## rg

- THE ROUGH GUIDE TO NUMBERS AND ALGEBRA.
- SET AXIOMS
- MILLION DOLLAR REWARD
- MORE SET THEORY
- ELITE SCHOOLS
- DNA AND PROTEINS
- FINDING YOURSELF ON THE MAP
- FUNCTION COMPOSITION
- GROUP AXIOMS
- RING AXIOMS
- BINOMIAL THEOREM
- PASCAL'S TRIANGLE
- RINGS AND MODULAR ARITHMETIC
- SWEET SIXTEEN
- SYMBOLIC MATHS
- KAMA SUTRA
- PYTHAGORAS THEOREM
- RANDOM WALKS, HERD IMMUNITY AND THE LOVE BUG VIRUS
- BRIDGE HANDS
- HISTORICAL CONTEXT
- REVOLUTIONARY RETRIBUTION: YEAR ZERO
- POLITICAL ECONOMY
- INDUSTRY AND EMPIRE
- AGE OF EXTREMES
- CORPORATE FASCISM
- THE CLASSIC CALCULATION
- PRIME TIME
- ULAM NUMBERS
- DYNAMICAL SYSTEMS
- D4X.EXE
- download readme file
- Download instructions
- Download DNA:self extracting WINDOWS application.
- Download RNA:self extracting WINDOWS application.
- math2546.tgz Linux version. c-source code
- Linux binary, fonts, scripts in case you have no compiler
- d4x.exe. 32-bit Windows program
- d4t.exe. 16-bit Windows program
- Windows Source code
- Mandelbrot Animation c-source code

####
THE ROUGH GUIDE TO NUMBERS AND ALGEBRA.

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                  Chaotic roads and excessive speed.

        Q: What is a number ?

        A: That depends. The largest numbers regularly used in calculations
    gives some measure of the level of civilisation parhaps. The invention
    of logarithms in a dank Scottish castle reveals strange pathways in
    man's progress from the caves. We expect chimpanzees to count banananas
    and cats to count meal times in the day, but only humans are concerned
    with the numerical values for the speed of light.

        The range A < x < B for a statistic x is important.

        0               Number with no name for most of human history
        1,2,3           Numbers worth names and honorific titles.
        6               Arrangements of three objects
        Small numbers   rare diseases, collectors items.
        10-99           Tens. Whites facing death penalty in USA.
        2^6             64  Number of I-Ching Hexagrams
        100-999         Hundreds. Black people facing death penalty in USA.
        1000-9999       Thousands. Bosnian men massacred at Srebenica.
        10000           Moon (Thai). Brique (French).
        100000          Lak (Urdu, Arabic).
        Millions        people, dollars, animals, diseases.
                        Jews killed by Nazis. Prison population of USA.
                        504475641 Surface of earth. km^2
        Billions        hyperinflation, micro-organisms, science figures.
                        Telephone numbers
        2^32            4294967296.  Number of Internet IP addresses.
                        Arrangements of 13 objects.
                        Size of PC Hard disk in bytes.
        Six Billion     World population in October, 1999.
        18    Digits    278237540975596768. Surface of Dyson sphere. km^2
        20-21 Digits    c^2 in metric. Bang for your Gram. Einstein.
                        Arrangements of 21 objects.
        68 digits       Shuffles of a standard 52 card pack.
        80 digits       Protons in Universe. 3/2*136*2^256. Eddington.
        > 80 Digits     Probabilities, possibilities.
        10^10 digits    Possibilities for human genome.
        Alef-zero       Cantor's name for countable infinity.

        Here c^2 refers to the square of the speed of light.

        Excercise: Try learning the counting numbers from 0 to 10 in a
    language which you do not know yet.

        Generally you cannot tell how the number is going to be stored in a
    computer. Privileged numbers such as Pi or friend's phone numbers are
    often remembered by people. Certain other numbers are better not stored
    but given generator functions. You don't care whether a right angle is
    ninety degrees or Pi/2 radians. Just keep three, four, five and a
    triangle.

        Euclid described prime numbers, which are numbers with no factors
    apart from one and themselves, and irrational numbers which are numbers
    which cannot be expressed as fractions. The English mathematian Wallis
    is said to have invented a symbol like a figure eight laying on its
    side to represent infinity, a number bigger than any ordinary counting
    number.

        Many numbers used to represent reality on computers are the result
    of sampling where a range of values is split into segments and a name
    or value is assigned to each segment. Colour displays and images are
    the most common example of this.

        Fan Dee, Fan Lotteree        Thai saying.

        Numbers games are another aspect of numbers. 'Running Numbers' has
    been dramatized by Hollywood films and folk-rock ballads. These
    emphasise a sort of struggle against 'state-control' of numbers.
    Buddhist monks in Thailand risk getting killed if they say too much
    about the next lottery number. There are cultures where almost everyone
    seems to bet on numbers. Next week's imaginary number becomes
    yesterday's real number. Just like an alcohol hangover. For many the
    lottery number is like the dreams of opium. One could say that Marx got
    it wrong and in place of the statement:-

        Religion is the Opium of the Masses
        Imaginary Number is the Opium of the Masses

        Q: What's that special number used in the Theory of Everything ?

        A: You mean numbers like 'forty two' which occur in the Hitch
    Hiker's Guide to the Universe series. There are many fundamental
    equations of physics, starting with Newton's equation for gravitation.
    All of these have values expressed in metric units although some
    organisations in the Anglo-Saxon world try to cling on to imperial
    units of weights and measures. Physical constants are usually given in
    'floating point notation': X= M E EXP or M E -EXP where 'M' is a
    decimimal number, normally between 1 and 10 and the 'E' stands for a
    power of 10, so that 27.5 could be written 2.75 E1 and 1/8 can be
    written 1.25 E-1.

        A selection of physical constants is given. The first group is taken
    from Appendix 2 of 'The Chemical Bond' by Linus Pauling.

        Velocity of light           c       2.99793 E10     cm/sec
        Electronic charge           e       4.8029  E-10    statcoulomb
        Mass of electron            m       9.1083  E-28    gram
        Planck's constant           h       6.6252  E-27    erg sec
        Avagadro's number           N       0.60229 E24     /mole
        Energy of 1 ev                      1.60206 E-12    erg
        Wavelength of 1 ev quantum          12397.67 Angstrom
        Mass of proton              M(p)    1.67239 E-24    gram
        Mass of neutron             M(n)    1.67470 E-24    gram
        Boltzmann's constant        k       1.3805  E-16    erg/deg
        Gas constant                R       1.9872          cal/(deg * mole)
        Energy of 1 gram mass               5.6100  E32     ev

        Q: What is Algebra  ?

        A: Algebra is a way of anticipating problems in calculations by
    letting symbols stand for unknown numbers. It is said to originate with
    the Arabs. There are many Arabic words similar to Algebra. These often
    mean force or coercion. The power form is also a name for God.

        'Modern Algebra' was invented in the 1800s, and one of the main
    advances was the interchange between numbers and functions or
    operators. The French mathematicians Fourier and Galois pioneered this
    line of thought. Algebra as tought in schools came from much earlier
    times. Italy saw the earliest European developments when Cardano and
    others came up with a method for solving cubic and quartic equations.

        Formal algebra came with the attempts to build mathematics around a
    system of axioms in the style of Euclid. David Hilbert and Bertrand
    Russell made these attempts fashionable in the early 1900s, but Kurt
    Godel showed that this approach had limitations during the 1930s. For
    most computer science applications the formal approach is quite good
    enough.

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SET AXIOMS

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        In the 1960s and 1970s the English started to teach 'set' theory in
    schools. This was often done in an ideological vacuum. The kids were
    not always confronted by such basic questions as race and class in this
    context. Nor were they necessarily drilled in hard questions on set
    theory which had previously been in the syllabus: calculations about
    permutations and combinations. The success of the British National
    Lottery shows just how far things have fallen.

        Sets are like packs of playing cards, boxes of chess pieces,
    selections of lottery numbers, or even populations loving, living and
    dying. Early mathematics teaching concentrated on the computational
    aspect of set theory, nowadays called Combinatorial Analysis. Horse
    races are an excellent example of sets, as are dictionaries.

        The maintenance of knowledge about particular sets is of great
    economic importance. A goldrush mentality fuels speculation on world
    stockmarkets as search engine companies go public. The ignorance of
    next week's Lottery Numbers is also important for profits of the
    operator. The order of winners in a horse race or the subsets of
    football matches with draws or high scores provides the paydirt for a
    global money extraction industry.

        A set of numbers can be written in brackets: {1 3 5 7 9} is the set
    of odd numbers less than ten. A set containing no elements at all is
    called the empty set and it is written {}. The set consisting of {0 1}
    is special, because 0 and 1 can be made to correspond to values 'true'
    and 'false'.

        Sets are connected with logic by using the phrase 'a is a member of
    A', or 'b belongs to class B'. Such statements are always either true
    or false in classical set theory. Fuzzy logic is a new form of set
    theory allowing for intermediate values between true and false.

        Modern Algebra uses certain conventions such as upper case for sets,
    and lower case for members. Many statements of logic can be translated
    to theorems in the algebra of sets. The most common are De Morgan's
    Laws. A union B or 'A u B' is defined as elements x which are members
    of A or B. A ^ B is defined as elements x which are in both A and B.
    A-B is the set of x where x is in A but not B. The number of elements
    in a set X may be written as c(X). For any set X and subsets A and B
    the following identities hold. Here '<=' stands for less than or equal
    to. If the intersection of two sets is empty then they are said to be
    disjoint.

        c(A^B) <= c(A) <= c(A u B) <= c(X)
        c(A^B) <= c(B) <= c(A u B) <= c(X)
        c(A u B) <= c(A) + c(B)
        c(A^B)   <= Min(c(A), c(B))       Here Min means minimum.

        c(A u B) = c(A) + c(B) - c(A^B)   Inclusion and Exclusion

        (A u B)^C  = (A^C)  u  (B^C)
        (A^B) u C  = (A u C)^(B u C)

        X - A u B = (X-A) ^ (X-B)         De Morgan's Laws
        X - A ^ B = (X-A) u (X-B)

        If A is a subset of a set X then it is possible to define a post-fix
    function called % with A%=100.0*c(A)/c(X) where '/' stands for divide.
    The '%' function is merely the ordinary percent calculation.

        Exercise:

        Q: 60% of housholds have a car and 95% have a TV. How many have
           neither ?
        A: At most 5%.

        Excercise:

        Q: 47.5% voted Democrat and 46.7% voted Republican so how many voted
        for neither ?
        A: At least 18.8%. You can't vote twice, but many did not vote at
        all. Divide the population into voters and non-voters. Let U be
        the proportion of non-voters. When U is close to 1 then a dictator
        selected by popular acclaim could be seen as the most democratic
        option.

        N=U+(1-U)*0.188     None
        R=(1-U)*0.467       Republican
        D=(1-U)*0.475       Democrat

####
MILLION DOLLAR REWARD

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        Excercise: Goldbach's conjecture.

        Let P={3 5 7 11 13 17 ...} be the set of odd primes and let E={6 8
    10 ..} be the set of even numbers greater than four. Then is E=P+P?
    Here P+P denotes the set formed by the sums of all pairs p+q with p,q
    in P. of P+P. It is already known that E=P+P*P*..P where P*P*P..P is
    some product of prime numbers. It is also known that every sufficiently
    large odd number is the sum of three odd primes. Vinogradov [1] proved
    this in 1937. It is also known, by computer search that any even number
    n is the sum of two primes for n < 400,000,000,000,000 (four hundred
    trillion in the American style).

        Goldbach outlined this conjecture in a letter to Euler in 1742. Now
    the publisher FABER is offering a million dollar reward for anyone who
    submits a proof of the conjecture, or a counter example before March
    15, 2002 [2].

####
MORE SET THEORY

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        Mathematicians invented sets long before computer lanuages evolved.
    Sets are easy to copy. As a tool of thought, they are unprotected by
    legalese or copyright. They are public domain stuff. Mathematical
    terminology alone seems a sufficient deterrent. Given two sets A and B
    it is possible imagine a table of pairs (a,b). The sets may be people
    and cars, or stock market shares and prices. The set of all possible
    products is written A x B and a typical element of the set A x B is the
    pair (a,b) with a in A and b in B. The set A x B is called the
    Cartesian Product. The number of elements of A x B is c(A) times c(B).
    If A and B are different horse races, then the prediction of a winner
    from each race is called a double, and is simply a member of the
    cartesian product. It is also possible to define cartesian products on
    more than two sets. The product formula can be generalised. In
    particular the size of the cartesian product of k copies of A is c(A)
    to the power k.

        For any given set X it is possible to define a 'relation' as a
    subset R of the cartesian product A x A. An equivalence relation
    satisfies three simple rules:

        Reflexive   If x in X then (x,x) in R.
        Symmetric   If (x,y) is in R then so is (y,x).
        Transitive  If (x,y) and (y,z) are in R then so is (x,z).

        Given an equivalence relation then it is possible any x in X to
    define a unique subset of X containg x: class(x)={y | (x,y) in R}. This
    is called the equivalence class of x with respect to R. Any two
    equivalence classes are disjoint, for if a in class(x)^class(y) then
    (a,x) and (a,y) are both in R, so (x,a) in R by symmetry, and therefore
    (x,y) in R and in fact class(x)=class(y). Since the reflexive rule
    implies that class(x) contains at least x as an element it follows that
    every element of x is in at least one equivalence class. The set of
    equivalence classes is often written X/R. Since the classes are
    disjoint, and every element is in just one equivalence class we can
    write X = union {C(x) with x in X/R}.

        Example:

        A person cannot exist in a modern state without being slotted into a
    cartesian product of some pair of sets. If a person is treated for a
    disease the bureaucracy maps that person into the cartesian product of
    (people x diseases). A relationship can be defined defined on the set
    of people by saying that (a,b) in R if a and b suffer from a given
    disease. Not only AIDS, but also multiple injuries will appear in
    clusters. Needless to say modern computers are incapable of doing
    justice to these types of databases. Analysis of these relationships is
    crucial in discussions on public financing.

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ELITE SCHOOLS

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        Example: Consider education, with the relation R being determined by
    school. Two people a,b are in this relation if there is a finite set of
    people a=x[0],x[1],x[2], .. x[n]=b where x[i],x[i+1] both went to the
    same school. If a person x did not go to school at all, then let (x,x)
    in R. Then the population can be broken into a disjoint union of
    equivalence classes. When inter-school transfers are common there may
    be just a single class. When the education of women is forbidden then
    there are at least two classes. In a global world these classes are
    unlikely to correspond to geography. A classless society could be seen
    as a world where everyone participated in the same distance learning
    scheme. Attempts to make everyone participate in the same school
    _system_ have been tried many times. When schools have been dominated
    by competing religions there have occassionally been problems. Northern
    Ireland is seen by many as a classic example. If teaching becomes a
    less attractive profession, and the cost of internet connectivity falls
    drastically then the uneducated could become replaced by the
    'media-educated'. Whoever dominates the media effictively controls
    education for large numbers of people.

        When a set is finite the class equation can be written:
                _
        c(X) = \   c(Xi)  where X1,X2,.. are the equivalence classes.
               /_

        Besides relations the subsets of A x B include the graphs of
    functions. A function from A to B written f:A->B is simply a subset G
    of A x B such that for each a in A there exists a single element b in B
    with (a,b) in G. For shares and prices the most common subset of the
    cartesian product is the price list, but other functions are possible:
    the price in two weeks time, for example, or the previous years low.

####
DNA AND PROTEINS

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        Example: Deoxyribonucleic acid, called DNA, is a molecule made up of
    components called adenine (A), cytosine (C), guanine (G) and thymine
    (T). Each of these components consists of a ring containing carbon and
    nitrogen and part of a sugar like molecule (ribose). As in many organic
    compounds the presence of nitrogen gives part of the DNA molecule an
    alkaline property, so the components are called bases, as opposed to
    acids. The base property helps in the formation of chemical bonds. The
    DNA molecule is a chain of any of the bases A,C,G and T in some
    particular order. The base pairs A,T and G,C also bond with each other
    in a natural way. RNA is similar to DNA except that thymine (T) is
    replaced by uracil (U). The AIDS virus is made up of RNA molecules.
    There are 16 possible base pairs and 64 triples. Scientists have
    determined that certain triples serve to encode amino acids, which are
    components of proteins and enzymes. There are about 20-22 amino acids
    which occur in the common proteins of life. The 64 base triples may be
    divided into equivalence classes according to the particular amino acid
    that the base encodes. Such triples are called codons.

        There are also some base triples whose function is not explained. It
    is conjectured, on very strong evidence, that parts of the human DNA
    sequence has been incorporated from orther organisms at earlier stages
    of evolution. DNA is important because it is passed from parents to
    offspring. Sometimes there are small random changes, but often these do
    not matter: the encoding from 64 bases to about 20 amino acids implies
    a considerable degree of redundancy. Some very common amino acids will
    have more than one codon. The genetic code is a mapping:-

                gc: CODONS -> Amino Acids

        The mapping is not one to one. Different organisms make the same
    proteins for similar functions, but they use slightly different base
    sequences. People can estimate a difference between DNA sequences for
    making a given protein by counting the different numbers of base pairs.
    If A and B are different DNA sequences, then it is possible to define a
    distance function d(A,B). For three bits of DNA A B and C the distance
    functions satisfy d(A,C) <= d(A,B) + d(B,C). This is the triangle
    inequality and it corresponds with our normal notions of space.

        DNA space is like normal space with its streets and alleyways.
    Species evolution is a set of trajectories in DNA space. A very tiny
    part of this story forms the major scenario for Julian Rathbone's
    political novel TRAJECTORIES. Einstein used Riemann Geometries to track
    real space, and similar tools from analysis and topology will be needed
    to explain DNA space.

        DNA space is a real battleground. AIDS, TB , Cholera and Malaria
    with its associated sickle-cell syndrome are some of the trajectories.
    Evolution on computers is usually modelled by some sort of function
    iteration and the mathematical discipline of dynamical systems can
    provide some of the theory to check out the results. The French
    mathematician Poincare invented dynamical systems about one hundred
    years ago.

        One of the most fascinating scientific ventures of the twentieth
    century was the human genome project. Human DNA is estimated to have
    about 2.1*10^10 bases. Each DNA molecule may therefore have over
    10^10^10 different configurations. Biologists often joke about GOD as
    the 'generator of diversity'. Following investment by some private
    companies the Human Genome Project is almost finished. In parallel
    efforts are being made to elucidate the DNA sequence of
    micro-organisms.

        This involves intensive number crunching and the employment of
    skilled scientists. Biotechnology research is controversial and it has
    become easy to mobilise demonstrations against some applications such
    as genetically modified foodstuff (GM foods).

        The real issue is the expense of the research, its financing and its
    public accessibilty. Riemann and Poincare published their results in
    academic journals but one hundred years later corporate lawyers are
    fighting to prevent the publication of knowledge of the geometry of DNA
    space. They are bahaving worse than the wretched priest cults that
    tried to monopilise predictions of the Nile flooding in the days of the
    pharoahs. The fact is that DNA space is so complicated that they need
    the help of the best minds in the world to sort out the navigation
    problems.

        The corporate elite that seeks to monopolise knowledge of DNA space
    are exactly the same types that persecuted Linus Pauling, the first
    person to explain protein structure. These business leaders who wish to
    monopolise knowledge have allies amongst many conservative religious
    people who also want to control access to knowledge.

        Just as Newton may have been concerned about stability of the
    heavenly bodies, so modern scientists are concerned about other types
    of orbits. These are trajectories in DNA space, and also trajectories
    in economic space. This last figure is really a consequence of human
    demography. Genes and DNA sequences need far more than mere human
    populations to express their differences. Micro-organisms have much
    larger populations than humans. This is good for evolution, because
    certain things must happen with very large populations. Just how the
    size of a population may force apparent coincidences is an application
    of the specialisation of set theory known a Ramsey Theory. The simplest
    statement in Ramsey theory concerns meetings with six people. In such a
    meeting there must be either a clique of at least three people who are
    mutually acquainted, or else there must be at least three people all of
    whom are complete strangers to each other.

####
FINDING YOURSELF ON THE MAP

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        DNA space may seem a very abstract entity, but other people's
    failure to understand this could get you locked up in jail for an
    indefinate period of time.

        The police take DNA samples from the general public following a
    particularly brutal and horrific sex murder. They then send the samples
    to a laboratory whose corporate owners have a particular financial
    interest in incarceration as an industry, just like Stalin's Gulag
    system of the 1930s and 1940s. The laboratory is staffed by underpaid
    and demoralised workers who use their computers to map areas of DNA
    space defined by samples found near the dead body. Tabloid
    sensationalism put the police under intense pressure to make an arrest,
    so they pick up the person with the nearest DNA match. The victim's
    brother, or a parent are a good guess, especially if the victim is in a
    minority group. An innocent man is condemned, and those who seek to be
    morally outraged look for other people to persecute.

        Modern states seek to gain credentials by cataloguing the DNA of
    criminal suspects. At least we know that DNA space is a little bit like
    Euclidean space.

####
FUNCTION COMPOSITION

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        As it happens functions got invented well before 'Set Theory'.
    Functions are like the act of copying even before the media was there.
    Functions were linked with procedures for evaluating numerical
    quantities, but methods like table look-up are perfectly good. If you
    have a graph G as a subset of A x B and you want to find the value for
    x just search the graph G for the correct pair (a,b) with a = x. Call
    the corresponding value f(x).

        If A,B,C are sets, and f:A->B and g:B->C are functions then it is
    possible to define a composition gf:A->C so that gf(x) = g(f(x)). A
    function is one-to-one, or injective if f(a)=f(b) only if a=b, and it
    is onto, or surjective if every b in B is the image f(a) of some a in
    A. A set is finite if c(A) is an ordinary number. In this case it is
    possible to define a function from A to the sequence {1,2,..c(A)}. This
    comes from any listing the different elements of A as a[1], a[2] ..
    a[n] etc. The graph of a function f:A->B can then just be written
    b[1],b[2]...b[n] where each b[i] is in B. That is the set of possible
    graphs is just the cartesian product of c(A) copies of B.

        If there is a function f:A->B which is 1-to-1 and onto then A and B
    are said to be in one to one corrspondence. Each b in b must be the
    image of a unique a in A with f(a)=b. Define the function g:B->A such
    that g(x) is the unique y with f(y)=x. Then g(f(y)=y for any y in A,
    and f(g(x))=x by definition of G. The function g is said to be an
    inverse of f. A set X is said to be countable if its elements can be
    put in 1-to-1 correspondance with the set of counting numbers
    {1,2,..n,...}.

        The set of positive whole numbers {1,2,3... etc} along with 0 and
    the negative numbers is a set. This set is usually noted Z, the symbol
    coming from the German word 'Zahlen' for numbers.

        Notice that Z has perfectly good 1-to-1 functions that are not onto.
    The function n->2*n is 1-to-1 but its image excludes the odd numbers.
    Similarly the set of positive numbers {1,2,..} has one to one functions
    which are not onto. The obvious choice is the sucessor function n->n+1.
    There is a hotel plan named after this fact. The 'Hilbert Hotel' can
    accomodate new guests even when all the rooms are full. It is also
    possible to double the population even when all the rooms are full. If
    a single guest arrives just tell every other guest to move up a room.
    If a huge crowd arrives, just tell everyone to move from room number n
    to number 2*n. Perhaps not very practical but at the end of the
    twentieth century many politicians still seem to think they can expand
    prison populations this way. If A and B are two countably infinite
    populations then the Hilbert Hotel can also accomodate the set of pairs
    A x B, with each pair in a single room. With a finite set of rooms,
    double occupancy eventually becomes necessary. This is called the
    Dirichlet Box principal. For a hotel of size N with N+1 guests, at
    least two must share.

        The set of functions from A to B is itself a set. If A has c(A)
    members and B has c(B) members then the size of this set, Map(A,B) is
    c(B)*c(B)*c(B)...*c(B) multiplied c(A) times. In particular the set of
    functions t:A->{0,1} has 2^c(A) elements. This is also the number of
    subsets of A. For each X which is a subset of A define the function
    tX:Z<-{0,1} where tX(z) = 1 if z is in X and 0 otherwise.

        When A is a finite set the set Map(A,A) of all functions f:A->A has
    c(A)^c(A) elements. if f:A->A is one to one, then the set sequence of
    function values f(a[1], f(a[2]).... f(a[n]) defines f where n=c(A).
    There are n ways of selecting b[1]=f(a[1]) but only n-1 choices for
    b[2]=f(a[2]) since the function is 1-to-1. Similarly there are only n-2
    ways of chosing b[2]=f(a[2]) and so on. The number of such mappings is
    n(n-1)(n-2)...2.1 or n factorial. This is often written n!, or
    n-shriek. This quantity is also known as the number of permutations of
    n objects. If s and t are two mappings of the set N={1,2,3..n} to
    itself then it is possible to define the composition of the two
    functions r=st by the formula r(i) = s(t{i)) for each i in N. The
    permution i:N->N given by i(k) = k for all k is called the identity
    permutation. For the set N the number of mappings in Map(N,N) is n^n.
    Given any mapping s in Map(N,N) and any k in N the set {k, s(k),
    s(s(k), s(s(s(k))), ,.m terms) is a subset of N. When n=C(n) is finite
    and m>n then two of the iterated functions (s^j1)(k) and (s^j2)(k) must
    be equal. This essentially means that the sequence k[n+1]=s(k[n])
    repeats itself after a certain number of terms. This set is called the
    orbit of k under the mapping s.

####
GROUP AXIOMS

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        A set of elements G is a group if the following axioms are true.

        G0: For any a,b in G there is a product,  ab in G
        G1: (ab)c = a(bc)       Associative axiom.
        G2: There is e in G with ae=ea=a  for all a in G.
        G3: For any a there exists a' such that aa' = a'a = e

        The element e such that ae=ea=a is called the identity element. The
    element a', corresponding to a such that a'a=aa'=e is called the
    inverse of a.

        Example: The set {1} with 1*1=1
        Example: Set of shuffles of N cards.
                 This is a non-commutative group.
        Example: Set of subsets of a set with A+B=(AuB)-A^B

        An Abelian group satisfies all of the axioms of a group, along with
    one further rule.

        G4: ab = ba for all a,b in G.  Commutative axiom.

        In the case of an Abelian group the law of composition is written as
    '+', and the identity element is called zero, and given the symbol '0'.
    It is said that the Hindus were the first to write about this feature
    of ordinary numbers. The additive inverse of an element a is usually
    written -a.

####
RING AXIOMS

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        A ring R is an Abelian group, with an additive law of composition
    '+', and an additive identity, 0. There is a further law of composition
    for pairs of elements r,s in R known as the product. The product of the
    elements r and s may be written rs. Rings satisfy the following axioms.

        R0:   For each r, s in R the product rs is in R.
        R1:   (rs)t = r(st)   Associative axiom
        R2:   (r+s)t= rt+st and r(t+s)=rt+rs        Distributive laws.

        Additionally a ring may have further axioms.

        R3:   There is an element 1 with r1=1r=r for all r in R.
        R4:   rs=sr                 Commutative axiom.

        Example: The set {0,1}
                 0+0=1+1=0 1+0=0+1=1; 1*1=1 and 0*1=0*0=1*0=0
                 This is often called Z2 or GF(2).

        Example: The counting numbers 1,2,3 etc along with zero and the
    negative numbers -1,-2,-3 etc. form a commutative ring with an identity
    element 1 under the usual laws of addition and multiplication.

        If R and S are two rings, the cartesian product R x S can be made
    into rings by defining componentwise addition and multiplication. The
    sum and product of the pairs (r1,s1) and (r2,s2) are (r1+r2,s1+s2) and
    (r1*r2, s1*s2) respectively. More generally if X is any set the set of
    functions f:X->R can be made into a ring by defining sums and products
    elementwise. if f and g are functions in Map(X,R) define (f+g)(x) =
    f(x)+g(x) and (f*g)(x) = f(x)*g(x). These definitions create the
    necessary ring structure. When R is the ring with two elements the set
    Map(X, Z2) is a ring. There is a 1-to-1 correspondance between elements
    of this ring and subsets of X. If A is a subset of X then define
    (tA)(s)=1 if s in A and 0 otherwise. If A and B are two subsets
    ((tA)*(tB))(s)=1 if and only if s in A^B and (tA+tB)(s)=1 if and only
    if s is in A or B but not both (s in A u B-A^B). This shows that the
    set of subsets of a set X may be made into a ring. This is known as the
    Boolean Algebra of the set X. Boolean Algebras are important in
    Mathematical Logic, and also integrated circuit design.

####
BINOMIAL THEOREM

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        In any ring with a 1 then the following identity holds. This is
    known as the binomial theorem.

          n                   2                3          n
     (1+x)  = 1 + nx + n(n-1)x  + n(n-1)(n-2) x  + ...  +x
                       ------     -----------
                         2!           3!

                                            k
                                      n!   x
         Here the general term is  --------
                                   k!(n-k)!
                          n!
         The quantity  -------  is known as the number of combinations
                       k!(n-k)!

         of k elements taken from n objects. It can be written as C[n,k].

         Proof:  By induction. Show that if this theorem is true for n, then
    it is true n+1, and therefore all n.
                               1
         With n = 1, then (1+x) =1+x.
         Given the identity for n  consider the expression:

              n+1            n                          2        3
         (1+x)   = (1+x)(1+x)  = (1+x)(1+C[n,1]x+C[n,2]x +C[n,3]x ....)
                             k
         The coefficient of x  is C[n,k]+C[n,k-1]. Writing this in full

         C[n,k]+C[n,k-1] =     n!        n!
                            ------   + --------------
                            k!(n-k)!   (k-1)!(n-k+1)!

                         =   n!             k
                           --------  (1 + -----)
                           k!(n-k)!       n-k+1

                        =    n!        n+1         (n+1)!
                          -------- *  ------   =  --------   = C[n+1,k]
                          k!(n-k)!    n+1-k       k!(n+1-k)

                                   n+1      ---           k
        So the expression for (1+x)    =    \    C[n+1,k]x
                                            /
                                            ---
        and the identity is proved.

        To come back to the product of rings, consider the simplest ring Z2,
    The Z2xZ2, Z2xZ2xZ2, .. etc form a family of rings. The members of
    Z2xZ2 may be written as the set of pairs {(0,0),(0,1),(1,0),(1,1)}.
    Alternatively the elements could be written {0,x,y,x+y} with z+z=0 and
    z*z=z for all z.

        The elements of Z2xZ2xZ2 may similarly be listed as 0, x, y, z, x+y,
    x+z, y+z, x+y+z. In terms of co-ordinate listing the quantities x,y,z
    stand for triples (1,0,0),(0,1,0),(0,0,1). The sums of any pair of
    these elements is a triple with 2 1s. The values x,y,z represent the
    vertices of a cube, and the set x+y,y+z,x+z are the next vertices as
    you move to the opposite vertex to (0 0 0). The number of points at
    distance d from the origin at (0 0 0) correspond to the binomial
    coefficients in (1+x)^3. For any number n the product Z2xZ2...xZ2 n
    times is exactly like the vertices of a hypercube of n-dimensions. The
    points can be represented as n-tuples (0 0 ..1 ..0) etc with 1s in
    certain positions less than n. The total number of elements, or vectors
    as they are known, is 2^n and there are n with single 1 co-ordinates,
    n(n-1)/2 with two non zero coordinates and so on. The general formula
    C[n,k] for k non-zero coordinates can be proved by induction similarly
    to the binomial theorem above.

                n+1                               n
        Let E=Z2    be the cartesian product of Z2 xZ2. A vector z in E may
    be written as an n-tuple x, followed by a or 0 1. The number of such
    (n+1)-tuples  with k non zero elements are made up from the C[n,k]
    n-tuples with the last co-ordinate zero (inductive hypothesis) and
    C[n,k-1] consisting of those (n+1)-tuples with k-1 of the first n
    co-ordinates being 1 and the last co-ordinate also 1. The formula
    C[n+1,k]=C[n,k]+C[n,k-1] gives the reqired proof.

        Example:
        The British National Lottery.
        A winning line is a set of six numbers from the set {1 .. 49}. There
    are 2^49 such subsets, but only those with six elements are in a draw.
    The six element sets correspond to 49-tuples in Z2 with just six non
    zero co-ordinates and there are just C[49 6] of these. This value is
    49.49-1...49-5 % 6! or as calculated by this formula

       (PRODUCT 49-i6) % PRODUCT 1+i6 = 13983816

        The rule of thumb is 14 million to one.

        Here PRODUCT X just gives a high precision product of a set of
    numbers X. The Index generator 'i' or 'iota' in APL simply gives a set
    of consecutive numbers. i6 is the set {0 1 2 3 4 5}. The chance of a
    random draw being correct is about 14 million to one, but the payout is
    often at much lower odds. The lottery operator accepts unlimited losses
    by the public, but the operator shares a single jackpot among the lucky
    winners when it loses.

####
PASCAL'S TRIANGLE

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        The binomial theorem gives a way of counting subsets of sets. It
    also holds true in any ring R. When R = Z, the set of integers the
    first fiew lines of the triangle look like this.

                                 1
                               1   1
                             1   2   1
                           1   3   3   1
                         1   4   6   4   1

         Each number is the sum of those immediately above.

         When R=Z2, the field of two elements, you get a triangle with holes
    in it. The study of objects with holes in them includes a specialised
    field of mathematics called topology. Early topologists included
    Hausdorff and Sierpinski. Their major concern at the time was the
    search for counter examples in theories of continuous functions. The
    early efforts were motivated by the search for space filling curves or
    conversely, well behaved 'perfect sets'.

####
RINGS AND MODULAR ARITHMETIC

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        The set of integers Z is a ring. For any non zero number m it is
    possible to define an equivalence relation on Z such that two numbers x
    and y are equivalent if and only if the difference x-y is divisible by
    m. For any number n > m it is possible to find numbers q and r such
    that

            n = mq + r with 0<r<m-1.

        q is the quotient and r is the remainder of division of n by m.
    Since n-r = mq we see that n is equivalent to r, a number which may
    take only one of m values. For given n the remainder r after division
    by m is often called n mod m. The number of equivalence classes is
    finite, and each equivalence class is generated by the set { r+mq | q
    in Z}, often written r+mZ. It is possible to define addition and
    multiplication on the set of equivalence classes.

        (x+mZ) + (y+mZ)       is (x+y mod m)+mZ
        (x+mZ) * (y+mZ)       is (x*y mod m)+mZ.

        These classes are called the integers modulo m, often denoted Zm.
    The ring of equivalence classes is also written Z/mZ.

####
SWEET SIXTEEN

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        Example:

        The values of integers are stored in a computer memory. Calculations
    are done with a processing unit. Most present day computers do
    arithmetic on the set Zm where m is a power of two. Old computers like
    the APPLE II had a 6502 processor which only did 8-bit arithmetic. The
    numbers 0-255 were divided into the numbers -127 through to 0 then 127.
    The value -128, represented as a byte like 10000000 was taken to be
    minus infinity. Normal additions and subtractions were carried out
    modulo 256.

        Apple's technical genius, Wozniak developed sixteen bit arithmetic
    by software. He called this sweet sixteen. But even this system has its
    limitations. Numbers greater in magnitude than 32000 cause problems.
    The computer hardware recognises equivalence classes of numbers, rather
    than the numbers themselves. Five kilometers looks like five miles.
    NASA programmers found this out the hard way. A recent Mars module got
    an excessively hard landing because of this type of mistake.

####
SYMBOLIC MATHS

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        Computers cannot really do accurate arithmetic. Programmers can make
    the computer do arithmetic to a high precision, but there will always
    be problems which cannot be cracked. All the people that received
    erroneous bills or bank statements are fully aware of this fact. If
    numbers cannot be dealt with in a satisfactory manner, then why not
    just use the computers to sort lists of names and addresses as in 1960s
    style sales ledger programs ? If the computer can manipulate text
    strings then it should also be possible to implement math systems which
    deal with all of the orders of infinity invented by Georg Cantor and
    others.

        Formal mathematics anticipated these developments in the 1930s.
    Church, Godel, Post and Turing contributed very much to the human
    achievements of the twentieth century.

        Every schoolchild should have opportunity to learn a little about
    symbol manipulation systems. The old fashioned business of solving
    equations is an excercise in doing this. But solving equations often
    makes use of rules which may not be accurately stated. The material
    will seem divorced from reality to many. Time to think is important.

        An equation for solution involves one or more symbols representing
    unknown quantities, usually numbers. If a pupil does not know how to
    divide, or possibly to extract square roots, then there seems little
    purpose in the excercise. That's where symbolic manipulation comes in.

####
KAMA SUTRA

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        To solve ax=b just write x=b/a. This is an example of a formula.
    Perhaps one of the best known recipes of formula is an Indian text
    called the Kama Sutra. It's a classical sex manual, and the literal
    translation is simply Love Formula. Sutra is comes from the Sanskrit
    root word for formula, while Kama corresponds to physical love.

        Maxwell's equations are formulae connecting the electromagnetic
    forces, and from these come the theory of radio and television and the
    ability for people to watch pornographic broadcasts worldwide. The
    formula is the connection.

        When thinking about formulae forget the calculations for a while and
    reach for the heavens.

        If R is a ring, it is possible to add symbols to the ring so that
    the ring axioms R0 to R4 are satisfied. Just add the symbol X and call
    the new ring R[X]. The element X satisfies 1.X=X.1=1 and the value X*X
    is written X^2. Multiplying n copies of X is written X^n. The elements
    of the ring R[X] are written as sums a[0]+a[1]X+a[2]X^2 etc. with only
    a finite number of terms. Addition of two ring elements is done
    componentwise, while multiplication is achieved by collecting equal
    powers of X.

        In the old days algebra was taught in schools by giving the pupils
    hundreds of excercises such as:

        Multiply 1+2x+5x^5 by 3x^2+2x^3.
        Factorise  6x^2+5x+1.

        These excercises were often chosen to avoid difficult theoretical
    points such as the non-existence of factors. There were often
    circumstances where the teachers would not have time to deal with these
    difficulties because of political and economic constraints. A media
    dominated world will see these difficulties considerably increased for
    the next generation of teachers.

        Excercise: Verify the following identities.

            (1-2a)(9-4a) = 9-22a+8a^2
            (3+14t)(14-11t) = 42+163t-154t^2
            (12+b)(3-13b) = 36-153b-13b^2
            (7c+6t)(2c+3t) = 14c^2+33ct+18t^2
            (2-5t)(1-t) = 2-7t+5t^2
            (-3c+4t)(-17c^2-13ct+5t^2) = 51c^3-29c^2t-67ct^2+20t^3
            (15+14z)(7+5z-z^2) = 105+173z+55z^2-14z^3
            (6-13a)(16+a+4a^2) = 96-202a+11a^2-52a^3
            (19t-9z)(7t+18z) = 133t^2+279tz-162z^2
            (13+4z)(17+19z+16z^2) = 221+315z+284z^2+64z^3
            (1+t)(3-5t) = 3-2t-5t^2
            (5-16c)(20+c) = 100-315c-16c^2
            (7+15c)(5-17c) = 35-44c-255c^2
            (9+14y)(3-21y-20y^2) = 27-147y-474y^2-280y^3
            (9b+11s)(-15b+7s) = -135b^2-102bs+77s^2
            (1+s)(19-6s) = 19+13s-6s^2
            (17c+9s)(10c-13s) = 170c^2-131cs-117s^2
            (-18c+x)(-13c^2+9cx+2x^2) = 234c^3-175c^2x-27cx^2+2x^3
            (7+9y)(2+13y) = 14+109y+117y^2
            (7+3s)(18+5s+11s^2) = 126+89s+92s^2+33s^3
            (1-11y)(14+13y) = 14-141y-143y^2
            (14-17x)(4-3x) = 56-110x+51x^2
            (8+17y)(5-6y) = 40+37y-102y^2
            (8-5z)(7+17z) = 56+101z-85z^2
            (7c+12x)(14c+3x) = 98c^2+189cx+36x^2
            (5-9b)(1-b) = 5-14b+9b^2
            (11-10b)(1-6b) = 11-76b+60b^2
            (18a+13y)(14a+11y) = 252a^2+380ay+143y^2
            (8s+15z)(7s-9z) = 56s^2+33sz-135z^2
            (-10x+y)(12x^2-4xy+13y^2) = -120x^3+52x^2y-134xy^2+13y^3
            (12-7c)(5-11c) = 60-167c+77c^2
            (5+9x)(10-7x) = 50+55x-63x^2
            (8b-11c)(b+16c) = 8b^2+117bc-176c^2
            (3+17z)(20+3z) = 60+349z+51z^2
            (2-15y)(19+10y) = 38-265y-150y^2
            (5c+18y)(3c+19y) = 15c^2+149cy+342y^2
            (7s+t)(s+t) = 7s^2+8st+t^2
            (-4b+9z)(-4b+3z) = 16b^2-48bz+27z^2
            (-6t+7y)(-2t+y) = 12t^2-20ty+7y^2
            (3-z)(1+z) = 3+2z-z^2
            (11b+18y)(9b-y) = 99b^2+151by-18y^2
            (4b+19x)(6b-x) = 24b^2+110bx-19x^2
            (2-a)(14+3a) = 28-8a-3a^2

        Some classical formulae attracted great controversy. The first was
    the formula for the area of a triangle. If A is the area of a triangle
    with sides a,b,c, then the quantities are related by the equation:

        A^2 = s(s-a)(s-b)(s-c) where s = (a+b+c)/2.

        Effectively the area is given as the square root of a number. This
    confounded some ancient philosophers, although the formula is quite
    accurate and practical. The formula may be re-arranged to eliminate s.
    In the past students and mathematicians would use pencil and paper to
    do the calculations, but nowadays it can be done on the computer.

        $ AOS"(a+b+c)(a+b-c)(a-b+c)(b+c-a)"
        -a^4+2a^2b^2+2a^2c^2-b^4+2b^2c^2-c^4

        Giving 16A^2= 2(a^2b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)

        Systems such as MAPLE, MATHEMATICA, MATLAB , TK-solve and so on can
    do most such calculations in an instant, while the user is given time
    to think if using the software which should accompany these notes.
    Whatever the reader may think of the calculations, the symmetry of the
    formula should be quite evident. The values a,b,c represent the lengths
    of sides, or distances in space. When a=b=c it is easy to see that
    16A^2=3a^4 or A= #sqrt(3/2)a^2 , where #sqrt stands for square root.

        In fact the formula as written looks valid for all sets of numbers.
    However if a,b,c represent the lengths of a normal triangle then we
    know that a<=b+c and the same for the other sides. This is certainly
    true in Euclidean space, and also DNA space.

        Two of the most famous formulae in history are:-

        (1) Pythagoras Theorem:  a^2=b^2+c^2 for right angled triangles.

        (2) Einstein's energy equation:  E=mc^2.

####
PYTHAGORAS THEOREM

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        Pythagoras Theorem is easy to prove by algebra. Given a right angled
    triangle with sides of length a,b,c with a<=b<=c construct a square of
    size a+b, and fit in four triangles and a square as shown below.

        ...b............................
        .     .. .                     .   Area of Square = (a+b)(a+b)
        .     .  . .                   b   Also the area is the sum of
        .    .      ......c.           .   four right angled triangles
        a   .               ........   .   and the center square of side c.
        .  .                        ....   Now the area of a right angled
        .  c                         . .   triangle is given by
        .  .                       .   .   Area = base x height/2.
        .  .                       .   .
        . .                        .   a   Large square   = a^2 + 2ab + b^2
        ..                        c    .                  = c^2 + 4(ab)/2
        . . ...                  .     .                  = c^2 + 2ab
        .      ....c             .     .
        b           .....       .      .    So a^2+2ab+b^2= c^2 + 2ab
        .                ..... .       .        a^2 + b^2 = c^2
        .......a...................b....

        This proof is reputed to have originated in China.

        The Triangle formula is a consequence of Pythagoras Theorem.

        +----------------------------------------------------------+
        |                                                          |
        |                       A.                                 |
        |                     ... ..                               |
        |                    .  .   ..                             |
        |                   .   .     ..                           |
        |                 ..    .       ...                        |
        |               c.      .          .. b                    |
        |               .       .y           ..                    |
        |             ..        .              ...                 |
        |            .          .                 ..               |
        |           .           .                   ..             |
        |         ..     x      .        a-x          ..           |
        |        B..............X........................C         |
        |                         a                                |
        |                                                          |
        +----------------------------------------------------------+

         The triangle ABC may be split by drawing a perpendicular line from
    A to its opposite side. If the numbers a,b,c represent the lengths of
    the sides and x,y the distances BX and AX then we have :

         Area = A = 1/2 * base * height = ay/2

         Pythagoras theorem can be applied to the triangles ABX and
         ACX giving equations for x and y.

         c^2 = x^2+y^2                (1)
         b^2=(a-x)^2+y^2              (2)

         It is possible to compute y in terms of a,b, and c.
         Subtract (2) from (1) giving

         c^2-b^2=-a^2+2ax    or  2ax = a^2+c^2-b^2

         For the area A the equation can be written

         4A^2 = a^2y^2 = a^2(c^2-x^2)
                       = a^2(c^2-((a^2+c^2-b^2)/2a)^2)
         16A^2         = (2ac)^2 - (a^2+c^2-b^2)^2
                       = (2ac+a^2+c^2-b^2)(2ac-a^2+b^2+c^2)
                       =((a+c)^2-b^2)(b^2-(a-c)^2)
                       =(a+b+c)(a-b+c)(b+c-a)(a+b-c)

####
RANDOM WALKS, HERD IMMUNITY AND THE LOVE BUG VIRUS

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        In May 2000 a Phillipino hacker released the Love Bug virus. More
    exactly the offending code was a 'Visual Basic Worm'. The worm was
    easily able to propogate amongst the computers of governments and large
    corporations. Following attacks on capitalism by demonstrators in
    Seattle, Washington, London and Chieng Mai governments and lawmakers
    became unduly sensitive and concentrate more forces on tracking school
    kids than they ever put into catching Osama Bin Laden.

        The Lovebug worm is a script written in Visual Basic which is
    activated when users of Microsoft Outlook open e-mail attachments. An
    e-mail attachment is an encoded package that comes along with a plain
    text message. There has never been any guarantee that e-mail
    attachments are wholesome. Indeed many in large organisations will
    exchange pornographic images and suchlike. It is rumoured that the
    Lovebug Worm cost it's victims up to one billion dollars in lost
    business. The world probably benefitted because much of this so called
    'business' is leading to environmental degredation and oppression of
    the poor. The writer, Ramel Ramirez did the world a favour.

        The ease of propogation of the worm highlights the incredible
    stupidity of corporate fascism. The genetic sequencers promise cure for
    malaria or AIDS by their understanding of DNA but the industrial
    complex is unlikely to deliver such results quickly. What they are
    really seeking is loads of money immediately, and the benefits to
    humanity will 'trickle down' towards the poor.

        Modern medicine has made vast advances in explaining immunity, but
    the explanations offer little consolation to individuals. What happens
    is that immunity works in whole populations [4]. Those who are immune
    to a disease act as a barrier to the propogation of the disease causing
    pathogen. In fact only a relatively small proportion of the population
    need immunity to drastically curtail disease propogation. This effect
    is called 'herd immunity'. Herd immunity was important in preventing
    the Love Bug worm doing any real harm. Users of computers whose
    managers were not addicted to Microsoft Office and it's cancerous
    overgrowths were completely unaffected.

        These ideas about immunity come from physics. Boltzmann and Maxwell
    contributed to the kinetic theory of gases which describes the
    movements of gas molecules as random walks with very predictable
    effects for the extremely large numbers of molecules involved. This
    random motion of molecules can also be observed with a microscope, and
    is named Brownian motion after the discoverer. Temperature is related
    to the velocity of motion of molecules.

        Mathematicians had been working on solutions of the heat equation
    since the 1820s. This equation and its solutions arise in areas of
    research ranging from the Theta Functions of Jacobi to Wall Street
    derivatives market.

        Following the success of the kinetic theory of gases the same sorts
    of calculations were applied to newly discovered particles such as
    protons and neutrons. These calculations were particularly important in
    predicting the chain reactions involved in nuclear fission. Neutron
    absorbers and reflectors became key components in nuclear detonators.

        The same models are applied to population dynamics. The modern
    interest in ecology and species diversity shows that people can be
    mobilised to express forceful views on possible absorbing and
    reflecting barriers in DNA space, but many of the most powerful players
    are sadly deficient in presentation skills. They also leave their own
    computer systems vulnerable to attack.

        Random walk explanations start from very simple models. The first
    model is the simple coin tossing game with sequences of heads or tails.
    A sequence THHTH.... etc can be translated into motion in many
    different ways, but the simplest is simply counting the number of
    occurences of one of the faces, which is considered a success. This
    gives an ever increasing sequence, which can be translated into the
    motion of a point just moving forever in one direction at an irregular
    velocity.

        It is possible to ask the probability that the point has moved k
    steps in n trials, and also where the point is most likely to be. With
    one trial then there is either no motion, or a displacement of one
    step. With probabilities of success and failure given the values p and
    q the number of successes S is 1 with probability p and 0 with
    probability q. This can be written pX+q where X is symbol. When p=q=1/2
    then just write 1+X. It then happens that all the probabilities of
    moving k steps in n trials can be obtained by reading of the
    coefficients in the series (1+X)^n, dividing by a factor of 2^n. There
    are many proofs of this in textbooks, but the idea is to reduce
    statements about motion to features of polynomial multiplication.
    Polynomials that correspond to movement or growth are often called
    generating functions. What physicists did was to go and take straight
    limits giving things like the bell curve much loved by certain
    statisticians and social scientists.

        Whatever the nature of the proof the theory works well in practice.
    Any player of backgammon or monopoly will be aware that seven is the
    most common total of two dice. The relative probabilities of the totals
    are given by reading off the coefficients of

             (x+x^2+x^3+x^4+x^5+x^6)(x+x^2+x^3+x^4+x^5+x^6)

              x^2+2x^3+3x^4+4x^5+5x^6+6x^7+5x^8+4x^9+3x^10+2x^11+x^12

        This gives seven as the most common outcome. Other features of
    generating functions are their ease of application. The rules don't
    change across different computer or language systems. As it happens
    genetics follows the same patterns. Going from one generation to the
    next various outcomes can be read off as coefficients in a polynomial.
    Many scientific processes are explained by taking limits of these
    coefficients, but DNA space is different. Just like physics has string
    theories, so undoubtedly we will see non-archimedian metrics applied to
    DNA space.

####
BRIDGE HANDS

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        In the game of Bridge each player gets 13 cards from a pack of 52.
    Decisions are made on point counts for high cards. The most frequently
    used system counts 4 for an Ace, 3 for a King, 2 for a Queen and 1 for
    a Jack. Zia Mahmood publicised a generating function for this point
    count in The Guardian.

        (1+y)^36 (1+xy)^4 (1+x^2y)^4 (1+x^3y)^4 (1+x^4y)^4

        Zia posed a question about these distributions in his Christmas
    competition which is published every year in the Guardian. He got
    several exhaustive analyses in his responses along with correct answers
    on bidding and playing hands. The formula was sent by Dr Jeremy Bygott
    of Oxford. The coefficient of y^13 gives the generating function for
    point count in a 13 card hand. This is a beautiful solution.

####
HISTORICAL CONTEXT

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        Q: What is an array ?

        Medieval times: Lords of the Array.

        An array is a set of numbers arranged in lines, often called rows or
    columns. This meaning of the word is relatively new.

        In feudal times the King could call on his vassals to provide arms
    and men in times of war. The managers of these mixtures of unwilling
    conscripts along with vain and boastful knights were called 'Lords of
    the Array'. The Chinese lost Hong Kong and Shanghai because they
    adhered to this system well into the nineteenth century. The British
    themselves fell victim of these feudal hangovers when a coterie of
    upper class generals, collectively termed 'The Donkeys' went and
    ordered hundreds of thousands of working class men to charge German
    machine gun emplacements during the First World War. These generals had
    learned nothing from the losses of the Chinese during the Opium wars of
    the 1840s and 1850s.

        In the 1700s Euler investigated the problem of '36 army officers'.
    There are six ranks of officer, and six regiments. The idea was to
    arrange these officers in a 6x6 square so there were no two from either
    the same rank or regiment standing in line.

        In 1782 Euler conjectured the problem was impossible.

        In 1900 G.Tarrey showed that Euler had been right through a brute
    force enumeration of all 6x6 latin squares.

        Godement's treatise on Algebra, written in the 1960s contained
    tables showing the number of bombs dropped by the Americans on
    Viet-Nam. An excercise asked the reader to verify the associative laws
    of addition by adding up items in these tables.

        More recently Ian Stewart described a military array problem in his
    regular Scientific American column [3]. The commander in chief (C-in-C)
    inspects his troops arranged into a certain number of squares. The
    C-in-C then takes his place in the army whose order of battle is a
    single huge phalanx, which also happens to be a square. They are
    unlikely to win the battle this way, but the size of the army is what
    counts here. The C-in-C inspects 61 equal squares of troops. The size
    of the army is the lowest integer solution to Y^2 = 1+61*X^2.

        The equation Y^2-M*X^2=1 is known as Pell's equation. It is a
    perennial favourite for maths puzzles.

        Excercise: Try the problem with 61 squares, then say ... 9349.

        Excercise: For the table top.

           You can arrange 84 match sticks to make a 6x6 square or a set
        of 28 triangles as shown. Try and make larger such arrangements.

             -- -- -- -- -- --                     /\
            |  |  |  |  |  |  |                    --
             -- -- -- -- -- --                    /\/\
            |  |  |  |  |  |  |                   ----
             -- -- -- -- -- --                   /\/\/\
            |  |  |  |  |  |  |                  ------
             -- -- -- -- -- --                  /\/\/\/\
            |  |  |  |  |  |  |                 --------
             -- -- -- -- -- --                 /\/\/\/\/\
            |  |  |  |  |  |  |                ----------
             -- -- -- -- -- --                /\/\/\/\/\/\
            |  |  |  |  |  |  |               ------------
             -- -- -- -- -- --               /\/\/\/\/\/\/\
                                             --------------

        Q: What are Tensors and Spinors ?

        A: You are talking about physics and applied math here.

        Tensors and spinors on are just special types of array which are
    used to explain 20th century physics. In a sense tensors are like
    mixtures of functions and numbers. Partial derivatives feature in many
    formulae involving tensors. If you are into pure maths, look at any
    good book on Quadratic Forms. These will give you the group theory.

####
REVOLUTIONARY RETRIBUTION: YEAR ZERO

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        Following the success of the French Revolution the new regime
    co-opted some of the best scientists of the day to supervise reform of
    the calendar. The new year was divided into months with names drawn
    from European folk tradition. The English made fun of this innovation
    by adding their own meanings. The new French years ran from September
    to August.

                       Table YEAR_ZERO
        Revolutionary                       English
        Month                               nick-name

        Vendemiere      vintage             Wheezy
        Brumaire        mist                Sneezy
        Frimaire        cold                Freezy
        Nivose          snow                Slippy
        Pluviose        rain                Drippy
        Ventose         wind                Nippy
        Germinal                            Showery
        Floreal         flowers             Flowery
        Prairial                            Bowery
        Messidor                            Wheaty
        Thermidore      heat                Heaty
        Fructidor       ripening            Sweety

        The French calendar was worked out by a committee of experts which
    in included the great mathematician Laplace. Months all had the same
    number of days. Extra days were added between some of the months to
    give public holidays. Every leap year would have an extra day which
    could celebrate the revolution.

    Laplace ensured that his role on the calendar committee was not too
    prominent because it bacame quite dangerous to express views for a
    while. Two other members of the committee went to the guillotine, and
    when Lavoisier the chemist was condemned his judge went on the record
    as saying that the revolution had no need for scientists. Nowadays
    Laplace is best known for Laplace's Equation which is of very great
    importance in electro-magnetic theory.

        The extremists of the time, called Jacobins,  brought in two
    important laws that acted as an influentual political model for the
    future.

        (1) Law of the Suspect.

        Any person suspected of not giving one hundred support to the new
    regime could be arrested and brought before a Revolutionary Tribunal.
    If that person were convicted of being a counter revolutionary then
    there was a mandatory death sentence.

        (2) Law of the maximum.

        The government could set maximum food prices by decree. Distributors
    who put up prices beyond the limit were obviously to be treated as
    saboteurs of the revolution.

        The Jacobins also wanted to abolish torture. Their death sentences
    made use of the newest technology of the time: a heavy free sliding
    blade invented by Dr Guillotine. There is some evidence that
    Guillotine's technology was rather more appropriate for executions than
    the new fad introduced by the Americans one hundred years later. During
    the 1890s debates between Siemans and Westinghouse over A.C. or D.C
    electric power systems the Americans tried out wiring up condemned
    criminals to the electrodes; they are still debating this application
    of technology one hundred years after that. A cynic, or maybe Theodore
    Kaczynski,  might well add that it is a pity that Laplace did not go to
    the guillotine as well as Lavoisier.

        It is fairly sure that Laplace was not the only astronomer who
    needed to be aware of the politics of the time. Most civilisations have
    had professional astronomers from ancient times. Some of them must have
    tried to get funding by predicting the fortunes of big-shots. Kepler
    himself is reputed to have cast a horoscope for Wallenstein, a famous
    European Warlord during the Thirty Years War (1618-48).

####
POLITICAL ECONOMY

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        Fibonacci  Model  1202. Leonardo de Pisa. Liber Abaci.

        Y[t+1] = Y[t]+Y[t-1]
        Single population. Two cohorts.

        Doubling time model. Widespread actuarial use from 1700s.
        Malthus [6] popularised the idea in the 1820s.

        dY/dt = K*Y

        Logistic Curve. Non-linear.

        dY/dt = aY-bY^2 = Y(a-bY)

        Volterra Equations. 1920s Two species.
        Predator and Prey.

        dX/dt = aX-bXY
        dY/dt = -cXY+pX

        The Volterra equations remain unsolved at the time of writing.
    Volterra himself was directly interested in these equations. His son in
    law analysed statistics on the Adriatic fish catch. Population
    modelling has generally not received much funding. There is much
    commercial and political pressure to supress this sort of knowledge.
    Industrial fisheries are a case in point. Scientists are generally
    ignored when they recommend prudence while robber barons are keen to
    exploit the benefits of science to follow fish shoals with satellite
    navigation aids. In the meantime the law enforcement agencies never use
    high technology to catch these kleptocratic industrial pirates but turn
    their guns on desperate people trying to either to migrate or to ship
    'illegal drugs'.

####
INDUSTRY AND EMPIRE

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                      The loom and the music box.

        The current day meaning of a list of numbers only became common
    towards the end of the last century when mathematicians started a
    systematic study of linear equations and invented matrices,
    determinants and tensors. The British mathematicians, Cayley and
    Sylvester, were well known contributors to this field.

        Ada, Countess of Lovelace [1], was an early pioneer of machine
    oriented calculations on arrays. One particular array taxed her mind.
    She worked on programming the calculation of the Bernoulli numbers,
    These are found in the power series expansion of cos(x)/sin(x), and
    x/(exp(x)-1).

        Babbage had made a theoretical design of a computer to do
    calculations and although his computer never worked the period saw a
    whole lot of inventions for storing programs. These early methods
    included rolls of material with holes in them, or rotating drums with
    pins sticking out. The two methods correspond to male and female, Yin
    and Yang, or N and P layers of semiconductors which stand for the
    absence or presence of electrons.

        The popular stored program machines of this era included textile
    machinery and clockwork music boxes. The Player Piano was perhaps the
    most elaborate such device. There were also early attempts to deceive
    the public into believing in 'chess playing computers' where the
    sponsor would hide a dimunitive chess master in an engine and get
    people to play chess against it.

####
AGE OF EXTREMES

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                 Shockwave Calculations.

        Einstein needed tensors in order to construct the Theory of
    Relativity. From that day the 'Lords of the Array' became completely
    reduced to the ranks of the proletariat. Arrays and matrices were only
    taught in advanced maths and physics courses, and they had more
    theoretical than practical use. Numbers were stored by writing them on
    paper.

        It was still easier to comission armies of manpower to handle vast
    calculations. Disney studios employed hundreds of artists for its
    animations. The Atom Bomb project used hundreds of human calculators to
    solve some of the differential equations.

        The 'National Emergency' of the Second World War saw great advances
    in computing and management science. The computing community split into
    two with 'Partial Differential Equations' opposed to 'Commercial Data
    Processing'. There were those who naively believed that increased
    computing power could make for a better society. The idea of economic
    planning had been OK in wartime, and food rationing schemes had
    sometimes been worked out on theoretical calculations. China and Russia
    both had governments which paid lip service to the planned economy.
    Whole systems of input-output equations had been invented to describe
    the economy. This was often called Marxism.

        Unfortunately for Soviet and Chinese statisticians these equations
    and the input data were too hot to handle because any data which
    reflected poorly on the performance of the regime would be supressed,
    and cause the statisticians to be imprisoned or shot.

        The earliest advocates of a planned economy often had to leave their
    countries in order to save their own lives. Karl Popper attempts to
    analyse the reason for this in his book 'The Open Society and it's
    Enemies'. Both Stalin's Russia and Hitler's Nazi Germany embraced state
    planning. Thinking men voted with their feet.

####
CORPORATE FASCISM

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                 Poverty of Data on the INTERNET.

        Nowadays statisticians do not run such high risks of being shot.
    Offending data is buried in ordure. There are plenty of subservient
    think-tanks to produce all sorts of idiotic reports to justify
    government policy. Nevertheless some data is so sensitive that people
    will be persecuted for revealing it. Drug prices, and food and drug
    safety issues are jealously guarded, and there are many half forgotten
    cases of alleged corporate manslaughter.

        Stanley Adams, a former Hoffman La-Roche employee, lost his liberty
    and his wife when he revealed the drug company's European price fixing
    arrangements. Planned profit is precious.

        Also in Europe there have been several suspicious deaths of
    frontline data collectors in the field of vetinary medecine. The
    additions of hormones and antibiotics to animal feed have raised
    concerns about health for decades, and the agri-business sector seems
    to have heeded these concerns by adding good old fashioned shit to the
    diets of the animals.

        The reader will not see many arrays in government reports, or
    elsewhere, including the Internet. More often reports are represented
    as graphs or bar charts with carefully concealed scaling information.
    The array that a scientist may want to access could be buried in
    megabytes of Post Script code.

        In real life data arrays are hard to accumulate. It needs some
    discipline to keep figures for a time series for example. Budget cuts
    and privatisation are the enemies of the modern statistician.
    Megamergers between rival pharmaceutical companies also contribute to
    the chaos. Any array has an associated size, normally the number of
    elements in the array. This is important. Sometimes new drugs and
    medical treatments are advocated even though there are more doctors
    doing the research than patients whose treatment is being evaluated.

        With the increasing power of computers it gets easier and easier to
    maintain statistics but most of them only reflect the enrichment of the
    elite: the profits of the big corporations and those of their immoral
    dealings which they wish to disclose to their old-boys club style
    regulators that run the World's capital markets. In the meantime those
    responsible for the data are cajoled into efficiently operating the
    machinary, rather than operating their brains.

####
THE CLASSIC CALCULATION

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        Bernoulli numbers just come from from long division by power series
    [3]. They can be used to estimate sums by integrals. The most famous
    formula in which they arise is Stirling's Approximation for n factorial
    (n! = 1 x 2 x 3 .... x n-1 x n). They come as coefficients in the
    expansion

                   oo
          x        ---         n
         ------ =  >     B[n] x
          x        ---       ---
         e  -1     n=1        n!

     Z<-N P_INV A
     |Inverse of series a[0]+a[1]x+a[2]x*x....
     |assume a[0]=1
     Z<-1 & N<-20 #ifndef"N"
     AA: ->(N<rZ)/XX
     Z<-Z,-+/(mZ)*(rZ)t1dA
     ->AA
     XX:

     B[  0]=               1  1/1
     B[  1]=            -0.5  -1/2
     B[  2]=      0.16666667  1/6
     B[  4]=     -0.03333333  -1/30
     B[  6]=      0.02380952  1/42
     B[  8]=     -0.03333333  -1/30
     B[ 10]=      0.07575758  5/66
     B[ 12]=     -0.25311355  -691/2730
     B[ 14]=       1.1666667  7/6
     B[ 16]=      -7.0921569  -3617/510
     B[ 18]=       54.971178  43867/798
     B[ 20]=      -529.12424  -174611/330
     B[ 22]=       6192.1232  854513/138
     B[ 24]=      -86580.253  -236364091/2730
     B[ 26]=       1425517.2  8553103/6
     B[ 28]=       -27298231  -23749461029/870
     B[ 30]=   6.0158087e+08  8615841276005/14322
     B[ 32]=  -1.5116316e+10  -7709321041217/510
     B[ 34]=   4.2961464e+11  2577687858367/6
     B[ 36]=  -1.3711655e+13  -26315271553053477373/1919190
     B[ 38]=   4.8833232e+14  2929993913841559/6
     B[ 40]=  -1.9296579e+16  -261082718496449122051/13530
     B[ 42]=   8.4169305e+17  1520097643918070802691/1806
     B[ 44]=  -4.0338072e+19  -27833269579301024235023/690
     B[ 46]=   2.1150749e+21  596451111593912163277961/282
     B[ 48]=  -1.2086627e+23  -5609403368997817686249127547/46410
     B[ 50]=   7.5008667e+24  495057205241079648212477525/66

        These numbers are calculated by inverting the series (exp(x)-1)/x.
    The method is found in the D4 script ada.d4f. Fast methods of
    evaluating these numbers were pioneered by Ramanujan who was reputed
    able to do such calculations in his head.

####
PRIME TIME

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        Zeta(k) is the sum 1/n^k for integers n >=0.

                    n=oo
                     __   1
         Zeta(s) =  \    ____
                    /__   s
                    n=1  n

        The zeros of Zeta(s) are distributed in a T shape in the complex
    plane. Some are on the negative x axis, y=0, x=-2n and others are in a
    strip 0<x<1, y  with infinitely many different possible values y. The
    value of Zeta(s) is also well known for values s=2n on the even
    numbers.

        The terms in the definition of Zeta(2) occur in the equations for
    quantum levels of spectral lines. More recently physicists have been
    checking resonent frequencies in dynamical systems and getting numbers
    that look like the distribution of the known zeros of the Zeta
    function.

        All of this work is motivated by a million dollar prize for a more
    precise characterisation of the zeros of the Zeta function. The Riemann
    hypothesis states that all the zeros not of the form s = -2n are of the
    form s = 1/2+it where t is real. As yet no one knows if this is true.

        During the search for a proof of Fermat's conjecture quite an
    elaborate theory of Adeles was constructed. All adeles corresponded to
    number fields and specialised zeta functions. These algebraic
    structures all had a topology and the interesting ones contained roots
    of unity.... solutions  to X^n - 1=0.  Chemists and physicists were
    used to working with poorly understood topologies to describe
    intractable results in real life. String theories are examples of weird
    topologies used to describe the real world.

        The theory of dynamical systems which gave fractals and explanations
    for chaos has been some advance on Galileo's theories on projectiles
    because it is adaptable to quantum theory.

        Quantum theory makes the maths of physics quite complicated if
    things like angles and space time co-ordinates are all integer
    sequences. Many states are impossible by easy calculations while other
    states may be subject to intractable calculations. Schrodinger's cat is
    either alive or dead so uncertainty is replaced by a wave function.

####
ULAM NUMBERS

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        Some very profound problems in mathematics have very simple
    definitions. A classic example is the '3n+1' problem or the Collatz
    Conjecture.

        Define a function f:N->N where N is the set of positive integers.

            f(n)  =  n/2 if n is even.
            f(n)  =  3n+1 if n is odd.

        For example 5 is odd so f(5)=3*5+1=16. f(16)=8, f(8)=4, f(4)=2 and
    f(2)=1. We also have f(1) = 4. For any function f and starting point x
    the sequence x, f(x), f(f(x)) .... is known as the orbit of x under f.
    In the case of the '3n+1' function these orbits are also known as 'Ulam
    Numbers'. The question concerns the eventual behaviour of such
    sequences: do these 'Ulam Numbers' always converge to the cycle 1 2 4 ?
    At the time of writing no one knows.

####
DYNAMICAL SYSTEMS

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        A dynamical system is a pair (X,f) where X is a set and f is a
    continuous function f:X->X. For example X may be co-ordinates and
    momenta of sun and planets and f a rule for computing the state of the
    system at time t+1 given its state at time t. For any given start x[0]
    the sequence given by x[n+1] = f(x[n]) gives the ultimate behaviour of
    the system. The French mathematician Poincare invented dynamical
    systems. Many important problems of maths, physics and even economics
    and ecology can be reduced to the study of dynamical systems.

        Since 1986 dynamical systems have served to illustrate the
    limitations of mathematics as a means of predicting the real world. The
    simple system z<-z^2+c defined on the complex number plane is enough to
    generate the Mandelbrot set. The fractal images that people see are the
    complement of the Mandelbrot set. The colours reflect the amount of
    computation required to show that the iteration z<-z^2+c diverges.

        The following fragment of C-language code generates the pixels
    of an image that can readily be found in books on fractals.

    <code>
     /* dynamic system z<-z^2+c with global parameters eps, bailout */

     int     j_orbit(cy, cx)
     double  cy, cx;
     {
     double  x = 0.0, y = 0.0;
     double  x1, y1, x2, y2;
     int     j;
     for(j = 0; j < bailout; j++) {
         x2= x*x; y2=y*y;
         if (4.1 < x2+y2) return j;
         x1 = cx + x2-y2;
         y1 = cy + 2*x*y;
         x = x1;
         y = y1;
         }
     return 0;
     }
    </code>

     References:
     [1] The Theory of Numbers, 4th edition.
     Hardy & Wright. pp19,22.

     [2] n=p1+p2(=$1m).
     David Ward.
     The Guardian, 18 March 2000

     [3] Ada and the First Computer
     Eugene Eric Kim & Betty Alexandra Toole,
     Scientific American, May 1999.

     [4] Plagues. Their Origin, history and future.
     Christopher Wills.
     Harper Collins 1996.

     [5] Prime Time
     Erica Klarreich
     New Scientist, 11 Nov 2000 #2264.

     [6] An Essay on the Principle of Population
     Malthus. Various editions from 1798-1830

####
D4X.EXE

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     (C)   Tony Goddard, Sheffield  2003
     +44(0)7944 764312
     http://d4maths.lowtech.org

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