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 +------------------------------------------------------------ Babelfish Online translation system from Systran. +------------------------------------------------------------ | 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 +------------------------------------------------------------ pilot Programmed Inquiry Learning Or Teaching. +------------------------------------------------------------ | 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. COUNT: 22 ENTRY ABSTRACT ALGEBRA Authors: started Oliver Knill: September 2003 Literature: Lecture notes +------------------------------------------------------------ | additive +------------------------------------------------------------ A function f: G to 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 cup 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 g h = h g 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 g N and operation (g N) (h N) = (g h) N which is defined because N was assumed to be normal. For example, if G is the group of additive integers and N=k N with an integer k, then G/N = Z_k 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 (a+b)+c = a+(b+c) associativity a+0 = a zero element forall a exists b a+b=0 inverse Examples: 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 Delta B = (A cup B) setminus (A cap B) form a group. The inverse of A is A itself because A Delta A = emptyset, the zero element is emptyset. +------------------------------------------------------------ | 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 h g 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 Delta and multiplication cap 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 x 2 matrices form a noncommutative ring the set of polyomials with real coefficients (x^2+pi x+2) * (x+5x) = 6 x^3 + 6 pi x^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 phi between two groups G,H is a map f: G to H which has the property phi(g*h) = phi(g) * phi(h) and phi(0)=0 for all elements g,h in 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 phi(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 phi(A)= det(A), we have phi(x*y) = phi(x) phi(y). +------------------------------------------------------------ | isomorphism +------------------------------------------------------------ An isomorphism phi 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 a_0+a_1 z+... + 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(sqrtd), 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(xi), where xi is a n'th root of 1. It is a field extension of degree phi(n), where phi(n) is the Euler function. +------------------------------------------------------------ | octonions +------------------------------------------------------------ The octonions can be written as linear combinations of elements e_0,e_1,e_2,...,e_7. The multiplication is determined by the multiplication table * 1 e_1 e_2 e_3 e_4 e_5 e_6 e_7 1 1 e_1 e_2 e_3 e_4 e_5 e_6 e_7 e_1 e_1 -1 e_4 e_7 -e_2 e_6 -e_5 -e_3 e_2 e_2 -e_4 -1 e_5 e_1 -e_3 e_7 -e_6 e_3 e_3 -e_7 -e_5 -1 e_6 e_2 -e_4 e_1 e_4 e_4 e_2 -e_1 -e_6 -1 e_7 e_3 -e_5 e_5 e_5 -e_6 e_3 -e_2 -e_7 -1 e_1 e_4 e_6 e_6 e_5 -e_7 e_4 -e_3 -e_1 -1 e_2 e_7 e_7 e_3 e_6 -e_1 e_5 -e_4 -e_2 -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 setminus 0 defined by phi(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 x n matrices with determinant 1 is a subgroup of the set of n x 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 setminus 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). This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 39 entries in this file. COUNT: 39 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 97-xx Mathematics education This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 1 entries in this file. COUNT: 1 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 +------------------------------------------------------------ royal There is no royal road to geometry. -- Euclid +------------------------------------------------------------ | 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) 10^4 One "myriad". The largest numbers, the Greeks were considering. 10^5 The largest number considered by the Romans. 10^10 The age of our universe in years. 10^22 Distance to our neighbor galaxy Andromeda in meters. 10^23 Number of atoms in two gram Carbon (Avogadro). 10^26 Size of universe in meters. 10^41 Mass of our home galaxy "milky way" in kg. 10^51 Archimedes's estimate of number of sand grains in universe. 10^52 Mass of our universe in kg. 10^80 The number of atoms in our universe. 10^100 One "googol". (Name coined by 9 year old nephew of E. Kasner). 10^153 Number mentioned in a myth about Buddha. 10^155 Size of ninth Fermat number (factored in 1990). 10^(10^6) Size of large prime number (Mersenne number, Nov 1996). 10^(10^7) 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^(10^51)) Inverse is chance to find yourself on Mars (quantum fluctuations) 10^(10^100) 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 +------------------------------------------------------------ obvious Mathematics consists essentially of : a) proving the obvious b) proving the not so obvious c) proving the obviously untrue 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 +------------------------------------------------------------ | barber +------------------------------------------------------------ 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 1+1 = 3, for large values of 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 PI= 3.14159265358979323846264338327950288419716939937510582097494459230781640628 +------------------------------------------------------------ | e +------------------------------------------------------------ e Euler E= 2.71828182845904523536028747135266249775724709369995957496696762772407663035 +------------------------------------------------------------ | 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 +------------------------------------------------------------ | recommendation +------------------------------------------------------------ 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 +------------------------------------------------------------ arithmetic God does arithmetic. -- K.F. Gauss +------------------------------------------------------------ | 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 4's 100359 5's 99548 6's 99800 7's 99985 8's 100106 9's --David Blatner, the joy of pi +------------------------------------------------------------ | 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 +------------------------------------------------------------ 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 +------------------------------------------------------------ | 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 +------------------------------------------------------------ | beauty +------------------------------------------------------------ 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. 2 a^2 = a^2 + ab, 5. 2 a^2 - 2 ab = a^2 + ab - 2 ab, 6. and 2 a^2 - 2 ab = a^2 - ab 7. Writing this as 2 (a^2 - a b) = 1 (a^2 - a b), 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 2 m^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 +------------------------------------------------------------ joke A good mathematical joke is better, and better mathematics than a dozen mediocre papers. -- John Edensor Littlewood +------------------------------------------------------------ | Leibniz +------------------------------------------------------------ Leibniz pi/4 = 1-1/3+1/5-1/7+1/9 .... -- Wilhelm von 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 ( mod p) PROOF. 1,2, ..., p-1 are roots of x^(p-1)==0 ( mod p). 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 +------------------------------------------------------------ lion How to catch a 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(kb because n! e and n!/k! were both integers. However, 0n 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 +------------------------------------------------------------ | rabbit +------------------------------------------------------------ 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" This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 124 entries in this file. COUNT: 124 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 g_1,...,g_n define an inductive structure (U,A,g_1,...,g_n) If all elements of U can be generated by repeated applications of the operations g_i on elements of A. Examples: (N,A=0,1,g_1(a,b)=a+b, g_2(a,b)=a*b defines an inductive structure. If U=N is the set of natural numbers, A=1,2,3 g_1(x,y)=3x-4,g_2(x,y,z)=7x+5y-z, then (U,A,g_1,g_2) define an inductive structure. +------------------------------------------------------------ | syntactic structure +------------------------------------------------------------ An inductive structure (U,A,g_1,...,g_n) is called a syntactic structure if it is uniquely readable that is if g_1(u_1,...,u_k) = g_2(v_1,...,v_l), then g_1=g_2,k=l and u_1=v_1,...,u_k=v_k. Example: if X is the set of finite words in the alphabet p,q,r,K,N and A=p,q,r. Define g_1(x,y) = Kxy and g_2(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 (n_i,t_i) which define transitions n_i to t_i. For example: N=S,T=K,N,p,q,r, G= S to p, S to q,S to r, S to KSS, S to 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) in G then |t| geq |n|. +------------------------------------------------------------ | context sensitive +------------------------------------------------------------ This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 10 entries in this file. COUNT: 10 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 +------------------------------------------------------------ amd Daemon which enables the NFS automount. +------------------------------------------------------------ | AMD +------------------------------------------------------------ AMD Advanced Micro Devices, Chip company. +------------------------------------------------------------ | 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 +------------------------------------------------------------ AT keyboard The standard keyboard used with the IBM compatible computer. +------------------------------------------------------------ | 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 second) which is 155.6 KBytes/sec Technology Speed mbit/s 56k modem 0.056 DSL varies cable varies T1 1.544 Ethernet 10.000 T3 44.736 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 see http://home.cfl.rr.com/cm3/speedtest7.htm 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 +------------------------------------------------------------ CASE: Computer Aided Software Engineering. +------------------------------------------------------------ | 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 +------------------------------------------------------------ DTML document template markup language. +------------------------------------------------------------ | DTP +------------------------------------------------------------ DTP Desktop publishing. +------------------------------------------------------------ | 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 +------------------------------------------------------------ EULA End user licence agreement. +------------------------------------------------------------ | 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 +------------------------------------------------------------ HTTP Hypertext Transfer Protocol. +------------------------------------------------------------ | HTTPD +------------------------------------------------------------ HTTPD Daemon to Apache webserver. +------------------------------------------------------------ | 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 +------------------------------------------------------------ IPFW IP firewall. +------------------------------------------------------------ | 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 +------------------------------------------------------------ NFS Network file system. +------------------------------------------------------------ | 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 +-----------