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Notes

[Notebook: DB 57 Language]

[Sunday 24 July 2005 - Saturday 30 July 2005]

[page 179]

Sunday 24 July 2005

I sit here feeling hungry but I cannot think of anything I want to eat. The hunger is a potential which can be reduced by an appropriate motion, ie change of energy state from hungry to not-hungry.

Energy = time rate change of state = ability to do work

Quantum mechanics: no change of state is possible without communication, hence the system is a network.

The core problems of religion surround birth and death. Religion = quantum field theory on the human domain.

The search for literary form is the search for stationary states (like this text) which can be used as basis states for constructing an 'energy matrix' (transformation matrix) which is isomorphic to the situation whose literary capture is sought. The procedure is isomorphic to the discovery of a suitable basis and Hamiltonian to give a quantum mechanical description of physical systems.

'On the similarities of quantum field theory and natural human language.'

String theory? String = sentence.

Monday 25 July 2005
Tuesday 26 July 2005

We wish to exclude 'action at a distance', so that everything is intelligible locally, ie all the relevant actors are present. Quantum mechanics is the result of such local peace. It tells us that local peaceful states can be represented as eigenvectors of a system of vectors and matrices, all of which are represented by ordered sets of symbols (strings)

LOCAL PEACE : LOCAL CONSISTENCY

LOCAL (all the relevant actors are present.

'Our mystic symbols, exp( -i omega), exp ( -1 / lambda)

[page 181]

We decode these symbols using infinite series, beginning with a definition of e, e = (1 + 1/n) n, as n approaches infinity.

Binomial expansion gives us . . .

exp ( -i n) describes a normalized little rotating arrow spiralling through space and/or time (Feynman QED). Feynman

Like an science, linguistics has its particular set of social, political, ethical and moral questions.

The probability of survival is biassed by a variable called fitness, values of which may be passed from generation to generation.

Sharp (specific, designed) tools require a minimum of force to be effective, and minimize collateral damage. This serves as a definition (ordering of words) to be tested ultimately by observation.

States interfere, that is they excite and inhibit one another. This excitation and inhibition is represented by the addition of two dimensional vectors represented as complex numbers.

One may either surf on speculative shares (which may or may not be better than a lottery) or widen one's interest to portfolios of portfolios . . . with sufficient breadth to track the overall growth (or not growth) of the system. Our abstractions can be couched in statistical terms. Statistics is our means of renormalization so that we can gain a finite view of an (effectively) infinite

[page 184]

problem. General strategies can be worked out statistically, but when we get down to individual instances, there is a probability that some cases will be far from the norm.

We observe quantum mechanical phenomena each of which is unique in the space {spacetime, form}, where form is the nature of the beast (electron, cyclone, galaxy . . . ) and spacetime is the 'volume' in which it is to be found. This ;volume' depends on both the spatial dimensions and the lifetime of the creature (particle) in question.

To express itself at any point, the world must solve the eigenvalue equation for that point. A solution of the eigenvalue equation is a vector (sentence) which responds to an operator (matrix, translation) as thought he operation were simply a number rather than a complex two dimensional symbol.

[picture of a matrix]

Every state has a complexity which is measured by the number of basis states (ie the dimensionality of the relevant Hilbert space). A further level of complexity is contained in the (continuous) set of complex numbers which serve at the answer to the question <i | psi>, where psi is any state vector and I is the vector (sentence) of a basis state. In this language, we may say that natural semantic metalanguage is a search for a set of stationary basis states which describe vectors in the space of human communication.

[page 183]

Teilhard de Chardin: Milieu and the Way of the Cross. Teilhard de Chardin The origin of consciousness brought with is awareness that life is tough, from which some have concluded that anything good must be bought with suffering.

Certainly we may see that people will not move unless motivated. Motivation comes as both sticks and carrots, way to get out of the fire and into the frying pan and from thee to somewhere as a physiologically optimum temperature. Optimum temperature = optimum energy per particle = stationary state. Optimum temperature is a function of both energy and entropy - how much energy is distributed over how many states.

I am reliable because I know I am unreliable ad therefore have a comprehensive error correcting system in place witch basically consists in a complete system of texts and the discipline necessary to keep them effectively up to date, ie I have to be able to pay all my debts as they fall due.

We would like to derive all physical laws form the properties of networks alone.

Our archetypal function is ?

Exstasis = standing outside, as the word stands outside the letter. Unlike de Chardin, the letter is in no way belittled, abased or emptied by being part of the word.

[page 184]

Pain, endurance vs productivity, skill. There are moments at any level of productivity, when maximum effort is required, but their frequency varies as 1/productivity.

The communication model sits perfectly with special relativity, since thee is delay in any communication due to the processing involved. As Lonergan noticed, some encodings and decodings (insights) require a long time (a lot of searching and reasoning - we use reason to constrain our searches) whereas others are so quick as to convince some (like Lonergan) that they are an entirely distinct species of process from the slow varieties. Quantum mechanics tells us, however, that even speed differences between 10 E -23 seconds and ten billion ears do not change the basis nature of encoding (emission) and decoding (absorption) processes.

Non relativistic quantum mechanics deals with process that are essentially instantaneous, where propagation delay is of no importance. This is rather lie the world of writers communicating by letter and working at such a pace that postal delays are irrelevant. Communication delays become important when time is of the essence and introduce uncertainties where the relevant features are changing faster than they can be dealt with. This situation serves to decouple source and sink, so localizing communication and causality. So it was in the old days of sailing ships, when a letter from the Viceroy in India to the King in England took six months to arrive and the reply six months to return, leaving the Viceroy to wait for direction and the king a year for replies to his questions.

[page 185]

It pisses me off a bit that I write more easily when I am stoned. Drug addiction? An interesting physio-psychological phenomenon? But so it is and I am resigned to sacrificing my lungs somewhat for free expression as to sacrificing the smoothness of my hands to my need for income and physical activity leading to certain satisfaction in achievement while my literary ambitions remain unfulfilled, while yet appearing to move with undiminished speed toward a desirable goal.

To make a mathematical theory we need a space, objects in that space and interactions between these objects. Thus we have a space of natural numbers, 1, 2, 3, objects in that space (in this case, the individual numbers themselves, and he arithmetical operations by which two numbers may interact to give a third.

Quantum mechanics exists in function space. Consider a set of natural numbers, 1, 2, 3, etc which we write N = {1, 2, 3, . . . n . . . |}. Such a set has a natural order - given any two natural numbers, one familiar with the numbers knows which comes first in the counting order. Now we may permute such a set by changing its order, so that instead of 1, 2, 3, we place 3 in the first place (for instance) and 1 in the second and two in the third. Such a permutation may also be called a mapping of the set onto itself, or a function.

A function on the natural numbers, then, consists of a set of correspondences between the natural numbers in their natural order (sometimes called the abscissa, x axis) and some other ordering of the numbers (the ordinate, y axis)

[page 186]

ORDER = ORDINAL
MEASURE = CARDINAL.

We take all units to be the same size (no matter how large the cardinal or complex their ordinal rank) so that we can assign a cardinal number or size to an ordinal number. The complex numbers have a two ply order which can be abstracted to 1 ply by taking the absolute value. In quantum mechanics, these absolute values represent probabilities.

We draw an analogy between speech and the emission of a photon; listening and the absorption of a photon.

NATURAL RELIGION : Theocracy II The Network, = 'Democratic'
The Hierarchy = 'Fascist"

Path integral method in Quantum mechanics : all paths are equal, but what about the phase, Feynman

PHASE can be integrated (counted). 4-momentum describes the rate of change of phase, which is measured by action : omega = Energy/Planck's constant.

Language and religion: quantum mechanic and networks both contain the idea that hidden processes govern the observable phenomena, and this view seems common to most religions.

A Turing machine models a function, converting elements from its input domain to its output range.

Resonance = union

Wednesday 27 July 2005
Thursday 28 July 2005
Friday 29 July 2005
Saturday 30 July 2005

 

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Further reading

Books

Ahmed, Shahab, What is Islam? The Importance of Being Islamic, Princeton University Press 2016 'What Is Islam? formulates a new conceptual language for analyzing Islam. It presents a new paradigm of how Muslims have historically understood divine revelation--one that enables us to understand how and why Muslims through history have embraced values such as exploration, ambiguity, aestheticization, polyvalence, and relativism, as well as practices such as figural art, music, and even wine drinking as Islamic. It also puts forward a new understanding of the historical constitution of Islamic law and its relationship to philosophical ethics and political theory.' 
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Chaitin, Gregory J, Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, World Scientific 1987 Jacket: 'Algorithmic information theory is a branch of computational complexity theory concerned with the size of computer programs rather than with their running time. . . . The theory combines features of probability theory, information theory, statistical mechanics and thermodynamics, and recursive function or computability theory. ... [A] major application of algorithmic information theory has been the dramatic new light it throws on Goedel's famous incompleteness theorem and on the limitations of the axiomatic method. . . .' 
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Feynman, Richard P, and Albert P Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill 1965 Preface: 'The fundamental physical and mathematical concepts which underlie the path integral approach were first developed by R P Feynman in the course of his graduate studies at Princeton, ... . These early inquiries were involved with the problem of the infinite self-energy of the electron. In working on that problem, a "least action" principle was discovered [which] could deal successfully with the infinity arising in the application of classical electrodynamics.' As described in this book. Feynman, inspired by Dirac, went on the develop this insight into a fruitful source of solutions to many quantum mechanical problems.  
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Feynman (1988), Richard, QED: The Strange Story of Light and Matter, Princeton UP 1988 Jacket: 'Quantum electrodynamics - or QED for short - is the 'strange theory' that explains how light and electrons interact. Thanks to Richard Feynmann and his colleagues, it is also one of the rare parts of physics that is known for sure, a theory that has stood the test of time. . . . In this beautifully lucid set of lectures he provides a definitive introduction to QED.' 
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Lorentz, Konrad, Behind the Mirror: A Search for a Natural History of Human Knowledge, Mariner Books 1978 'Nobel laureate Lorenz here investigates culture as a living system. From amoebas to humans, he traces the physiological mechanisms that direct behavior and thought. Translated by Ronald Taylor; Index. A Helen and Kurt Wolff Book.' 
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Polanyi, Michael, and Amaryta Sen (foreword), The Tacit Dimension, University Of Chicago Press 1966, 2009 Amazon product description: '“I shall reconsider human knowledge by starting from the fact that we can know more than we can tell,” writes Michael Polanyi, whose work paved the way for the likes of Thomas Kuhn and Karl Popper. The Tacit Dimension argues that tacit knowledge—tradition, inherited practices, implied values, and prejudgments—is a crucial part of scientific knowledge. Back in print for a new generation of students and scholars, this volume challenges the assumption that skepticism, rather than established belief, lies at the heart of scientific discovery.' 
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Teilhard de Chardin, Pierre, The Future of Man (translated by Norman Denny) , Borgo Press 1994 Amazon product description: 'Pierre Teilhard De Chardin was one of the most distinguished thinkers and scientists of our time. He fits into no familiar category for he was at once a biologist and a paleontologist of world renown, and also a Jesuit priest. He applied his whole life, his tremendous intellect and his great spiritual faith to building a philosophy that would reconcile Christian theology with the scientific theory of evolution, to relate the facts of religious experience to those of natural science. The Phenomenon of Man, the first of his writings to appear in America, Pierre Teilhard's most important book and contains the quintessence of his thought. When published in France it was the best-selling nonfiction book of the year.' 
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von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.' 
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Papers

Chaitin, Gregory J, "Randomness and Mathematical Proof", Scientific American, 232, 5, May 1975, page 47-52. 'Although randomness can be precisely defined and can even be measured, a given number cannot be proved random. This enigma establishes a limit in what is possible in mathematics'. back

Links

Alan Blinder and Michael Wines, Federal Appeals Court Strikes Down North Carolina Voter ID provision, ' “In holding that the legislature did not enact the challenged provisions with discriminatory intent, the court seems to have missed the forest in carefully surveying the many trees,” the appeals court wrote. “This failure of perspective led the court to ignore critical facts bearing on legislative intent, including the inextricable link between race and politics in North Carolina.” ' back

Ambrose Evans-Pritchard, IMF admits disastrous love affairwith the Euro led to immolation of Greece, 'At its root was a failure to grasp the elemental point that currency unions with no treasury or political union to back them up are inherently vulnerable to debt crises. States facing a shock no longer have sovereign tools to defend themselves. Devaluation risk is switched into bankruptcy risk. back

Aquinas 45, Whether this is a good definition of eternity, "The simultaneously-whole and perfect possession of interminable life"., I answer that, As we attain to the knowledge of simple things by way of compound things, so must we reach to the knowledge of eternity by means of time, which is nothing but the numbering of movement by "before" and "after". For since succession occurs in every movement, and one part comes after another, the fact that we reckon before and after in movement, makes us apprehend time, which is nothing else but the measure of before and after in movement. Now in a thing bereft of movement, which is always the same, there is no before or after. As therefore the idea of time consists in the numbering of before and after in movement; so likewise in the apprehension of the uniformity of what is outside of movement, consists the idea of eternity. Further, those things are said to be measured by time which have a beginning and an end in time, because in everything which is moved there is a beginning, and there is an end. But as whatever is wholly immutable can have no succession, so it has no beginning, and no end. Thus eternity is known from two sources: first, because what is eternal is interminable--that is, has no beginning nor end (that is, no term either way); secondly, because eternity has no succession, being simultaneously whole. back

Ben Doherty and Patrick Kingsley, Ferrovial staff 'risk prosecution' for managing Australian detention camps, 'Spanish infrastructure corporation Ferrovial, which is owned by one of the world’s richest families and the major stakeholder in Heathrow airport, has been warned by professors at Stanford Law School that its directors and employees risk prosecution under international law for supplying services to Australia’s camps on Nauru and Manus Island in Papua New Guinea.' back

Ben Terris, Debbie Wasserman Schultz was the ultimate Washington survivor. Until she wasn't, ' “When you’ve lived through breast cancer, it helps put the rest of this into perspective,” says Rep. Cheri Bustos (D-Ill.), a friend who lives just a few blocks from Wasserman Schultz in Washington. Bustos was a journalist in a past life and says she knows how “hit pieces” come together. During one of those troubling times, she gave Wasserman Schultz a book called “The Four Agreements: A Practical Guide to Personal Freedom” and pointed her to a passage she thought might be helpful: “Agreement 2: Don’t Take Anything Personally. Nothing others do is because of you. What others say and do is a projection of their own reality.” ' back

Bismuth-209 - Wikipedia, Bismuth-209 - Wikipedia, the free encyclopedia, 'Bismuth-209 was long thought to have the heaviest stable nucleus of any element, but in 2003, Noël Coron and his colleagues at the Institut d’Astrophysique Spatiale in Orsay, France, discovered that 209Bi undergoes alpha decay with a half-life of approximately 600 yottaseconds (1.9×1019 years), over a billion times longer than the current estimated age of the universe. Theory had previously predicted a half-life of 4.6×1019 years.' back

Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then the measured result will be one of the eigenvalues λ of A, and the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'. back

Chris Buckley, Liberal Magazine, 'Forced Into a Corner' by China, Girds for Battle, 'For 25 years, the monthly magazine Yanhuang Chunqiu has survived the ire of censors, protected by moderate-minded retired officials who used it to call for limited political liberalization, market overhauls and an honest reckoning with dark periods in the Communist Party’s past. But the magazine’s survival is in doubt as the authorities try to install editors who will march more closely to the political tune of President Xi Jinping. That would end the magazine’s relative independence, and editors have said they would rather let it die than continue compromised.' back

Christopher Kremmer, Global journalism needs global ethics, 'Like all big ideas, there is a lot to unpack. The concept of global values is itself a hornet’s nest of post-colonial anxieties and discontents. But Ward sensibly builds his argument about the need to transcend parochialism in journalism on a solid philosophical and political foundation, as embodied in the existing, internationally binding Universal Declaration of Human Rights adopted by the United Nations in 1948. If journalism is to recalibrate its moral compass this might not be a bad place to start.' back

David Smith, Michelle Obama's stirirng speech brings Democratic convention to tears, 'Her voice cracking with emotion, she continued: “And because of Hillary Clinton, my daughters, and all our sons and daughters, now take for granted that a woman can be president of the United States.” ' back

Fergus Hunter, Previous youth abuse revelations didn't piqure my interest: Nigel Scullion, 'Indigenous Affairs Minister Nigel Scullion has conceded that previous revelations about the shocking conditions inside the Northern Territory corrections system failed to "pique [his] interest sufficiently" and that he wasn't fully aware of the treatment of incarcerated Indigenous youth until seeing the vision aired by ABC's Four Corners on Monday night.' back

Francis Fukuyama, American Poliotical Decay or Renewal: The Meaning of the 2016 Election, 'The real story of this election is that after several decades, American democracy is finally responding to the rise of inequality and the economic stagnation experienced by most of the population. Social class is now back at the heart of American politics, trumping other cleavages—race, ethnicity, gender, sexual orientation, geography—that had dominated discussion in recent elections.' back

Gideon Lewis-Kraus, Could Hilary Clinton Become the Champion the the 99 Percent, '“Rewriting the Rules” got funding from the Ford Foundation, whose decision last year to refocus around the issue of inequality was influenced by Roosevelt, and whose president, Darren Walker, effused to me about Wong as an “incandescent leader” for the progressive movement. While written by Stiglitz, the paper was worked out in consultation with labor officials, academics, congressional staff members and — unusually for a think tank — advocates from places like Color of Change, Naral and the Black Civic Engagement Fund.' back

Reductio ad absurdum - Wikipedia, Reductio ad absurdum - Wikipedia, the free encyclopedia, 'Reductio ad absurdum (Latin: "reduction to absurdity"), also known as argumentum ad absurdum (Latin: argument to absurdity), is a common form of argument which seeks to demonstrate that a statement is true by showing that a false, untenable, or absurd result follows from its denial,[1] or in turn to demonstrate that a statement is false by showing that a false, untenable, or absurd result follows from its acceptance.' back

John C. Baez and Mike Stay, Physics, Topology, Logic and Computation: A Rosette Stone, 'Abstract In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology. Namely, a linear operator behaves very much like a `cobordism': a manifold representing spacetime, going between two manifolds representing space. This led to a burst of work on topological quantum field theory and `quantum topology'. But this was just the beginning: similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of `closed symmetric monoidal category'. We assume no prior knowledge of category theory, proof theory or computer science.' back

Julia Gillard, First Woman to First Woman, back

Konrad Lorenz - Wikipedia, Konrad Lorenz - Wikipedia, the free encyclopedia, 'Konrad Zacharias Lorenz (7 November 1903 – 27 February 1989) was an Austrian zoologist, ethologist, and ornithologist. He shared the 1973 Nobel Prize in Physiology or Medicine with Nikolaas Tinbergen and Karl von Frisch. He is often regarded as one of the founders of modern ethology, developing an approach that began with an earlier generation, including his teacher Oskar Heinroth. Lorenz studied instinctive behavior in animals, especially in greylag geese and jackdaws. Working with geese, he investigated the principle of imprinting, the process by which some nidifugous birds (i.e. birds that leave their nest early), bond with the first moving object that they saw within the first hours of hatching.' back

Leah Riches and James Bennett, We can't see the war for the memorials: balancing education and commemmoration, 'Spurred on by a centenary commemoration budget now in excess of half a billion dollars, the heroic narrative has become an unshakeable article of faith in the national psyche. Herein lies the root of the conflict between the business of commemorating and education about war and society; an important distinction explored at the University of Newcastle’s public symposium. To quote a popular aphorism, the truth is always the first casualty in war. It requires determination and courage to tell the truth in the face of resolute forgetting.' back

Louise Milligan and Andy Burns, George Pell subject to Victoria Police investigation into multiple allegations of sexual abuse, back

Mark Beeson, Asia's ineffective diplomacy makes life difficult for Australia, 'The problem for the region generally and for Australia in particular is that there are simply no regional organisations with the capacity or the political will to actually deal with key concerns like the rise of China. Other institutions such as the ASEAN Regional Forum, which ought to be the organisation par excellence with which to deal with such issues, are equally impotent because they subscribe to the ASEAN Way.' back

Menno Schilthuizen, Evolution Is Happening Faster Than We Think, 'For a long time, biologists thought evolution was a very, very slow process, too tardy to be observed in a human lifetime. But recently, we have come to understand that evolution can happen very quickly, as long as natural selection — the relative benefit that a particular characteristic bestows on its bearer — is strong. And where else to find such strong natural selection than in the heart of a big city? The urban environment is about as extreme as it gets.' back

Neil Greenwood, A new twist on an old fairytale as a daughter dances away from the patriarchy, back

P versus NP problem - Wikipedia, P versus NP problem - Wikipedia, the free encyclopedia, 'The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified (technically, verified in polynomial time) can also be solved quickly (again, in polynomial time). The underlying issues were first discussed in the 1950s, in letters from John Forbes Nash Jr. to the National Security Agency, and from Kurt Gödel to John von Neumann. The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field.' back

Prophet - Wikipedia, Prophet - Wikipedia, the free encyclopedia, 'In religion, a prophet is an individual who has claimed to have been contacted by the supernatural or the divine, and to speak for them, serving as an intermediary with humanity, delivering this newfound knowledge from the supernatural entity to other people. . . . [In] Deuteronomy 18:18, . . . God said, "...and I will put My words in his mouth, and he shall speak unto them all that I shall command him." ' back

Rick Morton, 'Faiure to fit in' feeding African-linked crime surge, 'African youths causing a crime wave in Victoria are the product of eroded family units, trauma and an Australian society which does not sanction corporal punishment as a means of keeping them in line, claims a community leader. Gangs of teenagers and young adults have been contributing to a surge in crimes, particularly in Melbourne’s east where carjackings, burglaries and theft are on the rise. But one of the defining characteristics of the young ­offenders is boldness and apparent indifferent to consequences.' back

Riesz-Fischer theorem - Wikipedia, Riesz-Fischer theorem - Wikipedia, the free encyclopedia, 'In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.. back

Satellite navigation - Wikipedia, Satellite navigation - Wikipedia, the free encyclopedia, 'A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows small electronic receivers to determine their location (longitude, latitude, and altitude/elevation) to high precision (within a few metres) using time signals transmitted along a line of sight by radio from satellites. The system can be used for navigation or for tracking the position of something fitted with a receiver (satellite tracking). The signals also allow the electronic receiver to calculate the current local time to high precision, which allows time synchronisation.' back

Sequence - Wikipedia, Sequence - Wikipedia, the free encyclopedia, 'In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n).' back

Series - Wikipedia, Series - Wikipedia, the free encyclopedia, 'In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence has defined first and last terms, whereas a series continues indefinitely.' back

Tom Hamburger and Karen Tumulty, WikiLeaks releases thousands of documents about Clinton and internal deliberations, 'The emails detail how bitter the relationship became, as the senator emerged as a real threat Clinton — the Democratic establishment's pick — and refused to abandon his bid as it became clear she was going to win.' back

Tom Switzer, Donald Trump may be ignorant but he has his finger on the pulse, 'Still, as Fukuyama points out, American voters have risen up against what they see as "a corrupt, self-defeating Establishment, turning to radical outsiders in the hopes of a purifying cleanse". If the US political system does not address the kinds of problems that Trump has highlighted, anxiety will roil American politics for a long time to come.' back

Wojciech Hubert Zurek, Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical, 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3)) 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 framework for the “wavepacket collapse”, designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates.' back

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