if the union of 

all elements in U is X. If U and V are open covers and V ⊂ U, then V is called a subcover of U. 

product space 

The [product space] between topological spaces is defined as (X × Y, O × P ), where X × Y is the set of all pairs 

(x, y), x ∈ X, y ∈ Y and O ×P is the coarsest topological space which contains all products A×B, where A ∈ O 

and B ∈ P . For example, if (X, O) = (Y, P ) are both the real line with the topology generated by d(x, y) = 

|x−y|, then the product space is homeomorphic to the plane with the metric d(x, y) = � (x1 − x2) 2 + (y1 − y2) 2 .

second countable 

A topological space is called [second countable], if it has a countable basis. Example. Every seperable metric 

space is second countable. Especially, every finite-dimensional Euclidean space is second countable. 

metrizable 

A topological space is called [metrizable] if there exists a metric d on the set X that induces the topology of X. 

Any regular space with a countable basis is metrizable. 

homotopic 

[homotopic] If f and g are continuous maps from the topological space X to a topological space Y , we say that 

f is homotopic to g if there is a continuous map F from X ×I to Y , such that F (x, 0) = f(x) and F (x, 1) = g(x) 

for all x. For example, the maps f(x) = x 2 and g(x) = sin(x) on the real line are homotopic, because we can 

define F (x, t) = (1 − t)x 2 + t sin(x). The maps f(x) = x and g(x) = sin(2πx) on the circle are not homotopic. 

While g is homotopic to the constant function h(x) = 0, the map f(x) can not be deformed to a constant 

without breaking continuity. 

induced topology 

The [induced topology] on a subset A of X, where (X, T ) is a topological spoace is the the topological space 

(A, {Y ∩ A}Y ∈T ). 

path homotopic 

[path homotopic] If f and g are and continuous homotopic maps from an interval to a space X, we say f and g 

are path homotopic if their images have the same end points. For instance, the maps f(x) = x 2 and g(x) = x 3 

are path homotopic on the closed interval from 0 to 1. The maps f(x) = 2x 2 and g(x) = x 3 are homotopic on 

the unit interval but not path homotopic. 

loop 

A [loop] is a path in a topological space that begins and ends at the same point. A loop is also called a closed 

curve. Loops play a role in definitions like simply connected: a topological space is simply connected if every 

loop is homotopic to a constant loop which is a fancy way telling that every closed path can be collapsed inside 

X to a point. 

fundamental group 

The [fundamental group] of a topological space at a point is the set of homotopy classes of loops based at that 

point.

Topologist’s Sine Curve 

The [Topologist’s Sine Curve] is the union S of the graph of the function sin(1/x) on the positive real axes R + 

with the y-axes. It an example of a topological space which is connected but not path-connected. Proof: if S 

were path-connected, there would exist a path r(t) = (x(t), y(t)) connecting the two points (0, 1) and (0, π). 

The set {t|r(t) ∈ S} is closed. Let T be the largest t in that set for which r(t) is in the y-axes. Then x(T)=0 

and r(t)=(x(t),sin(1/x(t)) for t > T . Because there are times tn > tn−1 > T, tn → T for which y(tn) = (−1) n , 

the function r(t) can not be continuous at t = T . 

Urysohn lemma 

The [Urysohn lemma] tells that if X is a normal space and A and B are disjoint closed subsets of X, then there 

exists a continuous map f from X to the unit interval such that f(x) = 0 for all x ∈ A, and f(x) = 1 for all 

x ∈ B. 

Proof: use the normality of X to construct a family Up of open sets of X indexed by the rational numbers P 

in the unit interval so that for p < q, the closure of Up is contained in Uq. These sets are simply ordered in the 

same way that their subscripts are ordered in the real line. Given some enumeration of the rationals, where 1 

and 0 are the first two elements of the enumeration, define U1 = X \ B. Because A is a closed set contained in 

the open set U, there is by normality an open set U0 such that A ⊂ U0 and the closure of U0 is a subset of U1. 

In general, let Pn denote the set consisting of the first n rational numbers in the sequence. Suppose that Up is 

defined for all rational numbers p in a set Pn, then p < q implies that the closure of Up is a subset of Uq. If r is 

the next rational number in the sequence; we define Ur: the set Pn+1 = Pn ∪ {r} is a finite subset of the unit 

interval and has a simple ordering induced by the ordering of the real line. In a finite simply ordered set, every 

element, other than the largest and smallest, has an immediate predecessor and and immediate successor. 0 is 

the smallest and 1 is the largest element of the simply ordered set Pn+1, and r /∈ {0, 1}. So r has an immediate 

predecessor p ∈ Pn+1 and an immediate successor q ∈ Pn+1. The sets Up and Uq are already defined, and the 

closure of Up is contained in Uq by the induction hypothesis. Because X is normal, we can find an open set 

Ur such that the closure of Up is contained in Ur and the closure of Ur is contained in Uq. Now the induction 

condition holds for every pair of elements of Pn+1. 



Index 

accumulation point, 7 

Alexander compactification, 1 

Alexander’s subbase theorem, 1 

arc-connected, 1 

Baire category, 1 

Baire category theorem, 1 

Baire space, 1 

ball, 2 

barrier function, 2 

basis, 2 

bicontinuous, 2 

boundary, 2 

bounded, 2 

Cantor middle set, 3 

Cantor set, 2 

closed set, 4 

closure, 3 

compact, 6 

connected, 4 

continuous, 4 

converges, 7 

countably compact, 6 

dense, 3 

directed, 7 

filter, 7 

Filter convergence, 7 

finer, 3 

fundamental group, 9 

Hausdorff, 5 

Hausdorff topology, 5 

homeomorphic, 4 

homeomorphism, 4 

homotopic, 9 

induced topology, 9 

interior, 7 

Koch curve, 8 

Lindeloef, 5 

locally compact, 6 

locally connected, 5 

loop, 9 

metric space, 8 

metrizable, 8, 9 

net, 8 

open cover, 8 

open set, 4 

paracompact, 6 

path homotopic, 9 

product space, 8 

relatively compact, 6 

11 

second countable, 9 

seperation axioms, 5 

topological space, 3 

Topologist’s Sine Curve, 10 

Urysohn lemma, 10

ENTRY ARTIFICIAL INTELLIGENCE - Department of Mathematics ...

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