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Notes

[Sunday 29 March 2009 - Saturday 4 April 2009]

[Notebook: DB 66 Turing Field]

Sunday 29 March 2009

[page 77]

Monday 30 March 2009

The enormous comnplexity of the current universal network may mask the underlying simplicity which we postulate to have its root in the initial singularity, the archetypical unit of memory. This is illustrated by the complexity of the Feynman diagrams needed to arrive at a precise computation of electrodynamic interactions.

[page 78]

The structure of the Universe, we suspect, is a product of communication which not only differentiates but also unites elements of the Universe, understood by analogy to the psychlogical model of the Trinity developed by Augustine and Aquinas Augustine, Aquinas. We map this onto Zurek's ideas about the quantum origin of quantized interactions using Shannon's ideas about eliminating error from communication. The process of error correction introduces delay while the encoding system waits for the source to emit enough symbols to construct a packet. This suggests that delay only enters the Universe with two component packets, since there is no way to understand delay when each packet is simply a single symbol. The single symbol epoch, the transmission of pure energy, we take to correspond in some way to gravitation, pure power transmission with no possibility of error. Somewhere here we can imagine the concept of the energy or frequency of a packet becoming meaningful.

Since light has a finite velocity, we can imagine that it dates from the epoch of the emergence of coding and delay, but photon-photon coupling at this point has the same strength as gravitation, ie it is simply the coupling of energy to energy. What comes next? Since photons seem to be among the simplest observable particles, the next epoch might witness the emergence of electrodynamics and charged particles. From the point of view of simplicity we see electrons as the emergent particles or process associated with this epoch and we imagine a Universe composed of photons, electrons and positrons. Electrons have rest mass, and so can be imagined as particles at rest. Since their interaction with photons requires potential and motion, we also see the rudiments of space, Lorentz transformtion and spin emerging in this epoch which corresponds in some way to the completed Trinity, three distinct 'personalities', electron, positron and photon in interaction. The

[page 79]

task is to distill the primordial interactions of these particles from the complex behaviour they manifest in the current Universe. At a guess, what we are looking for is comething like the CPT world of the 4-component Lorentz group which combines two directions in 'time', 2 directions in 'space' and two species of otherwise identical particle (particle and antiparticle) (Streater and Wightman page 9 sqq. Streater & Wightman

'A Lorentz transformation Lambda is a linear transformation, mapping space-time onto space-time which preserves the scalar product xmuymu.' In one dimension (time-energy) this can be seen as conservation of energy x0y0 constant. In to dimensions we have x0y0 - x1y1 is concerved, something that looks a bit like a Lagrangian.

Tuesday 31 March 2009

Coupling constant = rate of meeting x probability of interaction.

Quantum mechanically, probability of interaction falls off as some functcion of spatial distance which we imagine to be some function of logical distance.

Human coupling constant depends on individuals, proximity and duration of proximity, so gregarious people close for a long time are more likely to interact than the opposites.

Quantum field theory sets out to compute the traffic on various channels which are definedby various symmetries, selection rules and so on. Human etiquete also establishes various rules of communication, eg we cannot talk until we have been introduced, etc.

Similarly legislation is intened to increase the probability

[page 80]

of some events and desrease the probability of others. The structure of machines has similar purpose, guiding the process so that the probability of the the desired outcome (a crash free flight) has a probability very close to one.

The Universe is a system of many independent personalities talking to one another. We must ask why are there many personalities if it was initially just one; why do they communicate with one another? How do personalities differentiate? How do they communicate? How do we predict the behavious of a complex communicating community?

Veltmann says that quantum field theory is a mess but it gets the right answers. VeltmanWe say because it effectively treats the Universe as a network without taking a network view.

Events occur in a certain order, but quantum mechanics of itself cannot predict this order, it merely gives us the probability of one event following another. The event represented by phi is followed by psi with a probability |<phi | psi>|2. But to achieve a computation this is not enough. We must manipulate the probabilities in order to create an ordered cycle of events. All technology is comprised of ordered cycles of events.

This is the principal feature of quantum mechanics: of itself it cannot predict (or control) events.

ORDER requires SPACE (at every level of abstraction) so order and space are born together.

[page 81]

Misner, Thorne and Wheeler capture the essence of the scientific life: 'Here and elsewhere in science, as stressed not least by Henri Poincare, that view is out of date which used to say: "define your terms before you proceed". All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts they use (here B and E) [alphabet] and make statements about their concepts. Contrariwise, the absence of some body of theory, law and principle deprives one of the means properly to define or even use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concept, law, and method of measurement, -- forever inseparable -- are born into the world in union.' Misner, Thorne and Wheeler page 71.

CHAOS = NOT-ORDER = NOT-CYCLIC

Why did Darwin take so long to publish his book after he got home on the Beagle? Because he was faced with the big problem of synthesizing three years of data. I did my fieldwork in a Roman Catholic religious order where I had an in depth experience of an ancient and obsolete religion which has served as the data motivating the construction of a reasonable religion.

In the end our fundamental principle for guiding behaviour is the question What would a reasonable person do? The ideal reasonable person is one who is aware of all the issues at stake and . . . navigates a safe course through them all, or decides that one cannot go there. There are dangerous and forbidden zones in the space of human existence. A 'forbidden zone' is a volume in this space where there is a high probability of death. To die is for one's cycle to break, ie to cease cycling.

Quantum mechanics - sex, energy: 'disordered forces' ie uncontrolled. The fundamental religious questions are all the same: how do we control desire, energy and quantum mechanics? How are they controlled: after a period of search, people bond into a stable couples, and the relationship can be given weight by public vows.

[page 82]

What about religious vows? I vowed to curtail my three principal degrees of freedom: physical wealth and power are rejected by the vow of poverty; self determination by the vow of obedience, the foundation of militarism in which lives are expendable. Finally I vowed to cast sensuality, sexuality, reproduction and normal family life out of my life by the vow of chastity. The resulting human amputee was judged an ideal eunuch to administer the power of the Papacy. This is a system, but a system that fails to respect the nature of its components.

Every PERSON has a NATURE

QUANTUM MECHANICS = CHAOS (the entropy space in which the Universe grows, the womb of the Universe, constrained only by consistency = error correcting power). A cyclic system us one that does not make errors insofar as it always conserves itself.

Why do I work? To satisfy my desires, including the desire to work and get pay.

Quantum field theory introduces a four-space filled with operators which are points in infinite dimensional function space. In this way quantum field theory maps quantum mechanics onto spacetime through the Lorentz transformation induced by the observation that the velocity of light is the same for all observers because all observers observe the velocity of light in the same way, by measurements of time and distance. [Hilbert spaces are inertial spaces?]

[page 83]

Streater and Wightman page 96: 'Since in quantum mechanics observables are represented by hermitian operators which act on the Hilbert space of state vectors, one expects the anlogue in relativistic quantum mechanics of a classical observable field to be a set of hermitian operators defined at each point of space-time and having a well defined transformation law under the appropriate group.

GROUP = NOT-ORDERED CYCLE, ie the group structure does not constrain the order in which various transformations in the group are applied. The only constraint on transformation probabilities in a group of transitions is uncertainty, ie they happen one at a time, event by event. [what about continuous and differentiable groups?]

LIFE = "CONTROLLED RANDOM" PROCESS

Complex structures arise by the increase in probability of survival that the complexity of the system allows.

Quantum computation is a 'use' of quantum mechanics, devising systems that reproduce the quantum mechanical formalism with adequate fidelity to initiate a cascade of error control, ie improved error control improving error control, rather like the evolution of the public health system designed to control error (ill-health) in the human population. Eventually the concept of public health will extend to every species and element of our common environment.

USE = ORDER (as I use a spanner by taking it through a cycle of motion in space-time). Quantum field theory gives every point in four dimensional

[page 84]

spacetime an infinite dimensional 'interior' represented by a Hilbert space. In this way each Hilbert space obtains a spacetime address. Every point in spacetime is the address (name) of a state vector.

NAME = ADDRESS

Name and address simply divides address into two parts, one more specific and one more generic which between them enable the identification of an individual. Hilbert spaces [are] the memory in which the nature of the person at each spacetime point is written.

Wednesday 1 April 2009

Quantum mechanics deals in possibilities and probabilities, not in order.

Streater & Wightman page 96: '. . . the theory of fields can be expressed in terms of certain associated distributions, the vacuum expectation values of products of field operators.'

Such products characterize a path in the path integral approach, which is interpreted as a network approach, the path being the continuous limit of the operations of the nodes the signal goes through on its way from initial to final state. Each node is thus conceived as an infinitesimal (close to identity) operator, and the totality of them conceived as the product that carries from the initial operator (with a certain set of observable probabilities) to the final, with a different set of probabilities.

[page 85]

We express the properties of the theory in terms of the properties of vacuum expectation values.

'It was recognized early in the analysis of field measurement for the electromagnetic field in quantum electrodynamics that, in their dependence on a space-time point, the components of fields are more singular than ordinary functions.'

Since continuity carries no information.

BROKEN LIFE CYCLE = EXTINCTION

Quantum mechanically indistinguishable does not mean not spatially distinguished, insofar as electrons do occupy different spacetime locations even though quantum mechanics sees them as 'identical'.

What we are looking for is a physical representation (ie a representation governed by conservation laws / symmetries of the symmetric network which is isomorphic to the good bits of quantum field theory).

The show must go on, the cycle (however defined) has to be completed in a way that the end of one cycle can becomes the beginning of another. Each cycle is an ordered group of transformations leading to the identity, so that the cyclic 'particle' has extended existence. My basic period is my lifetime, L, and all other periods within me are less than my lifetime, with corresponding frequencies.

FREQUENCY = 1/PERIOD, or k/PERIOD

[page 86]

A good theory, like special relativity, is very obvious once one has 'seen' it but rather obscure before that. We can think of understanding in terms of decoding, where we detect the meaning in the message. In everyday human communications with regular communicants this is perceptibly instantaneous, in more arcane areas like the nature of the world, it seems to need centuries of collective effort, which can be seen from a distance as a collection of rather random walk[s] toward understanding the behaviour (personality) of the world.

The PERSONALITY of NATURE

Hazewinkel Duality Principle Hazewinkel

Algebraic System

Group: 'The concept of group is historically one of the first examples of abstract algebraic systems and served, in many respects, as a model for the restructuring of other mathematical systems at the turn of the twentieth century, as a result of which the concept of a mathematical system (a structure) has become a fundamental concept in mathematics.'

Group has one binary operation which 1. associative; 2. has a unit; 3. has an inverse

Hazwinkel volume 2 page 900 'The deep connections between the properties of permutation groups and those of equations were pointed out by N H Abel (1824) and E Galois (1830). Niels Hendrik Abel - Wikipedia, Evariste Galois - Wikipedia

A physical group is a physical incarnation of an abstract

[page 87]

group, ie one has to render the abstract (logical) group physical to see it actually in action in the permutation of electrons in an atom or the behaviour of a machine or some such. In a physical Universe dynamics and formalism are the two constraints on action, ie between them define actions as possible and assign probabilities to the possibilities by invoking various symmetries or constraints.

Cayley: Any finite group can be represented by permutations. Cayley's theorem - Wikipedia

'Cayley. . . conceived a group as a system which is defined by its generating elements and defining relations. The final step in this development was the Erlangen program of F Klein (1872), who based the classification of geometries on the concept of a transformation group.' Felix Klein - Wikipedia

'1895 Lie defined a group as a set of transformations that is closed under an operation that is associative, admits a unit element and inverse elements.' Sophus Lie - Wikipedia

Hazewinkel 2:901: 'Group theory plays . . . a role in physics. Thus the state of a physical system is represented in quantum mechanics by a point in an infinite dimensional vector space. If the physical system changes from one state into another its representative point undergoes some linear transformation. The ideas of symmetry and the theory of group representations are of prime importance here.'

Thursday 2 April 2009

A cycle requires an error free return to the starting point.

[page 88]

ORDER = Non-Abelian group = non-commutative.

Is the essence of quantum mechanics non-commutative multiplication, or does this not arise until we enter quantum field theory?

Alexandroff page 20: 'Permutations are considered as mappings of a finite set onto itself.' Alexandroff

A permutation on n objects is spcified by a function ak = f(k) where k is the index of the element to be replaced by ak. A machine to construct a particular permutation must remember f(k) for all k, or an algorithm to compute f(k). So we have computation vs memory, and postulate that there is a sweet spot in there somewhere that corresponds to an extreme of action, ie of the time integral of the Lagrangian.

We can divide permutations into two classes named by sign, or odd and even. Oscillator 1 21212 . . . .

Alexandroff page 24; 'The group of all even permutations on n elements is called the alternating group on n elements and is denoted by An.'

Computer instructions forma a group? Any two instructions combined give another inbstruction? Not necessarily.

. . .

[page 89]

Friday 3 April 2009

How do we understand a magpie singing as part of god? The magpie itself is a set of self sustaining cycles whose tendency to error is controlled by the magpie's 'ingestion' of low entropy from the sun. The song is a message to other magpies which is part of the larger cycle of magpie reproduction and the work of obtaining the necessities for life from its environment. An individual magpie dies if its cycle of life is broken. Magpies become extinct if the cycle of reproduction is broken.

A guess: the Universe is a physical realization of mathematics subject to the constraint of limited resources, ie limited number and definition of symbols. It is driven by the Cantor potential to diversify.

We treat the initial singularity as an isolated quantum system and consider the Universe of experience to evolve within this system.

All we need to construct a transfinite Hilbert space is a measure which gives convergence at every transfinite cardinal dimension of space. The measure is a bit like the Dirac delta, insofar as we take aleph(n) x 1/aleph(n) = 1.

We may see the marital constraint on free love as a measure to secure the resources necessary for bringing any children conceived to self-sufficient majority. Self-sufficient does not mean doing everything for oneself, but contributing as much work to the overall system as one takes product out of it, by some measure.

[page 90]

MEASURE = CARDINALIZATION of an ORDER. So I measure an immensely complex organism like a tree with a simple unit like a cubic metre, used to measure volume of millable timber. Similarly the census merely counts us as individual units. Counting is made possible by the quantization of communication, so that the world can be seen as discrete events.

What time does a clock on a photon measure? To any observer it looks stopped. We are asking a question about the rest frame of a photon (what is the time recorded in the rest frame of a photon?) which does not exist. Place and time have no meaning for a photon, in other words it antedates them. So electric charge, ie coupling constant (cross section) for creation and annihilation of photons.

Then we imagine electromagnetism as the hardware layer of the strong force. Improved algorithms increase the probability of interaction to the point where we have confinement and asymptotic freedom, 8 gluons, 6 quarks etc. How does this happen? Write some software, ie the Turing machine that corresponds to the colour field when a set of them are placed in a network. The gluon network.

Let us guess that Einstein's equations are constrained by the fact that there is no external constraint on the Universe and yet spacetime is somehow bounded, being inside the initial singularity. If we imagine the initial singularity as a hollow sphere, we can see all particle geodesics originating on the 'boundary' of the sphere. Can we say that the Universe is a closed set, one which contains its own boundary? This boundary is the complex structure of any spacelike slice through

[page 91]

the Universe, is the fixed points in universal dynamics. Our boundary lies where local consistency meets local inconsistency.

According to [Einstein], space and time -- which must be put together as space-time -- are curved near heavy masses. And it is the attempt of things to go along straight lines in curved space-time which makes them move the way they do.

So the orbit of the moon is a quasi-periodic straight line, whose shape is determined by the mass / energy environment in which it moves.

It seems reasonable to be angry with the Church, which has trapped billions of people in misery during its long life and is probably responsible for about a million excess deaths per annum do to its inhuman policies, particularly with respect to gender, sexuality, reproduction and sexually transmitted diseases. It is as bad as malaria, and worth wiping out.

The initial singularity is an inertial frame with respect to itself. Within this frame, we can have a non-inertial = accelerated frame. We know that a frame is accelerated when it emits a material message, like a rocket or a radio transmitter.

Thermodynamics and statistical mechanics are the theories of a network where all that is required is secure transmission of messages, nothing about meaning. Ie physics deals with the hardware layer and the error correction layer, but goes no further. Here we are in the territory of biology.

[pages 92 and 93 blank]

[page 94]

Saturday 4 April 2009

In the physical realization of set theory correspondences are realized by computations. This statement in effect renders mathematics dynamic. Instead of the simple formal assumption that a corresponds to b, we now have to find a dynamic mechanism for the establishment and maintenance of such correspondences [bonds, which lower energy].

We cannot measure time at equilibrium because a steady state has no 'ticking' features which can be identified as units of time. So equilibrium is outside time. Time becomes established by fluctuations in the network that become self sustaining with fixed frequency.

Huang page 5: 'A thermodynamic quantity is said to be extensive if it is proportional to the amount of substance in the system under consideration and it is said to be intensive if it is independent of the amount of substance in the system under consideration. It is an important empirical fact that to a good approximation thermodynamic quantities are either extensive or intensive."Huang

EXTENSIVE = CARDINAL
INTENSIVE = ORDINAL (?)

page 10: Second law of thermodynamics. Second law of thermodynamics - Wikipedia

page 13: Absolute scale of temperature can be established by a string of Carnot engines all performing the same amount of work, each of which shares input and output temperatures with the engines above and below it. The Carnot engines discretize the temperature scale by an energy unit delta W per engine, thus

[page 95]

allowing temperature measurement by counting. A harmonic oscillator.

The major unsolved problem of quantum field theory, the absence of a bound state. Amazon customer review [by] Carlson [of] Peskin and Schroeder. Peskin & Schroeder

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Alexandroff, P S, and (translated by Hazel Perfect and G M Petersen, An Introduction to the Theory of Groups, Blackie and Son Limited 1959-1963 back
Aquinas, Thomas, Summa Theologica (translated by Fathers of the English Dominican Province), Tabor Publishing 1981 'Brother Thomas raised new problems in his teaching, invented a new method, used new systems of proof. To hear him teach a new doctrine, with new arguments, one could not doubt that God, by the irradiation of this new light and by the novelty of this inspiration, gave him the power to teach, by the spoken and written word, new opinions and new knowledge.' (William of Tocco, T's first biographer) 
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Augustine, Saint, and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augistine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.  
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Hazewinkel, Michiel, and (managing editor), Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' 
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Huang, Kerson, Statistical Mechanics, John Wiley 1987 'Preface: ... The purpose of this book is to teach statistical mechanics as an integral part of theoretical phyiscs, a discipline that aims to describe all natural phenomena on the basis of a single unifying theory. This theory, at present, is quantum mechanics. ... Before the subject of statistical mechanics proper is presented, a brief but self contained discussion of thermodynamics and the classical kinetic theory of gases is given. The order of this devlopment is imperative, from a pedagogical point of view, for two reasons. First, thermodynamics has successfully described a large part of macroscopic experience, which is the concern of statistical mechanics. It has done so not on the basis of molecular dynamics but on the basis of a few simple and intuitive postulates stated in everyday terms. If we first falimiarize ourselves with thermodynamics, the task of statistical mechanics reduces to the explanation of thermodynamics. Second, the classical kinetic theory of gases is the only known special case in which thermodynics can be derived nearly from first principles, ie, molecular dynamics. A study of this special case will help us to understand why statstical mecahnics sorks.' 
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Peskin, Michael E, and Dan V Schroeder, An Introduction to Quantum Field Theory, Westview Press 1995 Amazon Product Description 'This book is a clear and comprehensive introduction to quantum field theory, one that develops the subject systematically from its beginnings. The book builds on calculation techniques toward an explanation of the physics of renormalization.'  
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Streater, Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2005 Amazon product description: ' PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.' 
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Papers
Shannon, Claude E, "The mathematical theory of communication", Bell System Technical Journal, 27, , July and October, 1948, page 379-423, 623-656. 'A Note on the Edition Claude Shannon's ``A mathematical theory of communication'' was first published in two parts in the July and October 1948 editions of the Bell System Technical Journal [1]. The paper has appeared in a number of republications since: • The original 1948 version was reproduced in the collection Key Papers in the Development of Information Theory [2]. The paper also appears in Claude Elwood Shannon: Collected Papers [3]. The text of the latter is a reproduction from the Bell Telephone System Technical Publications, a series of monographs by engineers and scientists of the Bell System published in the BSTJ and elsewhere. This version has correct section numbering (the BSTJ version has two sections numbered 21), and as far as we can tell, this is the only difference from the BSTJ version. • Prefaced by Warren Weaver's introduction, ``Recent contributions to the mathematical theory of communication,'' the paper was included in The Mathematical Theory of Communication, published by the University of Illinois Press in 1949 [4]. The text in this book differs from the original mainly in the following points: • the title is changed to ``The mathematical theory of communication'' and some sections have new headings, • Appendix 4 is rewritten, • the references to unpublished material have been updated to refer to the published material. The text we present here is based on the BSTJ version with a number of corrections.. back
Zurek, Wojciech Hubert, "Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical", Physical Review A, 76, 5, 16 November 2007, page 052110-1--5. Abstract: 'Measurements transfer information about a system to the apparatus and then, further on, to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide a framework for 'wave-packet collapse', designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment — the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them — into becoming a witness.'. back
Links
Action (physics) - Wikipedia Action (physics) - Wikipedia, the free encyclopedia 'In physics, the action is a particular quantity in a physical system that can be used to describe its operation in an alternative manner to the usual differential equation approach. The action is not necessarily the same for different types of system. The contemporary action approach for physical systems yields the same results as those found using differential equations to describe the system, but only requires the states of the physical variable to be specified at two points, called the initial and final states. The values of the physical variable at all intermediate points may then be determined by 'minimizing' the action.' back
Aquinas 160 Summa: I 27 1 Is there procession in God? 'Our Lord says, "From God I proceeded" (Jn. 8:42).' back
Australian Magpie - Wikipedia Australian Magpie - Wikipedia 'The Australian Magpie (Cracticus tibicen) is a medium-sized black and white passerine bird of the family Artamidae native to Australia and southern New Guinea.' back
Cayley's theorem - Wikipedia Cayley's theorem - Wikipedia, the free encyclopedia 'In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G). Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.' back
Dirac delta function - Wikipedia Dirac delta function - Wikipedia, the free encyclopedia 'The Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function ?(x) that has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1. In the context of signal processing it is often referred to as the unit impulse function. Note that the Dirac delta is not strictly a function. While for many purposes it can be manipulated as such, formally it can be defined as a distribution that is also a measure.' back
Felix Klein - Wikipedia Felix Klein - Wikipedia, the free encyclopedia 'Felix Christian Klein (25 April 1849 – 22 June 1925) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.' back
Niels Hendrik Abel - Wikipedia Niels Hendrik Abel - Wikipedia, the free encyclopedia "Niels Henrik Abel (August 5, 1802 – April 6, 1829) was a noted Norwegian mathematician[1] who proved the impossibility of solving the quintic equation in radicals.' back
Publication of Darwin's theory - Wikipedia Publication of Darwin's theory - Wikipedia, the free encyclopedia 'The publication of Darwin's theory brought into the open Charles Darwin's ideas of evolution through natural selection, the culmination of more than twenty years of work.' back
Second law of thermodynamics - Wikipedia Second law of thermodynamics - Wikipedia - The free encyclopedia 'The second law of thermodynamics is an expression of the universal law of increasing entropy, stating that the entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.' back
Sophus Lie - Wikipedia Sophus Lie - Wikipedia 'Marius Sophus Lie (pronounced [liː], as "Lee") (17 December 1842 - 18 February 1899) was a Norwegian-born mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations. back
Thomas Aquinas Summa Theologica Thomas Aquinas: The medieval theological classic online. :'Because the doctor of Catholic truth ought not only to teach the proficient, but also to instruct beginners (according to the Apostle: As unto little ones in Christ, I gave you milk to drink, not meat -- 1 Cor. 3:1-2), we purpose in this book to treat of whatever belongs to the Christian religion, in such a way as may tend to the instruction of beginners. We have considered that students in this doctrine have not seldom been hampered by what they have found written by other authors, partly on account of the multiplication of useless questions, articles, and arguments, partly also because those things that are needful for them to know are not taught according to the order of the subject matter, but according as the plan of the book might require, or the occasion of the argument offer, partly, too, because frequent repetition brought weariness and confusion to the minds of readers.' back

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