Notes
Sunday 8 March 2020 - Saturday 14 March 2020
[Notebook: DB 84 Pam's Book]
Sunday 8 March 2020
[page 191]
Monday 9 March 2020
Jammer page 373: von Neumann: 'it is not possible to formulate the laws of quantum mechanics in a complete and consistent way without reference to human consciousness.' So how did the world work before we came along? Max Jammer: The Conceptual Development of Quantum Mechanics
page 380: Helmholtz: ' ". . . each quality or property of a thing is really nothing else but its capacity for exercising certain effects on other things. . . . it can never depend upon the action of one agent alone but exists only in relation to and dependent on the nature of some second object which is acted upon' [everything influences everything else, at least gravitationally; all that changes is the strength of the coupling constant].
page 381: 'the language of quantum mechanics is a language of interactions and not attributes; processes and not properties are the elements of its syntax.
EPR paradox "If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to the physical quantity." EPR Paradox - Wikipedia
No mention of entanglement or spooky action at a distance, but following Bell evidence for non-locality and evidence that simple non-relativistic quantum mechanics precedes the emergence of space and the emergence / evolution of the universe. Here we can let quantum mechanics rest for a while and turn to discrete mathematics which exists in effect outside of space-time in the world of disembodied logic [discrete, no space-time continuum required in principle, although it must be written out in space-time]. Bell state - Wikipedia
[page 192]
Tuesday 10 March 2020
Everything is inertial, modulated by mass and binding. So when I push something that is completely free it accelerates, but when I push against a wall nothing moves because I am coupled to the floor which is coupled to the wall which couplings exactly balance the force I am exerting on the wall.
In the 'spiritual' dimension I am pushing against the huge mass of falsehood and indifference embodied in traditional theologies and religions.
Gravitational potential shapes the universe, but where does it come from? Let us guess that the only possible source is consistency and the most basic constraints on consistency are to be found in inertial space, photons and charge. Maybe not, but one likely source seems to be the creation of energy and the need for three dimensional space for universal non interfering [error free] communication. This is in a sense contrary to quantum mechanics which relies on interference for the creation of structure from different frequencies of energy [3D space seems to be a means of excluding quantum mechanical behaviour by preventing interference]?? There is something here but it is far from clear. Why are electrical phenomena 1040 times stronger than gravitational phenomena, which is to ask why are electrical potentials so much stronger than gravitational [potentials] [and we could ask the same questions about the weak and strong potentials]. Something to do with the bifurcation of charge into positive and negative, whereas gravitation has only one species of charge, mass-energy [we might also see it as an algorithmic thing, electrodynamics using a much more powerful algorithm that gravitation, ie position significant vs unary numbers?]. Unary numeral system - Wikipedia
[page 193]
Maybe the gravitational potential has something to do with the spooky action at a distance, ie entanglement arising from the fact that the universe began from a one particle (singlet) state identical to the initial singularity.
A digital movie imitates the world to some extent by using cameras, microphones and other equipment to record it in the first place and then transform all the shots into something akin to the director's conception. Although natural dynamics has some role here it is often greatly modified but we can somehow imagine the underlying dynamic system of the universe producing the experiences we experience as we go through our day to day lives. Although most of quantum field theory is written in terms of continuous mathematics, all practical calculations, whether electronic or done with pencil and paper are eventually completed digitally after various algebraic processes [usually based on discrete logic] are invoked to get the fundamental ideas in computable form.
Wednesday 11 March 2020
My first few days doing a BSc at Adelaide university have more or less convinced me that I have very little to learn going this way, so I have half a mind to withdraw, which I can do by 16/3 without penalty. This may be partly motivated by my family situation. I miss my children and grandchildren very badly and feel sorry that it is all based on a false accusation. In a way it makes me feel more theological and Christlike, following the Christian [delusion] that Jesus won a great victory by being crucified for a crime he did not commit. A clear expression of the amazing flexibility of human thought.
[page 194]
What physicists, rather like film directors, novelists and creative artists in general are trying to do is construct model that yield the outputs we observe in our feeling about the universe [whether these feelings are formed by social situations, laboratory work of life in general]. Some of these feelings like [those induced by] the readings of measuring instrument may be clearly labelled objective, but others are mediated through the nature we have received by billions of years of evolution and though we accept that at least in their original context they have survival value, it is sometimes very hard to see how they help us in our current condition.
Our understanding of logical model is rooted in the logical connectives and and not from which we may construct large systems of software which couple specific inputs to specific outputs through instances of the deterministic form of computing machinery invented by Turing and his predecessors and successors. At the root of much of my thought, expressed in my thesis, is the idea that we can model the whole world with a set of deterministic machines loosely connected in a network structure which introduces uncertainty into the system because simple deterministic machines cannot control their own connection into random networks. I suppose I despair somewhat (at the moment) that I cannot take this idea further, but I must rely on the invincible potential expressed in the slogan "where there is a will there is a way" that lies at the heart of politics. From this point of view the breakthrough I seek is to explain the origin of potential. Clearly it has something, for living creatures, to do with sunlight and entropy, but how did it first come into the universe [and make the big bang go off]? This must be the research project that lies behind the websites creationrules.com, actionrules.com and something
[page 195]
to do with potential that I have yet to register, potentialcreates maybe, or lust4life, lust-4-life, the gospel according to god [the universe].
Thursday 12 March 2020
Hawking and Ellis: What I am looking for is an account of gravitational collapse in order to get a clue about gravitational expansion / complexification of the universe. Hawking & Ellis: The Large Scale Structure of Space-Time
page 1: '. . . we shall take the local physical laws that have been experimentally determined and we shall see what these laws imply about the large scale structure of the universe.
'In fact most of our results will be independent of the detailed nature of the physical laws but will involve certain general properties such as the description of space-time by a pseudo-Riemannian geometry and the positive definiteness of energy density.' It may be possible to describe the space-time manifold by considering it as a computer network: Prolegomena 3.4.3: Prolegomenon to scientific theology
'. . . gravity is by far the weakest [fundamental interaction] (the ratio Gm2/e2 of gravitational to electrical force between two electrons is about 10-40). Nevertheless it plays a dominant role in shaping the large scale structure of the universe.
page 2: [Because] Gravity . . . is always attractive. Thus the gravitational fields of all the particles in a body add up to produce a field which, for sufficiently large bodies, dominates over all other forces.
[page 197]
'Not only is gravity the dominant force on a large scale, it is the only force that affects every particle in the same way.'
'It has also been observed that light is deflected by gravitational fields, Since it is thought that no signals can travel faster than the velocity of light, this means that gravity determines the causal structure of the universe, ie it determines which events in space-time can be causally related to one another.' Of course the bulk of local casual influences [which are massive], like my body travel much more slowly than light.
' . . . if a sufficiently large amount of matter were concentrated in some region, it could deflect light going out from the region so much that it was in fact dragged back inwards. This was recognised in 1798 by Laplace . . .. ' Black hole - Wikipedia
Hawking & Ellis page 2: Penrose closed trapped surface relies on the distortion of space-time by mass: 'Since the matter inside a sufficiently massive body cannot travel faster than the velocity of light, it will be trapped in a region whose boundary decreases to zero in a finite time. . . . We shall show that in such a situation a space-time singularity must occur if certain reasonable conditions hold.' A geometric argument with or without real dynamic consequences.
page 3: Two situations: [one] collapse of a massive star. Other 'Recent observations of the microwave background indicate that the universe contains enough matter to cause a time-reversed closed trapped surface.' Why [and how] time reversed? Why doesn't the universe just stay a black hole? Much of this, it seems, just depends on Riemann and wishful
[page 198]
thinking, but it seems to match observation, at least on a cosmic scale.
The basic source of power seems to lie in numbers, all gravitational fields (or muscle fibres) pulling in the same direction.
The usual question, am I getting anywhere. Generating lots of questions and becoming more dissatisfied with the status quo but short of answers. One reason for my decision to withdraw from physics and biology courses it that it will be just another run through familiar territory and I am more interested than ever in trying to connect the physical and spiritual worlds in a unified picture by exploiting the potential of the logical network outlined in my honours thesis, something that seems to be a framework powerful [large] enough to connect the whole range of human experience from fundamental physics to love and war. Long ago [1987] this work started as a theory of peace and I have now set up a triplet of websites, lust-4-life to deal with potential and motivation, action rules to deal with the construction of physical and social capital and creationsaves to help navigate a course away from hell toward heaven. A theory of Peace (1987): Lecture 1: Mathematical Theology
The unitarity of quantum mechanics and the tensor calculus of relativistic mechanics maintained by continuous deterministic computational processes means that all these processes proceed at constant entropy and are therefore not creative. Hawking and Ellis find that gravitation inevitably merges geodesics thus reducing the complexity of gravitational dynamics toward
[page 199]
the initial singularity. This process operating in reverse may be seen as increasing the entropy of the universe by ultimately giving every stable particle its own geodesic, but this cannot be achieved by a continuous deterministic process but requires the establishment of critical points at which geodesics bifurcate creating new particles. In quantum mechanical terms this is equivalent to the breach of unitarity by observation. What I need to know is enough mathematics to encapsulate these ideas in a publishable theory.
Friday 13 March 2020
After much foostering around, partly motivated by bushfires, coronavirus, the collapse of the stock market and my failure to get a hearing in the Adelaide department of philosophy I have finally decided that my true vocation is mathematical theology, following the Manifesto I broadcast on 2BOB Radio in 1987. So I have to face up to a need for an intensive study of mathematics and will withdraw from first years physics and biology. So lust-4-life gains a new subtitle: the quest for mathematical theology and becomes to some extent as history of mathematics as the hypothesis generating wing of science. creationsaves then becomes the celebration of applied algorithms to improve our lot and actionrules goes to the tasks of implementing our mathematical discoveries. So I say lust-4-life a manifesto.
Saturday 14 March 2020
Hawking and Ellis page 4: 'In chapter 3 a formulation of the general theory of relativity is given in terms of three postulates
[page 200]
about a mathematical model of space-time. This model is a manifold M with a metric g of Lorentz signature. The physical significance of the metric is give by the first two postulates: those of local causality and of local conservation of energy-momentum. These postulates are common to both the General and Special Theories of Relativity and so are supported by the experimental evidence for the latter theory. The third postulate, the field equations for the metric g, is less well experimentally established. However most of our results will depend only on the property of the field equation that gravity is attractive for positive matter densities.'
One important task is to get 'behind' these assumptions to understand why they are the case, and the present reading of this book is fundamentally a search for clues to this quest. This, I hope, will reveal important insights into the relationship between gravitation and quantum theory which will serve as inputs into an explanation of the creation of the universe from the initial singularity that emerges from the three postulates listed. In other words, I am seeking a clear conceptual model of how the universe has grown. Much of modern cosmology accepts the 'big bang' as given and seeks to explain it in terms of the idea that where there is a sufficient local concentration of energy material particles will form which are subject to the other three physical interactions, strong, weak and electromagnetic
[page 201]
'In chapter 4 we discuss the significance of curvature by considering its effect on families of timelike and null geodesics. . . The curvature can be interpreted as differential tidal forces which induces relative acceleration between neighbouring geodesics. . . . One can show . . . that this leads to focal or conjugate points where neighbouring geodesics intersect . . ..'
page 6: 'These results are used in chapter 8 to establish the existence of singularities under certain conditions.'
'In chapter 5 we describe a number of exact solutions to Einstein's equations. These solutions are not realistic in that they all possess exact symmetries. . . .. In particular, the highly symmetrical cosmological models nearly all possess space-time singularities. . . . It will be one of our main objects to show [that such singularities also exist in more realistic models].'
'In chapter 6 we discuss the causal structure of space-time [as dictated by the finite velocity of light]. . .. The causal structure of spacetime can be used to define a boundary or edge to space-time.'
In chapter 8 we discuss the definition of space-time singularities. This presents certain difficulties because one cannot regard the singular point as being part of the space-time manifold.' Nevertheless it seems that the manifold emerges from the singularities, which is a central question for the creation of the universe. Continuous mathematics cannot cross this barrier, which suggests we turn instead to logical continuity.
page 8: 'In chapter 9 we show that singularities may occur near that stars more than 1½ times the mass of the Sun in the final stages of their evolution.'
[page 202]
Hawking & Ellis page 11: 'A manifold is essentially a space which is locally similar to Euclidean space in that it can be covered with coordinate patches [which may overlap]. This permits differentiation to be defined but does not distinguish intrinsically between different coordinate systems. Thus the only concepts defined by a manifold are those that are independent of the choice of coordinate system.'
Differentiation = communication. Coordinate patch = source; differential connection between sources is the communication link which is independent of the coordinate systems but dependent on the codecs operating between sources.
Locally Lipschitz: Lipschitz continuity - Wikipedia
page 13: Hausdorf: Hausdorff space - Wikipedia
page 14: paracompact Paracompact space - Wikipedia
'All manifolds considered will be paracompact C∞ Hausdorf manifolds without boundary.'
What I really need to do is bring the same confidence, competence and certainty that have marked my fifty years as a physical builder to my late life career as an intellectual builder, and the backbone to this new structure is mathematics, so well deployed in classical physics and engineering, but a bit dodgy when we come to the insistence among quantum mechanics and relativists that continuous mathematics is real, trustworthy and the key to nature.
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Further readingBooks
Feynman, Richard P, and Robert B Leighton et al, The Feynman Lectures on Physics (volume 1) : Mainly Mechanics, Radiation and Heat, Addison Wesley 2011 Foreword: 'This book is based on a course of lectures in introductory physics given by Prof. R P Feynman at the California Institute of Technology during the academic year 1961-62. ... The lectures constitute a major part of a fundamental revision of the introductory course, carried out over a four year period. ... The need for a basic revision arose both from the rapid development of physics in recent decades and from the fact that entering freshmen have shown a stewady incrase in mathematical ability as a result of improvements in high school mathematical course content.'
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Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time, Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.'
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Jammer, Max, The Conceptual Development of Quantum Mechanics, McGraw Hill 1966 ' This book is a critical analysis, based on a broad physical, historical, and philosophical study, of how empirical phenomena led to the renunciation of classical physics and how experimental research -- combined with mathematical thought and philosophical speculation -- opened entirely novel perspective. It offers a thorough, detailed exposition of the various phases of the development of quantum theory, analyzes the logical dependence of each stage on those preceding it, and leads the reader from the very beginning of this intellectual process to the front line positions of current foundational research in physics. "... the only full history of the Quantum Theory." -- Dr. F. Hund, Institute for Theoretical Physics, Gottingen University.'
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Lonergan, Bernard J F, Insight: A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '. . . Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding'
Amazon
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Wigner, Eugene, Symmetries and Reflections: Scientific Essays , MIT Press 1970 Jacket: 'This volume contains some of Professor Wigner's more popular papers which, in their diversity of subject and clarity of style, reflect the author's deep analytical powers and the remarkable scope of his interests. Included are articles on the nature of physical symmetry, invariance and conservation principles, the structure of solid bodies and of the compound nucleus, the theory of nuclear fission, the effects of radiation on solids, and the epistemological problems of quantum mechanics. Other articles deal with the story of the first man-made nuclear chain reaction, the long term prospects of nuclear energy, the problems of Big Science, and the role of mathematics in the natural sciences. In addition, the book contains statements of Wigner's convictions and beliefs as well as memoirs of his friends Enrico Fermi and John von Neumann.
Eugene P. Wigner is one of the architects of the atomic age. He worked with Enrco Fermi at the Metallurgical Laboratory of the University of Chicago at the beginning of the Manhattan Project, and he has gone on to receive the highest honours that science and his country can bestow, including the Nobel Prize for physics, the Max Planck Medal, the Enrico Fermi Award and the Atoms for Peace Award. '.
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Papers
Steane, Andrew M, "Efficient Fault-tolerant quantum computing", Nature, 399, 6732, 13 May 1999, page 124-126. 'Quantum computing—the processing of information according to the fundamental laws of physics—offers a means to solve efficiently a small but significant set of classically intractable problems. Quantum computers are based on the controlled manipulation of entangled quantum states, which are extremely sensitive to noise and imprecision; active correction of errors must therefore be implemented without causing loss of coherence. Quantum error-correction theory has made great progress in this regard, by predicting error-correcting 'codeword' quantum states. But the coding is inefficient and requires many quantum bits, which results in physically unwieldy fault-tolerant quantum circuits. Here I report a general technique for circumventing the trade-off between the achieved noise tolerance and the scale-up in computer size that is required to realize the error correction. I adapt the recovery operation (the process by which noise is suppressed through error detection and correction) to simultaneously correct errors and perform a useful measurement that drives the computation. The result is that a quantum computer need be only an order of magnitude larger than the logic device contained within it. For example, the physical scale-up factor required to factorize a thousand-digit number is reduced from 1,500 to 22, while preserving the original tolerated gate error rate (10 -5) and memory noise per bit (10-7). The difficulty of realizing a useful quantum computer is therefore significantly reduced.'. 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 . 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 &mdash the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them &mdash into becoming a witness.'. back |
Links
Alexis Morgan & Jai McAllister, Patyegarang was Australia's first teacher of Aboriginal language, colonisation-era notebooks show, ' Against the backdrop of early colonisation, on Sydney Harbour's shoreline, an extraordinary exchange took place between a young Aboriginal woman and a First Fleet Lieutenant.
. . .
Patyegarang, a young Gamaraigal woman who spoke the Gadigal language, would prove crucial to the survival of her Sydney-based native tongue.
William Dawes, an English Lieutenant and astronomer, recorded the pair's conversations, in what remains today the only known first-hand accounts of the Gadigal language.' back |
Bell state - Wikipedia, Bell state - Wikipedia, the free encyclopedia, 'The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. . . .
A Bell state is defined as a maximally entangled quantum state of two qubits. The qubits are usually thought to be spatially separated (held by Alice and Bob, respectively, to use quantum cryptography terms). Nevertheless they exhibit perfect correlations which cannot be explained without quantum mechanics.' back |
Black hole - Wikipedia, Black hole - Wikipedia, the free encyclopedia, ' Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace.[12] The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; it was during the 1960s that theoretical work showed they were a generic prediction of general relativity.' back |
Chris Impey, How technology can combat the rising tide of fake science, ' At the University of Arizona, we have trained neural nets on handpicked popular articles about climate change and biological evolution, and the neural nets are 90% successful in distinguishing wheat from chaff. With a quick scan of a site, our neural net can tell if its content is scientifically sound or climate-denial junk. After more refinement and testing we hope to have neural nets that can work across all domains of science.' back |
David Wethe & Chris Martin, The Greening of a City Built on Oil, '. . . Houston has been purposefully going green for more than a decade. The municipal government gets 92% of all the power it uses in the buildings it owns from wind and solar and has been on the U.S. Environmental Protection Agency’s list of the largest renewable power users since 2016. . . . The big objective: carbon neutrality by 2050. . . . A big catalyst of the renewable advance was the wind. Texas, which spent more than $7 billion building out a wind-power transmission system, is by far the biggest producer of it, with the ability to produce as much electricity as 30 nuclear reactors. That’s more than double the size of the next biggest market, in Iowa. Wind turbines supplied 20% of the state’s electricity last year, surpassing coal in January.' back |
EPR Paradox - Wikipedia, EPR Paradox - Wikipedia, the free encyclopedia, 'In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. "EPR" stands for Einstein, Podolsky, and Rosen, who introduced the thought experiment in a 1935 paper to argue that quantum mechanics is not a complete physical theory.' back |
Hausdorff space - Wikipedia, Hausdorff space - Wikipedia, the free encyclopedia, ' In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. .' back |
Joan of Arc - Wikipedia, Joan of Arc - Wikipedia, the free encyclopedia, 'Saint Joan of Arc (French: Jeanne d'Arc[ c. 1412[ – 30 May 1431) is a national heroine of France and a Catholic saint. A peasant girl born in eastern France, she led the French army to several important victories during the Hundred Years' War, claiming divine guidance, and was indirectly responsible for the coronation of Charles VII. She was captured by the English, tried by an ecclesiastical court, and burned at the stake when she was nineteen years old. Twenty-four years later, the Holy See reviewed the decision of the ecclesiastical court, found her innocent, and declared her a martyr. She was beatified in 1909 and later canonized in 1920.' back |
Job, Job, from the holy Bible, King James Version, 'About the electronic version
Job, from The holy Bible, King James version
Bible, King James
Creation of machine-readable version: Kraft, Robert A.
Conversion to TEI.2-conformant markup: ca. 125 kilobytes Oxford Text Archive Oxford University Computing Service, 13 Banbury Road, Oxford, OX2 6NN, UK
This version available from the University of Virginia Library
Charlottesville, Va.
Available from: Oxford Text Archive' back |
John Grace, Sunak's wholesale spaffing sends Tories into rapture, back |
Joseph Stalin - Wikipedia, Joseph Stalin - Wikipedia, the free encyclopedia, 'Joseph Stalin (born Ioseb Besarionis dze Jughashvili – 18 December 1878[1] – 5 March 1953) was the General Secretary of the Communist Party of the Soviet Union's Central Committee from 1922 until his death in 1953. In the years following Lenin's death in 1924, he rose to become the leader of the Soviet Union.' back |
Landauer, Irreversibility and Heat Generation in the Computing Process, 'Abstract: 'It is argued that computing machines inevitably involve devices which perform logical functions that do not have a single-valued inverse. The logical irreversibility is associated with physical irreversibility, and requires a minimum heat generation, per machine cycle, typically of the order of kT for each irreversible function. The dissipation serves the purpose of standardizing signals and making them independent of their exact logical history. Two simple, but representative, models of bistable devices are subjected to a more detailed analysis of switching kinetics to yield the relationship between speed and energy dissipation, and to estimate the effects of errors induced by thermal fluctuations.' back |
Le Corbusier - Wikipedia, Le Corbusier - Wikipedia, the free encyclopedia, 'Charles-Édouard Jeanneret-Gris, who chose to be known as Le Corbusier (October 6, 1887 – August 27, 1965), was a Swiss-French architect, designer, urbanist, writer and also painter, who is famous for being one of the pioneers of what now is called Modern architecture or the International Style. He was born in Switzerland and became a French citizen in his 30s.
He was a pioneer in studies of modern high design and was dedicated to providing better living conditions for the residents of crowded cities. His career spanned five decades, with his buildings constructed throughout central Europe, India, Russia, and one each in North and South America. He was also an urban planner, painter, sculptor, writer, and modern furniture designer.' back |
Lipschitz continuity - Wikipedia, Lipschitz continuity - Wikipedia, the free encyclopedia, ' In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (or modulus of uniform continuity). For instance, every function that has bounded first derivatives is Lipschitz.' back |
Mark, Gospel of Mark, back |
Matthew, Gospel of Matthew, back |
Mohammed Hanif, 'How Do I Know Where Your Socks Are, ' Yet look into the inbox of any woman in public life, and you will see a lot of unsolicited pictures of men’s genitalia or references to it. And for a young woman to hold up a placard saying to stop doing that is seen as a threat to the national moral order.
What these placards have achieved is remarkable. They have taken the discussion about women’s rights from Parliament, the courts and the workplace straight into the kitchen and the bedroom. They have challenged the most sacred institution of all: family. More than 1,000 women are murdered in the name of honor in Pakistan every year, the overwhelming majority of them by relatives.' back |
Paracompact space - Wikipedia, Paracompact space - Wikipedia, the free encyclopedia, ' In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.' back |
Peter Kirby, Early Christian Writings: New testament, Apocrypha, Gnostics, Church Fathers, 'Early Christian Writings is the most complete collection of documents from the first two centuries with translations and commentary. Includes the New Testament, Apocrypha, Gnostics, and Church Fathers. The "Early Christian Writings: New Testament, Apocrypha, Gnostics, Church Fathers" site is copyright © 2001-2006 Peter Kirby. Permission is given to link to any HTML file on the Early Christian Writings site.' back |
Pigeonhole sort - Wikipedia, Pigeonhole sort - Wikipedia, the free encyclopedia, ' Pigeonhole sorting is a sorting algorithm that is suitable for sorting lists of elements where the number of elements (n) and the length of the range of possible key values (N) are approximately the same. It requires O(n + N) time.' back |
Unary numeral system - Wikipedia, Unary numeral system - Wikipedia, the free encyclopedia, ' The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol representing 1 is repeated N times.
For example, in this system the number 0 would be represented as the empty string and the numbers 1, 2, 3, 4, 5, 6... would be represented as:
1, 11, 111, 1111, 11111, 111111...' back |
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