1.
Quantum mechanics
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Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms, in one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. This phase is known as old quantum theory, according to Planck, each energy element is proportional to its frequency, E = h ν, where h is Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, lower energy/frequency means increased time and vice versa, photons of differing frequencies all deliver the same amount of action, but do so in varying time intervals. High frequency waves are damaging to human tissue because they deliver their action packets concentrated in time, the Copenhagen interpretation of Niels Bohr became widely accepted. In the mid-1920s, developments in mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons
2.
Nick Herbert (physicist)
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Nick Herbert is an American physicist and author, best known for his book Quantum Reality. Herbert studied Engineering Physics at the Ohio State University, graduating in 1959 and he received a Ph. D. in physics from Stanford University in 1967 for work on nuclear scattering experiments. After a one-year teaching job at Monmouth College in Illinois, Herbert held a number of posts in industry. He was also senior physicist at Smith-Corona Marchant Corporation in Palo Alto, California where he developed a new theory of xerographic process and worked on early developments in ink jet printing. While employed in industry, Herbert was part of the Fundamental Fysiks Group at Lawrence Berkeley National Laboratory, founded in May 1975 by Elizabeth Rauscher and George Weissmann. During the 1970s and 1980s, Herbert and Saul-Paul Sirag organized a yearly Esalen Seminar on the Nature of Reality, with Richard Shoup of Xerox PARC, Herbert constructed a Metaphase Typewriter, a quantum operated device whose purpose was to communicate with disembodied spirits. Despite many tests, including an attempt to contact the spirit of Harry Houdini on the anniversary of his birth. Herbert supports a holistic interpretation of quantum physics and he has argued for quantum animism in which mind permeates the world at every level. Werner Krieglstein wrote regarding his quantum animism, In 1981, Herbert proposed FLASH, of this proposal, quantum computing pioneer Asher Peres wrote, I was the referee who approved the publication of Nick Herbert’s FLASH paper, knowing perfectly well that it was wrong. I explain why my decision was the one, and I briefly review the progress to which it led. Chief among the results that Peres claimed stemmed from a refutation of Herberts proposal was the theorem, proved by Wootters, Zurek. Quantum Reality, Beyond the New Physics, faster Than Light, Superluminal Loopholes in Physics. Elemental Mind, Human Consciousness & the New Physics
3.
Doubleday (publisher)
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Doubleday is an American publishing company founded as Doubleday & McClure Company in 1897 that by 1947 was the largest in the United States. It published the work of mostly U. S. authors under a number of imprints, in 2009 Doubleday was merged with Knopf Publishing Group to form the Knopf Doubleday Publishing Group. The firm was founded as Doubleday & McClure Company in 1897 by Frank Nelson Doubleday, one of their first bestsellers was The Days Work by Rudyard Kipling. Other authors published by the company in its early years include W. Somerset Maugham, theodore Roosevelt, Jr. later served as a vice-president of the company. In 1900, the company became Doubleday, Page & Company when Walter Hines Page joined as a new partner, in 1922, the founders son, Nelson Doubleday, joined the firm. In 1910, Doubleday, Page, and Co. moved its operations, the Doubleday company purchased much of the land on the east side of Franklin Avenue, and estate homes were built for many of its executives on Fourth Street. In 1916, company co-founder and Garden City resident Walter Hines Page was named Ambassador to Great Britain, in 1927, Doubleday merged with the George H. Doran Company, creating Doubleday, Doran, then the largest publishing business in the English-speaking world. In 1946, the company became Doubleday and Company, Nelson Doubleday, Jr. resigned as president, but continued as chairman of the board until his death on January 11,1949. Douglas Black took over and was president from 1946 to 1963, by 1947, Doubleday was the largest publisher in the US, with annual sales of over 30 million books. In 1980, the company bought the New York Mets baseball team and it defeated the Boston Red Sox to win the World Series in 1986 in a classic 7-game contest. The company had offices in London and Paris and wholly owned subsidiaries in Canada, Australia, and New Zealand, with joint ventures in the UK and the Netherlands. Doubleday sold the company to Bertelsmann in 1986, and teamed up with minority owner Fred Wilpon to buy the Mets in his own name. In 1988, portions of the firm part of the Bantam Doubleday Dell Publishing Group. In late 2008 and early 2009, the Doubleday imprint was merged with Knopf Publishing Group to form the Knopf Doubleday Publishing Group, image Books, Catholic Books—still a Doubleday unit as part of Doubleday Religious Publishing Nan A. Talese/Doubleday, a literary imprint established in 1990. Talese, the publisher and editorial director, is a senior vice president of Doubleday
4.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
5.
Dewey Decimal Classification
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The Dewey Decimal Classification, or Dewey Decimal System, is a proprietary library classification system first published in the United States by Melvil Dewey in 1876. It has been revised and expanded through 23 major editions, the latest issued in 2011 and it is also available in an abridged version suitable for smaller libraries. It is currently maintained by the Online Computer Library Center, a cooperative that serves libraries. OCLC licenses access to a version for catalogers called WebDewey. The Decimal Classification introduced the concepts of relative location and relative index which allow new books to be added to a library in their location based on subject. Libraries previously had given books permanent shelf locations that were related to the order of acquisition rather than topic, the classifications notation makes use of three-digit Arabic numerals for main classes, with fractional decimals allowing expansion for further detail. Using Arabic numerals for symbols, it is flexible to the degree that numbers can be expanded in linear fashion to cover aspects of general subjects. A library assigns a number that unambiguously locates a particular volume in a position relative to other books in the library. The number makes it possible to find any book and to return it to its place on the library shelves. The classification system is used in 200,000 libraries in at least 135 countries, the major competing classification system to the Dewey Decimal system is the Library of Congress Classification system created by the U. S. Melvil Dewey was an American librarian and self-declared reformer and he was a founding member of the American Library Association and can be credited with the promotion of card systems in libraries and business. He developed the ideas for his classification system in 1873 while working at Amherst College library. He applied the classification to the books in library, until in 1876 he had a first version of the classification. In 1876, he published the classification in pamphlet form with the title A Classification and Subject Index for Cataloguing and Arranging the Books and he used the pamphlet, published in more than one version during the year, to solicit comments from other librarians. It is not known who received copies or how many commented as only one copy with comments has survived, in March 1876, he applied for, and received copyright on the first edition of the index. The edition was 44 pages in length, with 2,000 index entries, comprised 314 pages, with 10,000 index entries. Editions 3–14, published between 1888 and 1942, used a variant of this same title, Dewey modified and expanded his system considerably for the second edition. In an introduction to that edition Dewey states that nearly 100 persons hav contributed criticisms, one of the innovations of the Dewey Decimal system was that of positioning books on the shelves in relation to other books on similar topics
6.
Library of Congress Classification
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The Library of Congress Classification is a system of library classification developed by the Library of Congress. It is used by most research and academic libraries in the U. S. the Classification is also distinct from Library of Congress Subject Headings, the system of labels such as Boarding schools and Boarding schools—Fiction that describe contents systematically. The classification was invented by Herbert Putnam in 1897, just before he assumed the librarianship of Congress, with advice from Charles Ammi Cutter, it was influenced by his Cutter Expansive Classification, the Dewey Decimal System, and the Putnam Classification System. It was designed specifically for the purposes and collection of the Library of Congress to replace the fixed location system developed by Thomas Jefferson, by the time Putnam departed from his post in 1939, all the classes except K and parts of B were well developed. LCC has been criticized for lacking a theoretical basis, many of the classification decisions were driven by the practical needs of that library rather than epistemological considerations. Although it divides subjects into broad categories, it is essentially enumerative in nature and that is, it provides a guide to the books actually in one librarys collections, not a classification of the world. In 2007 the Wall Street Journal reported that in the countries it surveyed most public libraries, the National Library of Medicine classification system uses the initial letters W and QS–QZ, which are not used by LCC. Some libraries use NLM in conjunction with LCC, eschewing LCCs R for Medicine, others use LCCs QP–QR schedules and include Medicine R. Subclass AC – Collections. Collected works Subclass AE – Encyclopedias Subclass AG – Dictionaries and other reference works Subclass AI – Indexes Subclass AM – Museums. Collectors and collecting Subclass AN – Newspapers Subclass AP – Periodicals Subclass AS – Academies, directories Subclass AZ – History of scholarship and learning. The humanities Subclass B – Philosophy Subclass BC – Logic Subclass BD – Speculative philosophy Subclass BF – Psychology Subclass BH – Aesthetics Subclass BJ – Ethics Subclass BL – Religions, rationalism Subclass BM – Judaism Subclass BP – Islam. Seals Subclass CE – Technical Chronology, calendar Subclass CJ – Numismatics Subclass CN – Inscriptions. Former Soviet Republics – Poland Subclass DL – Northern Europe, maps Subclass GA – Mathematical geography. Cartography Subclass GB – Physical geography Subclass GC – Oceanography Subclass GE – Environmental Sciences Subclass GF – Human ecology, anthropogeography Subclass GN – Anthropology Subclass GR – Folklore Subclass GT – Manners and customs Subclass GV – Recreation. Leisure Subclass H – Social sciences Subclass HA – Statistics Subclass HB – Economic theory, demography Subclass HC – Economic history and conditions Subclass HD – Industries. Labor Subclass HE – Transportation and communications Subclass HF – Commerce Subclass HG – Finance Subclass HJ – Public finance Subclass HM – Sociology Subclass HN – Social history, Social reform Subclass HQ – The family. Marriage, Women and Sexuality Subclass HS – Societies, secret, benevolent, races Subclass HV – Social pathology. Municipal government Subclass JV – Colonies and colonization, International migration Subclass JX – International law, see JZ and KZ Subclass JZ – International relations Subclass K – Law in general
7.
Ontology
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Ontology is the philosophical study of the nature of being, becoming, existence or reality as well as the basic categories of being and their relations. Although ontology as an enterprise is highly hypothetical, it also has practical application in information science and technology. Some philosophers, notably of the Platonic school, contend that all refer to existent entities. Other philosophers contend that nouns do not always name entities, between these poles of realism and nominalism, stand a variety of other positions. An ontology may give an account of which refer to entities, which do not, why. Principal questions of ontology include, What can be said to exist, into what categories, if any, can we sort existing things. What are the meanings of being, what are the various modes of being of entities. Various philosophers have provided different answers to these questions, one common approach involves dividing the extant subjects and predicates into groups called categories. Such an understanding of ontological categories, however, is merely taxonomic, what does it mean for a being to be. Is existence a genus or general class that is divided up by specific differences. Which entities, if any, are fundamental, how do the properties of an object relate to the object itself. What features are the essential, as opposed to merely accidental attributes of a given object, how many levels of existence or ontological levels are there. Can one give an account of what it means to say that an object exists. Can one give an account of what it means to say that an entity exists. What constitutes the identity of an object, when does an object go out of existence, as opposed to merely changing. Do beings exist other than in the modes of objectivity and subjectivity, i. e. is the subject/object split of modern philosophy inevitable. e. Being, that which is, which is the present participle of the verb εἰμί, eimí, i. e. to be, I am, and -λογία, -logia, i. e. logical discourse. The first occurrence in English of ontology as recorded by the OED came in a work by Gideon Harvey, Archelogia philosophica nova, or, New principles of Philosophy
8.
Interpretations of quantum mechanics
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An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing and this question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. The definition of quantum theorists terms, such as functions and matrix mechanics. Although the Copenhagen interpretation was originally most popular, quantum decoherence has gained popularity, thus the many-worlds interpretation has been gaining acceptance. The authors reference a similarly informal poll carried out by Max Tegmark at the Fundamental Problems in Quantum Theory conference in August 1997, in Tegmarks poll, the Everett interpretation received 17% of the vote, which is similar to the number of votes in our poll. A general law is a regularity of outcomes, whereas a causal mechanism may regulate the outcomes, a phenomenon can receive interpretation either ontic or epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception, in a broad sense, scientific theory can be viewed as offering scientific realism—approximately true description or explanation of the natural world—or might be perceived with antirealism. A realist stance seeks the epistemic and the ontic, whereas an antirealist stance seeks epistemic, in the 20th centurys first half, antirealism was mainly logical positivism, which sought to exclude unobservable aspects of reality from scientific theory. The instrumentalist view is carried by the quote of David Mermin, Shut up and calculate. Other approaches to conceptual problems introduce new mathematical formalism. In classical field theory, a property at a given location in the field is readily derived. In Heisenbergs formalism, on the hand, to derive physical information about a location in the field, one must apply a quantum operation to a quantum state. Schrödingers formalism describes a waveform governing probability of outcomes across a field, yet how do we find in a specific location a particle whose wavefunction of mere probability distribution of existence spans a vast region of space. The act of measurement can interact with the state in peculiar ways. Yet quantum decoherence grants that all the possibilities can be real, Quantum entanglement, as illustrated in the EPR paradox, seemingly violates principles of local causality. Complementarity holds that no set of physical concepts can simultaneously refer to all properties of a quantum system. For instance, wave description A and particulate description B can each describe quantum system S, as now well known, the origin of complementarity lies in the non-commutativity of operators that describe quantum objects. Since the intricacy of a system is exponential, it is difficult to derive classical approximations
9.
Ultraviolet catastrophe
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The term ultraviolet catastrophe was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 derivation of the Rayleigh–Jeans law. Since the first appearance of the term, it has also used for other predictions of a similar nature, as in quantum electrodynamics. An example, from Masons A History of the Sciences, illustrates multi-mode vibration via a piece of string, as a natural vibrator, the string will oscillate with specific modes, dependent on the length of the string. In classical physics, a radiator of energy will act as a natural vibrator, and, since each mode will have the same energy, most of the energy in a natural vibrator will be in the smaller wavelengths and higher frequencies, where most of the modes are. According to classical electromagnetism, the number of modes in a 3-dimensional cavity. This therefore implies that the power per unit frequency should follow the Rayleigh–Jeans law. Planck derived the form for the intensity spectral distribution function by making some strange assumptions. In particular, Planck assumed that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta, of energy, E quanta = h ν = h c λ, where h is Plancks constant. Plancks assumptions led to the form of the spectral distribution functions. Albert Einstein solved the problem by postulating that Plancks quanta were real physical particles—what we now call photons and he modified statistical mechanics in the style of Boltzmann to an ensemble of photons. Einsteins photon had a proportional to its frequency and also explained an unpublished law of Stokes. Many popular histories of physics, as well as a number of physics textbooks, in that version, the catastrophe was first noticed by Planck, who developed his formula in response. That Plancks proposal happened to provide a solution for it was realized only later, wien approximation Vacuum catastrophe Kroemer, Herbert, Kittel, Charles. Cohen-Tannoudji, Claude, Diu, Bernard, Laloë, Franck
10.
Werner Heisenberg
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Werner Karl Heisenberg was a German theoretical physicist and one of the key pioneers of quantum mechanics. He published his work in 1925 in a breakthrough paper, in the subsequent series of papers with Max Born and Pascual Jordan, during the same year, this matrix formulation of quantum mechanics was substantially elaborated. In 1927 he published his uncertainty principle, upon which he built his philosophy, Heisenberg was awarded the Nobel Prize in Physics for 1932 for the creation of quantum mechanics. He was a principal scientist in the Nazi German nuclear weapon project during World War II and he travelled to occupied Copenhagen where he met and discussed the German project with Niels Bohr. Following World War II, he was appointed director of the Kaiser Wilhelm Institute for Physics and he was director of the institute until it was moved to Munich in 1958, when it was expanded and renamed the Max Planck Institute for Physics and Astrophysics. He studied physics and mathematics from 1920 to 1923 at the Ludwig-Maximilians-Universität München, at Munich, he studied under Arnold Sommerfeld and Wilhelm Wien. At Göttingen, he studied physics with Max Born and James Franck and he received his doctorate in 1923, at Munich under Sommerfeld. He completed his Habilitation in 1924, at Göttingen under Born, at the event, Bohr was a guest lecturer and gave a series of comprehensive lectures on quantum atomic physics. There, Heisenberg met Bohr for the first time, and it had a significant, Heisenbergs doctoral thesis, the topic of which was suggested by Sommerfeld, was on turbulence, the thesis discussed both the stability of laminar flow and the nature of turbulent flow. The problem of stability was investigated by the use of the Orr–Sommerfeld equation and he briefly returned to this topic after World War II. Heisenbergs paper on the anomalous Zeeman effect was accepted as his Habilitationsschrift under Max Born at Göttingen, in his youth he was a member and Scoutleader of the Neupfadfinder, a German Scout association and part of the German Youth Movement. In August 1923 Robert Honsell and Heisenberg organized a trip to Finland with a Scout group of this association from Munich, Heisenberg arrived at Munich in 1919 as a member of Freikorps to fight the Bavarian Soviet Republic established a year earlier. Five decades later he recalled those days as youthful fun, like playing cops and robbers and so on, from 1924 to 1927, Heisenberg was a Privatdozent at Göttingen. His seminal paper, Über quantentheoretischer Umdeutung was published in September 1925 and he returned to Göttingen and with Max Born and Pascual Jordan, over a period of about six months, developed the matrix mechanics formulation of quantum mechanics. On 1 May 1926, Heisenberg began his appointment as a university lecturer and it was in Copenhagen, in 1927, that Heisenberg developed his uncertainty principle, while working on the mathematical foundations of quantum mechanics. On 23 February, Heisenberg wrote a letter to fellow physicist Wolfgang Pauli, in his paper on the uncertainty principle, Heisenberg used the word Ungenauigkeit. In 1927, Heisenberg was appointed ordentlicher Professor of theoretical physics and head of the department of physics at the Universität Leipzig, in his first paper published from Leipzig, Heisenberg used the Pauli exclusion principle to solve the mystery of ferromagnetism. Slater, Edward Teller, John Hasbrouck van Vleck, Victor Frederick Weisskopf, Carl Friedrich von Weizsäcker, Gregor Wentzel, in early 1929, Heisenberg and Pauli submitted the first of two papers laying the foundation for relativistic quantum field theory
11.
Paul Dirac
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Paul Adrien Maurice Dirac OM FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions, Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger for the discovery of new productive forms of atomic theory. He also made significant contributions to the reconciliation of general relativity with quantum mechanics and he was regarded by his friends and colleagues as unusual in character. Albert Einstein said of him This balancing on the path between genius and madness is awful. He is regarded as one of the most significant physicists of the 20th century, Paul Adrien Maurice Dirac was born at his parents home in Bristol, England, on 8 August 1902, and grew up in the Bishopston area of the city. His father, Charles Adrien Ladislas Dirac, was an immigrant from Saint-Maurice, Switzerland and his mother, Florence Hannah Dirac, née Holten, the daughter of a ships captain, was born in Cornwall, England, and worked as a librarian at the Bristol Central Library. Paul had a sister, Béatrice Isabelle Marguerite, known as Betty, and an older brother, Reginald Charles Félix, known as Felix. Dirac later recalled, My parents were terribly distressed, I didnt know they cared so much I never knew that parents were supposed to care for their children, but from then on I knew. Charles and the children were officially Swiss nationals until they became naturalised on 22 October 1919, Diracs father was strict and authoritarian, although he disapproved of corporal punishment. Dirac had a relationship with his father, so much so that after his fathers death, Dirac wrote, I feel much freer now. Charles forced his children to speak to him only in French, when Dirac found that he could not express what he wanted to say in French, he chose to remain silent. Dirac was educated first at Bishop Road Primary School and then at the all-boys Merchant Venturers Technical College, the school was an institution attached to the University of Bristol, which shared grounds and staff. It emphasised technical subjects like bricklaying, shoemaking and metal work and this was unusual at a time when secondary education in Britain was still dedicated largely to the classics, and something for which Dirac would later express his gratitude. Dirac studied electrical engineering on a City of Bristol University Scholarship at the University of Bristols engineering faculty, shortly before he completed his degree in 1921, he sat the entrance examination for St Johns College, Cambridge. He passed, and was awarded a £70 scholarship, but this short of the amount of money required to live. Instead he took up an offer to study for a Bachelor of Arts degree in mathematics at the University of Bristol free of charge and he was permitted to skip the first year of the course owing to his engineering degree. In 1923, Dirac graduated, once again with first class honours, along with his £70 scholarship from St Johns College, this was enough to live at Cambridge. From 1925 to 1928 he held an 1851 Research Fellowship from the Royal Commission for the Exhibition of 1851 and he completed his PhD in June 1926 with the first thesis on quantum mechanics to be submitted anywhere
12.
Richard Feynman
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For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sinichirō Tomonaga, received the Nobel Prize in Physics in 1965. Feynman developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, during his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time. Along with his work in physics, Feynman has been credited with pioneering the field of quantum computing. Tolman professorship in physics at the California Institute of Technology. They were not religious, and by his youth, Feynman described himself as an avowed atheist, like Albert Einstein and Edward Teller, Feynman was a late talker, and by his third birthday had yet to utter a single word. He retained a Brooklyn accent as an adult and that accent was thick enough to be perceived as an affectation or exaggeration – so much so that his good friends Wolfgang Pauli and Hans Bethe once commented that Feynman spoke like a bum. The young Feynman was heavily influenced by his father, who encouraged him to ask questions to challenge orthodox thinking, from his mother, he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, maintained a laboratory in his home. When he was in school, he created a home burglar alarm system while his parents were out for the day running errands. When Richard was five years old, his mother gave birth to a brother, Henry Philips. Four years later, Richards sister Joan was born, and the moved to Far Rockaway. Though separated by nine years, Joan and Richard were close and their mother thought that women did not have the cranial capacity to comprehend such things. Despite their mothers disapproval of Joans desire to study astronomy, Richard encouraged his sister to explore the universe, Joan eventually became an astrophysicist specializing in interactions between the Earth and the solar wind. Feynman attended Far Rockaway High School, a school in Far Rockaway, Queens, upon starting high school, Feynman was quickly promoted into a higher math class. A high-school-administered IQ test estimated his IQ at 125—high, but merely respectable according to biographer James Gleick and his sister Joan did better, allowing her to claim that she was smarter. Years later he declined to join Mensa International, saying that his IQ was too low, physicist Steve Hsu stated of the test, I suspect that this test emphasized verbal, as opposed to mathematical, ability. Feynman received the highest score in the United States by a margin on the notoriously difficult Putnam mathematics competition exam
13.
Path integral formulation
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The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. Unlike previous methods, the path integral allows a physicist to easily change coordinates between very different canonical descriptions of the quantum system. Another advantage is that it is in easier to guess the correct form of the Lagrangian of a theory. Possible downsides of the approach include that unitarity of the S-matrix is obscure in the formulation, the path-integral approach has been proved to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach go away. The Schrödinger equation is an equation with an imaginary diffusion constant. The basic idea of the integral formulation can be traced back to Norbert Wiener. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 article, the complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian as a starting point, in quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a later time differs from the state at the current time by the result of acting with the Hamiltonian operator. For states with an energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle. The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames, so the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics. The Hamiltonian is a function of the position and momentum at one time, the Lagrangian is a function of the position now and the position a little later. The relation between the two is by a Legendre transform, and the condition that determines the classical equations of motion is that the action has an extremum, in quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with discretization in time, the Legendre transform becomes ϵ H = p − ϵ L and p = ∂ L ∂ q ˙, where the partial derivative with respect to q ˙ holds q fixed. The inverse Legendre transform is ϵ L = ϵ p q ˙ − ϵ H, where q ˙ = ∂ H ∂ p, and the partial derivative now is with respect to p at fixed q
14.
Wave function
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A wave function in quantum physics is a description of the quantum state of a system. The wave function is a probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a function are the Greek letters ψ or Ψ. The wave function is a function of the degrees of freedom corresponding to some set of commuting observables. Once such a representation is chosen, the function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom. Other discrete variables can also be included, such as isospin and these values are often displayed in a column matrix. According to the principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions. The Schrödinger equation determines how wave functions evolve over time, a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name wave function, and gives rise to wave–particle duality, however, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. The integral of this quantity, over all the degrees of freedom. This general requirement a wave function must satisfy is called the normalization condition, since the wave function is complex valued, only its relative phase and relative magnitude can be measured. In 1905 Einstein postulated the proportionality between the frequency of a photon and its energy, E = hf, and in 1916 the corresponding relation between photon momentum and wavelength, λ = h/p, the equations represent wave–particle duality for both massless and massive particles. In the 1920s and 1930s, quantum mechanics was developed using calculus and those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing wave mechanics. Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, Schrödinger subsequently showed that the two approaches were equivalent. However, no one was clear on how to interpret it, at first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with the scattering of a wave packet representing a particle off a target. While a scattered particle may scatter in any direction, it not break up
15.
Signal processing
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According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digitalization or digital refinement of techniques can be found in the digital control systems of the 1940s and 1950s. Feature extraction, such as understanding and speech recognition. Quality improvement, such as reduction, image enhancement. Including audio compression, image compression, and video compression and this involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators, non-linear circuits include compandors, multiplicators, voltage-controlled filters, voltage-controlled oscillators and phase-locked loops. Continuous-time signal processing is for signals that vary with the change of continuous domain, the methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology was a predecessor of digital processing, and is still used in advanced processing of gigahertz signals. Digital signal processing is the processing of digitized discrete-time sampled signals, processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, other typical operations supported by the hardware are circular buffers and look-up tables. Examples of algorithms are the Fast Fourier transform, finite impulse response filter, Infinite impulse response filter, nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains. Nonlinear systems can produce complex behaviors including bifurcations, chaos, harmonics
16.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
17.
Waveform
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A waveform is the shape and form of a signal such as a wave moving in a physical medium or an abstract representation. In many cases the medium in which the wave propagates does not permit a direct observation of the true form, in these cases, the term waveform refers to the shape of a graph of the varying quantity against time. An instrument called an oscilloscope can be used to represent a wave as a repeating image on a screen. To be more specific, a waveform is depicted by a graph that shows the changes in a signals amplitude over the duration of recording. The amplitude of the signal is measured on the y -axis, most programs show waveforms to give the user a visual aid of what has been recorded. If the waveform is of low or high height, the recording was most likely conducted under conditions with a low or high input volume, respectively. From this example, it follows that the represented by the waveform is affected by both the input signal and conditions under which it is recorded. A periodic waveforms include these while t is time, λ is wavelength, the amplitude of the waveform follows a trigonometric sine function with respect to time. Square wave = { a, mod λ < duty − a and this waveform is commonly used to represent digital information. A square wave of constant period contains odd harmonics that decrease at −6 dB/octave, triangle wave =2 a π arcsin sin 2 π t − ϕ λ. It contains odd harmonics that decrease at −12 dB/octave, sawtooth wave =2 a π arctan tan 2 π t − ϕ2 λ. This looks like the teeth of a saw, found often in time bases for display scanning. It is used as the point for subtractive synthesis, as a sawtooth wave of constant period contains odd. Other waveforms are often called composite waveforms and can often be described as a combination of a number of waves or other basis functions added together. The Fourier series describes the decomposition of periodic waveforms, such that any periodic waveform can be formed by the sum of a set of fundamental, finite-energy non-periodic waveforms can be analyzed into sinusoids by the Fourier transform. AC waveform Arbitrary waveform generator Spectrum analyzer Waveform monitor Waveform viewer Wave packet Yuchuan Wei, common Waveform Analysis, A New And Practical Generalization of Fourier Analysis. Springer US, Aug 31,2000 Waveform Definition Hao He, Jian Li, Waveform design for active sensing systems, a computational approach. Solomon W. Golomb, and Guang Gong, Signal design for good correlation, for wireless communication, cryptography, and radar
18.
Bandwidth (signal processing)
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Bandwidth is the difference between the upper and lower frequencies in a continuous set of frequencies. It is typically measured in hertz, and may refer to passband bandwidth, sometimes to baseband bandwidth. Passband bandwidth is the difference between the upper and lower frequencies of, for example, a band-pass filter, a communication channel. In the case of a filter or baseband signal, the bandwidth is equal to its upper cutoff frequency. A key characteristic of bandwidth is that any band of a given width can carry the amount of information. For example, a 3 kHz band can carry a telephone conversation whether that band is at baseband or modulated to some higher frequency, Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the range occupied by a modulated carrier signal. An FM radio receivers tuner spans a range of frequencies. A government agency may apportion the regionally available bandwidth to broadcast license holders so that their signals do not mutually interfere, each transmitter owns a slice of bandwidth. For different applications there are different precise definitions, which are different for signals than for systems. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a level of performance. A less strict and more practically useful definition will refer to the frequencies beyond which frequency response is small, small could mean less than 3 dB below the maximum value, or more rarely 10 dB below, or it could mean below a certain absolute value. As with any definition of the width of a function, many definitions are suitable for different purposes, in some contexts, the signal bandwidth in hertz refers to the frequency range in which the signals spectral density is nonzero or above a small threshold value. That definition is used in calculations of the lowest sampling rate that will satisfy the sampling theorem, the threshold value is often defined relative to the maximum value, and is most commonly the 3dB point, that is the point where the spectral density is half its maximum value. The word bandwidth applies to signals as described above, but it could apply to systems. To say that a system has a certain bandwidth means that the system can process signals of that bandwidth, or that the system reduces the bandwidth of a white noise input to that bandwidth. If the maximum gain is 0 dB, the 3 dB bandwidth is the range where the gain is more than −3 dB. This is also the range of frequencies where the gain is above 70. 7% of the maximum amplitude gain
19.
Uncertainty principle
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The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928. Heisenberg offered such an effect at the quantum level as a physical explanation of quantum uncertainty. Thus, the uncertainty principle actually states a fundamental property of quantum systems, since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their research program. These include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems, applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers. The uncertainty principle is not readily apparent on the scales of everyday experience. So it is helpful to demonstrate how it applies to more easily understood physical situations, two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, a nonzero function and its Fourier transform cannot both be sharply localized. In matrix mechanics, the formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value, for example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. According to the de Broglie hypothesis, every object in the universe is a wave, the position of the particle is described by a wave function Ψ. The time-independent wave function of a plane wave of wavenumber k0 or momentum p0 is ψ ∝ e i k 0 x = e i p 0 x / ℏ. In the case of the plane wave, | ψ |2 is a uniform distribution. In other words, the position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. The figures to the right show how with the addition of many plane waves, in mathematical terms, we say that ϕ is the Fourier transform of ψ and that x and p are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, One way to quantify the precision of the position and momentum is the standard deviation σ. Since | ψ |2 is a probability density function for position, the precision of the position is improved, i. e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i. e. increased σp. Another way of stating this is that σx and σp have a relationship or are at least bounded from below
20.
Blind men and an elephant
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The story of the blind men and an elephant originated in the Indian subcontinent from where it has widely diffused. It is a story of a group of men who touch an elephant to learn what it is like. Each one feels a different part, but only one part and they then compare notes and learn that they are in complete disagreement. It is a parable that has crossed between many religious traditions and is part of Jain, Buddhist, Sufi, Hindu and Bahá’í lore, the tale later became well known in Europe, with 19th century American poet John Godfrey Saxe creating his own version as a poem. The story has been published in books for adults and children. In various versions of the tale, a group of men touch an elephant to learn what it is like. Each one feels a different part, but only one part and they then compare notes and learn that they are in complete disagreement. The stories differ primarily in how the body parts are described, how violent the conflict becomes and how the conflict among the men. In some versions, they stop talking, start listening and collaborate to see the full elephant, when a sighted man walks by and sees the entire elephant all at once, the blind men also learn they are all blind. While ones subjective experience is true, it may not be the totality of truth, If the sighted man were deaf, he would not hear the elephant bellow. A Jain version of the story says that six men were asked to determine what an elephant looked like by feeling different parts of the elephants body. A king explains to them, All of you are right, the reason every one of you is telling it differently is because each one of you touched the different part of the elephant. So, actually the elephant has all the features you mentioned, the ancient Jain texts often explain the concepts of anekāntavāda and syādvāda with the parable of the blind men and an elephant, which addresses the manifold nature of truth. This parable resolves the conflict, and is used to illustrate the principle of living in harmony with people who have different belief systems, and this is known as the Syadvada, Anekantvada, or the theory of Manifold Predications. Two of the references to this parable are found in Tattvarthaslokavatika of Vidyanandi. Mallisena uses the parable to argue that immature people deny various aspects of truth, deluded by the aspects they do understand, due to extreme delusion produced on account of a partial viewpoint, the immature deny one aspect and try to establish another. This is the maxim of the blind and the elephant, Mallisena also cites the parable when noting the importance of considering all viewpoints in obtaining a full picture of reality. The Buddha twice uses the simile of blind men led astray, in the Canki Sutta he describes a row of blind men holding on to each other as an example of those who follow an old text that has passed down from generation to generation
21.
Niels Bohr
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Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research, although the Bohr model has been supplanted by other models, its underlying principles remain valid. He conceived the principle of complementarity, that items could be analysed in terms of contradictory properties. The notion of complementarity dominated Bohrs thinking in science and philosophy. Bohr founded the Institute of Theoretical Physics at the University of Copenhagen, now known as the Niels Bohr Institute, Bohr mentored and collaborated with physicists including Hans Kramers, Oskar Klein, George de Hevesy, and Werner Heisenberg. He predicted the existence of a new element, which was named hafnium, after the Latin name for Copenhagen. Later, the element bohrium was named after him, during the 1930s, Bohr helped refugees from Nazism. After Denmark was occupied by the Germans, he had a meeting with Heisenberg. In September 1943, word reached Bohr that he was about to be arrested by the Germans, from there, he was flown to Britain, where he joined the British Tube Alloys nuclear weapons project, and was part of the British mission to the Manhattan Project. After the war, Bohr called for cooperation on nuclear energy. He had a sister, Jenny, and a younger brother Harald. Jenny became a teacher, while Harald became a mathematician and Olympic footballer who played for the Danish national team at the 1908 Summer Olympics in London. Bohr was a footballer as well, and the two brothers played several matches for the Copenhagen-based Akademisk Boldklub, with Bohr as goalkeeper. Bohr was educated at Gammelholm Latin School, starting when he was seven, in 1903, Bohr enrolled as an undergraduate at Copenhagen University. His major was physics, which he studied under Professor Christian Christiansen and he also studied astronomy and mathematics under Professor Thorvald Thiele, and philosophy under Professor Harald Høffding, a friend of his father. This involved measuring the frequency of oscillation of the radius of a water jet, Bohr conducted a series of experiments using his fathers laboratory in the university, the university itself had no physics laboratory. To complete his experiments, he had to make his own glassware and his essay, which he submitted at the last minute, won the prize. He later submitted a version of the paper to the Royal Society in London for publication in the Philosophical Transactions of the Royal Society
22.
John Archibald Wheeler
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John Archibald Wheeler was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II, Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission. Together with Gregory Breit, Wheeler developed the concept of Breit–Wheeler process, Wheeler earned his doctorate at Johns Hopkins University under the supervision of Karl Herzfeld, and studied under Breit and Bohr on a National Research Council fellowship. In 1939 he teamed up with Bohr to write a series of using the liquid drop model to explain the mechanism of fission. He returned to Princeton after the war ended, but returned to government service to help design, for most of his career, Wheeler was a professor at Princeton University, which he joined in 1938, remaining until his retirement in 1976. At Princeton he supervised 46 PhDs, more than any other professor in the Princeton physics department, Wheeler was born in Jacksonville, Florida on July 9,1911 to librarians Joseph Lewis Wheeler and Mabel Archibald Wheeler. He was the oldest of four children, having two brothers, Joseph and Robert, and a younger sister, Mary. Joseph earned a Ph. D. from Brown University and a Master of Library Science from Columbia University, Robert earned a Ph. D. in geology from Harvard University and worked as a geologist for oil companies and at colleges. Mary studied library science at the University of Denver and became a librarian and they grew up in Youngstown, Ohio, but spent a year in 1921 to 1922 on a farm in Benson, Vermont, where Wheeler attended a one-room school. After they returned to Youngstown he attended Rayen High School, after graduating from the Baltimore City College high school in 1926, Wheeler entered Johns Hopkins University with a scholarship from the state of Maryland. He published his first scientific paper in 1930, as part of a job at the National Bureau of Standards. He earned his doctorate in 1933 and his dissertation research work, carried out under the supervision of Karl Herzfeld, was on the Theory of the Dispersion and Absorption of Helium. He received a National Research Council fellowship, which he used to study under Gregory Breit at New York University in 1933 and 1934, and then in Copenhagen under Niels Bohr in 1934 and 1935. In a 1934 paper, Breit and Wheeler introduced the Breit–Wheeler process, the University of North Carolina at Chapel Hill made Wheeler an associate professor in 1937, but he wanted to be able work more closely with the experts in particle physics. He turned down an offer in 1938 of a professorship at Johns Hopkins University in favor of an assistant professorship at Princeton University. Although it was a position, he felt that Princeton. He remained a member of the faculty there until 1976, werner Heisenberg subsequently developed the idea of the S-matrix in the 1940s. In doing so he was led to introduce a unitary characteristic S-matrix, Wheeler did not develop the S-matrix, but joined Edward Teller in examining Bohrs liquid drop model of the atomic nucleus
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Wheeler's delayed choice experiment
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Wheelers delayed choice experiment is actually several thought experiments in quantum physics, proposed by John Archibald Wheeler, with the most prominent among them appearing in 1978 and 1984. Some interpreters of these experiments contend that a photon either is a wave or is a particle, Wheelers intent was to investigate the time-related conditions under which a photon makes this transition between alleged states of being. His work has been productive of many revealing experiments, however, he himself seems to be very clear on this point. Either it was a wave or a particle, either it went both ways around the galaxy or only one way, actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago to be is to be perceived and this line of experimentation proved very difficult to carry out when it was first conceived. Nevertheless, it has proven very valuable over the years since it has led researchers to provide increasingly sophisticated demonstrations of the duality of single quanta. As one experimenter explains, Wave and particle behavior can coexist simultaneously, Wheelers delayed choice experiment refers to a series of thought experiments in quantum physics, the first being proposed by him in 1978. Another prominent version was proposed in 1983, all of these experiments try to get at the same fundamental issues in quantum physics. According to the complementarity principle, a photon can manifest properties of a particle or of a wave, what characteristic is manifested depends on whether experimenters use a device intended to observe particles or to observe waves. When this statement is applied very strictly, one could argue that by determining the type one could force the photon to become manifest only as a particle or only as a wave. Detection of a photon is a process because a photon can never be seen in flight. A photon always appears at some highly localized point in space, suppose that a traditional double-slit experiment is prepared so that either of the slits can be blocked. If both slits are open and a series of photons are emitted by the then a interference pattern will quickly emerge on the detection screen. The interference pattern can only be explained as a consequence of wave phenomena, if only one slit is available then there will be no interference pattern, so experimenters may conclude that each photon decides to travel as a particle as soon as it is emitted. One way to investigate the question of when a photon decides whether to act as a wave or a particle in an experiment is to use the interferometer method. If the apparatus is changed so that a beam splitter is placed in the upper-right corner. Experimenters must explain these phenomena as consequences of the nature of light. They may affirm that each photon must have traveled by both paths as a wave else that photon could not have interfered with itself
24.
David Bohm
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To complement it, he developed a mathematical and physical theory of implicate and explicate order. In this, his epistemology mirrored his ontology, due to his Communist affiliations, Bohm was the subject of a federal government investigation in 1949, prompting him to leave the United States. He pursued his career in several countries, becoming first a Brazilian. Bohm was born in Wilkes-Barre, Pennsylvania, United States, to a Hungarian Jewish immigrant father, Samuel Bohm, and he was raised mainly by his father, a furniture-store owner and assistant of the local rabbi. Despite being raised in a Jewish family, he became an agnostic in his teenage years, Bohm attended Pennsylvania State College, graduating in 1939, and then the California Institute of Technology, for one year. He then transferred to the physics group directed by Robert Oppenheimer at the University of California, Berkeley. Bohm lived in the neighborhood as some of Oppenheimers other graduate students. He was active in Communist and Communist-backed organizations, including the Young Communist League, the Campus Committee to Fight Conscription, during World War II, the Manhattan Project mobilized much of Berkeleys physics research in the effort to produce the first atomic bomb. Though Oppenheimer had asked Bohm to work with him at Los Alamos, Bohm remained in Berkeley, teaching physics, until he completed his Ph. D. in 1943 by an unusual circumstance. According to Peat, the calculations that he had completed proved useful to the Manhattan Project and were immediately classified. Without security clearance, Bohm was denied access to his own work, not only would he be barred from defending his thesis, to satisfy the university, Oppenheimer certified that Bohm had successfully completed the research. Bohm later performed theoretical calculations for the Calutrons at the Y-12 facility in Oak Ridge, after the war, Bohm became an assistant professor at Princeton University, where he worked closely with Albert Einstein. In May 1949, the House Un-American Activities Committee called upon Bohm to testify because of his previous ties to suspected Communists, Bohm invoked his Fifth amendment right to refuse to testify, and refused to give evidence against his colleagues. In 1950 Bohm was arrested for refusing to answer HUACs questions and he was acquitted in May 1951, but Princeton had already suspended him. His request to go to Manchester received Einsteins support but was unsuccessful, Bohm then left for Brazil to assume a professorship of physics at the University of São Paulo at Jayme Tiomnos invitation, and on Einsteins and Oppenheimers recommendation. During his early period, Bohm made a number of significant contributions to physics, particularly quantum mechanics, as a post-graduate at Berkeley, he developed a theory of plasmas, discovering the electron phenomenon known now as Bohm-diffusion. His first book, Quantum Theory, published in 1951, was received by Einstein. But Bohm became dissatisfied with the interpretation of quantum theory he had written about in that book
25.
Walter Heitler
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Walter Heinrich Heitler was a German physicist who made contributions to quantum electrodynamics and quantum field theory. He brought chemistry under quantum mechanics through his theory of valence bonding and he then became an assistant to Max Born at the Institute for Theoretical Physics at the Georg-August University of Göttingen. Heitler completed his Habilitation, under Born, in 1929, in that year, he was let go by the university because he was Jewish. These institutes were located at the LMU, under Arnold Sommerfeld, the University of Göttingen, under Max Born, furthermore, Werner Heisenberg and Born had just recently published their trilogy of papers which launched the matrix mechanics formulation of quantum mechanics. These papers immediately put the personnel at the theoretical physics institutes onto applying these new tools to understanding atomic. In Zurich, with Fritz London, Heitler applied the new mechanics to deal with the saturable, nondynamic forces of attraction and repulsion, i. e. exchange forces. Their valence bond treatment of this problem, was a landmark in that it brought chemistry under quantum mechanics, the application of quantum mechanics to chemistry would be a prominent theme in Heitlers career. While Heitler was at Göttingen, Adolf Hitler came to power in 1933, with the rising prominence of anti-Semitism under Hitler, Born took it upon himself to take the younger Jewish generation under his wing. In doing so, Born arranged for Heitler to get a position that year as a Research Fellow at the University of Bristol, at Bristol, Heitler was a Research Fellow of the Academic Assistance Council, in the H. H. With Bethe, he published a paper on production of gamma rays in the Coulomb field of an atomic nucleus. Heitler also contributed to the understanding of cosmic rays, as well as predicted the existence of the electrically neutral pi meson, in 1936, Heitler published his major work on quantum electrodynamics, The Quantum Theory of Radiation, which marked the direction for future developments in quantum theory. The book appeared in many editions and printings, even being translated in Russian, after the fall of France in 1940, Heitler was briefly interned on the Isle of Man for several months. At Dublin, Heitlers work with H. W. Peng on radiation damping theory, during the 1942-1943 academic year, Heitler gave a course on elementary wave mechanics, during which W. S. E. Hickson took notes and prepared a finished copy. These notes were the basis for Heitlers book Elementary Wave Mechanics, Introductory Course of Lectures, a new edition was published as Elementary Wave Mechanics in 1945. This version was revised and republished many times, as well as being translated into French and Italian and published in 1949, a further revised version appeared as Elementary Wave Mechanics With Applications to Quantum Chemistry in 1956, as well as in German in 1961. Schrödinger resigned as Director of the School for Theoretical Physics in 1946, in 1958, Heitler held the Lorentz Chair for Theoretical Physics at the University of Leiden. While in Zurich, after years, he began writing on the philosophical relationship of science to religion. His books were published in German, English, and French, Physics eats chemistry with a spoon. S. E
26.
Quantum entanglement
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Measurements of physical properties such as position, momentum, spin, and polarization, performed on entangled particles are found to be appropriately correlated. Later, however, the predictions of quantum mechanics were verified experimentally. Recent experiments have measured entangled particles within less than one hundredth of a percent of the time of light between them. According to the formalism of theory, the effect of measurement happens instantly. It is not possible, however, to use this effect to transmit information at faster-than-light speeds. Research is also focused on the utilization of entanglement effects in communication and computation, the counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen. In this study, they formulated the EPR paradox, an experiment that attempted to show that quantum mechanical theory was incomplete. They wrote, We are thus forced to conclude that the description of physical reality given by wave functions is not complete. However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered and he shortly thereafter published a seminal paper defining and discussing the notion, and terming it entanglement. Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, Einstein later famously derided entanglement as spukhafte Fernwirkung or spooky action at a distance. The EPR paper generated significant interest among physicists and inspired much discussion about the foundations of quantum mechanics, until recently each had left open at least one loophole by which it was possible to question the validity of the results. However, in 2015 the first loophole-free experiment was performed, which ruled out a class of local realism theories with certainty. The work of Bell raised the possibility of using these super-strong correlations as a resource for communication and it led to the discovery of quantum key distribution protocols, most famously BB84 by Charles H. Bennett and Gilles Brassard and E91 by Artur Ekert. Although BB84 does not use entanglement, Ekerts protocol uses the violation of a Bells inequality as a proof of security, in entanglement, one constituent cannot be fully described without considering the other. Quantum systems can become entangled through various types of interactions, for some ways in which entanglement may be achieved for experimental purposes, see the section below on methods. Entanglement is broken when the entangled particles decohere through interaction with the environment, for example, as an example of entanglement, a subatomic particle decays into an entangled pair of other particles. For instance, a particle could decay into a pair of spin-½ particles. The special property of entanglement can be observed if we separate the said two particles
27.
Many-worlds interpretation
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The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternate histories and futures are real, the theory is also referred to as MWI, the relative state formulation, the Everett interpretation, the theory of the universal wavefunction, many-universes interpretation, or just many-worlds. The original relative state formulation is due to Hugh Everett in 1957, later, this formulation was popularized and renamed many-worlds by Bryce Seligman DeWitt in the 1960s and 1970s. The decoherence approaches to interpreting quantum theory have been explored and developed. MWI is one of many multiverse hypotheses in physics and philosophy and it is currently considered a mainstream interpretation along with the other decoherence interpretations, collapse theories, and hidden variable theories such as the Bohmian mechanics. Before many-worlds, reality had always viewed as a single unfolding history. Many-worlds, however, views reality as a tree, wherein every possible quantum outcome is realised. Many-worlds reconciles the observation of events, such as random radioactive decay. In Dublin in 1952 Erwin Schrödinger gave a lecture in which at one point he jocularly warned his audience that what he was about to say might seem lunatic. He went on to assert that when his Nobel equations seem to be describing several different histories, they are not alternatives and this is the earliest known reference to the many-worlds. The idea of MWI originated in Everetts Princeton Ph. D, the phrase many-worlds is due to Bryce DeWitt, who was responsible for the wider popularisation of Everetts theory, which had been largely ignored for the first decade after publication. Under scrutiny of the environment, only pointer states remain unchanged, other states decohere into mixtures of stable pointer states that can persist, and, in this sense, exist, They are einselected. These ideas complement MWI and bring the interpretation in line with our perception of reality, Deutsch is dismissive that many-worlds is an interpretation, saying that calling it an interpretation is like talking about dinosaurs as an interpretation of fossil records. As with the interpretations of quantum mechanics, the many-worlds interpretation is motivated by behavior that can be illustrated by the double-slit experiment. When particles of light are passed through the slit, a calculation assuming wave-like behavior of light can be used to identify where the particles are likely to be observed. Yet when the particles are observed in experiment, they appear as particles. Everetts Ph. D. work provided such an alternative interpretation. e, the state of the observer and the observed are correlated after the observation is made. This led Everett to derive from the unitary, deterministic dynamics alone the notion of a relativity of states, thus the appearance of the objects wavefunctions collapse has emerged from the unitary, deterministic theory itself
28.
Hugh Everett III
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Hugh Everett III was an American physicist who first proposed the many-worlds interpretation of quantum physics, which he termed his relative state formulation. Discouraged by the scorn of other physicists for MWI, Everett ended his physics career after completing his Ph. D, afterwards, he developed the use of generalized Lagrange multipliers for operations research and applied this commercially as a defense analyst and a consultant. He was married to Nancy Everett née Gore and they had two children, Elizabeth Everett and Mark Oliver Everett, who became frontman of the musical band Eels. Born in 1930, Everett was born and raised in the Washington, Everetts parents separated when he was young. Initially raised by his mother, he was raised by his father and stepmother from the age of seven, Everett won a half scholarship to St Johns College, a private military high school in Washington DC. From there he moved to the nearby Catholic University of America to study engineering as an undergraduate. While there he read about Dianetics in Astounding Science Fiction, although he never exhibited any interest in Scientology, he did retain a distrust of conventional medicine throughout his life. During World War II his father was fighting in Europe as a lieutenant colonel on the general staff. After World War II, Everetts father was stationed in West Germany, father and son were both keen photographers and took hundreds of pictures of West Germany being rebuilt. Reflecting their technical interests, the pictures were almost devoid of people, Everett graduated from The Catholic University of America in 1953 in chemical engineering, although he had completed sufficient courses for a mathematics degree as well. Everett then received a National Science Foundation fellowship that allowed him to attend Princeton University for graduate studies and he started his studies at Princeton in the Mathematics Department working on the then-new field of game theory under Albert W. Tucker, but slowly drifted into physics. In 1953 he started taking his first physics courses, notably Introductory Quantum Mechanics with Robert Dicke, during 1954, he attended Methods of Mathematical Physics with Eugene Wigner, although he remained active with mathematics and presented a paper on military game theory in December. He passed his examinations in the spring of 1955, thereby gaining his Masters degree. In his third year at Princeton Everett moved into an apartment which he shared with three friends he had made during his first year, Hale Trotter, Harvey Arnold and Charles Misner, Arnold later described Everett as follows, He was smart in a very broad way. I mean, to go from chemical engineering to mathematics to physics and spending most of the buried in a science fiction book, I mean. It was during this time that he met Nancy Gore, who typed up his Wave Mechanics Without Probability paper, Everett married Nancy Gore, the next year. The long paper was retitled as The Theory of the Universal Wave Function. Wheeler himself had traveled to Copenhagen in May,1956 with the goal of getting a favorable reception for at least part of Everetts work, but in vain
29.
John von Neumann
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John von Neumann was a Hungarian-American mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics, physics, economics, computing, and statistics. He published over 150 papers in his life, about 60 in pure mathematics,20 in physics, and 60 in applied mathematics and his last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain. His analysis of the structure of self-replication preceded the discovery of the structure of DNA, also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939, on the ergodic theorem, Princeton, 1931–1932. During World War II he worked on the Manhattan Project, developing the mathematical models behind the lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. Von Neumann was born Neumann János Lajos to a wealthy, acculturated, Von Neumanns place of birth was Budapest in the Kingdom of Hungary which was then part of the Austro-Hungarian Empire. He was the eldest of three children and he had two younger brothers, Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa was a banker, who held a doctorate in law and he had moved to Budapest from Pécs at the end of the 1880s. Miksas father and grandfather were both born in Ond, Zemplén County, northern Hungary, johns mother was Kann Margit, her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in apartments above the Kann-Heller offices in Budapest. In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph, the Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town, the appellation was chosen in reference to Margaret, Neumann János became Margittai Neumann János, which he later changed to the German Johann von Neumann. Von Neumann was a child prodigy, as a 6 year old, he could multiply and divide two 8-digit numbers in his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly, the 6 year old von Neumann asked her, formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins, Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. A copy was contained in a private library Max purchased, One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor. Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911 and this was one of the best schools in Budapest, part of a brilliant education system designed for the elite
30.
Garrett Birkhoff
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Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory, the mathematician George Birkhoff was his father. The son of the mathematician George David Birkhoff, Garrett was born in Princeton and he began the Harvard University BA course in 1928 after less than seven years of prior formal education. Upon completing his Harvard BA in 1932, he went to Cambridge University in England to study mathematical physics, while visiting the University of Munich, he met Carathéodory who pointed him towards two important texts, Van der Waerden on abstract algebra and Speiser on group theory. From these facts can be inferred the number and quality of Birkhoffs papers published by his 25th year, during the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane, substantially advanced American teaching and research in abstract algebra. In 1941 he and Mac Lane published A Survey of Modern Algebra, Mac Lane and Birkhoffs Algebra is a more advanced text on abstract algebra. A number of papers he wrote in the 1930s, culminating in his monograph, Lattice Theory and his 1935 paper, On the Structure of Abstract Algebras founded a new branch of mathematics, universal algebra. During and after World War II, Birkhoffs interests gravitated towards what he called engineering mathematics, during the war, he worked on radar aiming and ballistics, including the bazooka. In the development of weapons, mathematical questions arose, some of which had not yet been addressed by the literature on fluid dynamics, Birkhoffs research was presented in his texts on fluid dynamics, Hydrodynamics and Jets, Wakes and Cavities. Birkhoff, a friend of John von Neumann, took a close interest in the rise of the electronic computer. Birkhoff supervised the Ph. D. thesis of David M. Young on the solution of the partial differential equation of Poisson. Extending the results of Young, the Birkhoff-Varga collaboration led to publications on positive operators. Birkhoffs research and consulting work developed computational methods besides numerical linear algebra, Birkhoff published more than 200 papers and supervised more than 50 Ph. D. s. He was a member of the National Academy of Sciences and the American Academy of Arts and he was a Guggenheim Fellow for the academic year 1948–1949 and the president of the Society for Industrial and Applied Mathematics for 1966–1968. He won a Lester R. Ford Award in 1974, Birkhoff, Garrett, Lattice theory, American Mathematical Society Colloquium Publications,25, Providence, R. I. American Mathematical Society, ISBN 978-0-8218-1025-5, MR598630 ——, Mac Lane, Saunders, A Survey of Modern Algebra, peters, ISBN 1-56881-068-7 ——, Hydrodynamics, A study in logic, fact, and similitude, Greenwood Press ——, Zarantonello, E. H. Robertson, Edmund F. Garrett Birkhoff, MacTutor History of Mathematics archive, University of St Andrews
31.
Distributive lattice
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In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely, every lattice is – up to isomorphism – given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of theory or of universal algebra. Both views and their correspondence are discussed in the article on lattices. In the present situation, the description appears to be more convenient, A lattice is distributive if the following additional identity holds for all x, y. Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins and it is a basic fact of lattice theory that the above condition is equivalent to its dual, x ∨ = ∧ for all x, y, and z in L. In every lattice, defining p≤q as usual to mean p∧q=p, a lattice is distributive if one of the converse inequalities holds, too. More information on the relationship of this condition to other distributivity conditions of order theory can be found in the article on distributivity. A morphism of lattices is just a lattice homomorphism as given in the article on lattices. Because such a morphism of lattices preserves the structure, it will consequently also preserve the distributivity. Distributive lattices are ubiquitous but also rather specific structures, as already mentioned the main example for distributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Further examples include, The Lindenbaum algebra of most logics that support conjunction and disjunction is a lattice, i. e. and distributes over or. Every Boolean algebra is a distributive lattice, every Heyting algebra is a distributive lattice. Especially this includes all locales and hence all open set lattices of topological spaces, also note that Heyting algebras can be viewed as Lindenbaum algebras of intuitionistic logic, which makes them a special case of the above example. Every totally ordered set is a lattice with max as join and min as meet. The natural numbers form a lattice by taking the greatest common divisor as meet. This lattice also has a least element, namely 1, which serves as the identity element for joins
32.
Boolean algebra (structure)
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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets and it is also a special case of a De Morgan algebra and a Kleene algebra. The term Boolean algebra honors George Boole, a self-educated English mathematician, booles formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons, the first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whiteheads 1898 Universal Algebra, Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoffs 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing, a Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. It follows from the last three pairs of axioms above, or from the axiom, that a = b ∧ a if. The relation ≤ defined by a ≤ b if these equivalent conditions hold, is an order with least element 0. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤, the first four pairs of axioms constitute a definition of a bounded lattice. It follows from the first five pairs of axioms that any complement is unique, the set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra, one obtains another Boolean algebra with the same elements, furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression. The smallest element 0 is the empty set and the largest element 1 is the set S itself, starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra. This construction yields a Boolean algebra and it is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra, interval algebras are useful in the study of Lindenbaum-Tarski algebras, every countable Boolean algebra is isomorphic to an interval algebra. For any natural n, the set of all positive divisors of n, defining a≤b if a divides b
33.
Polarizer
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A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass and blocks light waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam of well-defined polarization, the common types of polarizers are linear polarizers and circular polarizers. Polarizers are used in many techniques and instruments, and polarizing filters find applications in photography. Polarizers can also be made for other types of electromagnetic waves besides light, such as waves, microwaves. The simplest linear polarizer is the wire-grid polarizer, which consists of many fine parallel wires that are placed in a plane perpendicular to the incident beam. Electromagnetic waves which have a component of their electric fields aligned parallel to the wires will induce the movement of electrons along the length of the wires, for waves with electric fields perpendicular to the wires, the electrons cannot move very far across the width of each wire. Therefore, little energy is reflected and the incident wave is able to pass through the grid, in this case the grid behaves like a dielectric material. Overall, this causes the wave to be linearly polarized with an electric field that is completely perpendicular to the wires. The hypothesis that the waves slip through the gaps between the wires is incorrect, for practical purposes, the separation between wires must be less than the wavelength of the incident radiation. In addition, the width of each wires should be compared to the spacing between wires. Therefore, it is easy to construct wire-grid polarizers for microwaves, far-infrared. In addition, advanced techniques can also build very tight pitch metallic grids. Since the degree of polarization depends little on wavelength and angle of incidence, certain crystals, due to the effects described by crystal optics, show dichroism, preferential absorption of light which is polarized in particular directions. They can therefore be used as linear polarizers, the best known crystal of this type is tourmaline. However, this crystal is used as a polarizer, since the dichroic effect is strongly wavelength dependent. Herapathite is also dichroic, and is not strongly coloured, but is difficult to grow in large crystals, a Polaroid polarizing filter functions similarly on an atomic scale to the wire-grid polarizer. It was originally made of microscopic herapathite crystals and its current H-sheet form is made from polyvinyl alcohol plastic with an iodine doping. Stretching of the sheet during manufacture causes the PVA chains to align in one particular direction, valence electrons from the iodine dopant are able to move linearly along the polymer chains, but not transverse to them
34.
Join and meet
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In a partially ordered set P, the join and meet of a subset S are respectively the supremum of S, denoted ⋁S, and infimum of S, denoted ⋀S. In general, the join and meet of a subset of an ordered set need not exist. Join and meet can also be defined as a commutative, associative, if a and b are elements from P, the join is denoted as a ∨ b and the meet is denoted a ∧ b. Join and meet are symmetric duals with respect to order inversion, the join/meet of a subset of a totally ordered set is simply its maximal/minimal element. A partially ordered set in all pairs have a join is a join-semilattice. Dually, an ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice, a lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a lattice, in which not all pairs have a meet or join. Let A be a set with a partial order ≤, an element z of A is the meet of x and y, if the following two conditions are satisfied, z ≤ x and z ≤ y. For any w in A, such that w ≤ x and w ≤ y, we have w ≤ z. If there is a meet of x and y, then it is unique, since if both z and z′ are greatest lower bounds of x and y, then z ≤ z′ and z′ ≤ z, if the meet does exist, it is denoted x ∧ y. Some pairs of elements in A may lack a meet, either since they have no lower bound at all, or since none of their lower bounds is greater than all the others. If all pairs of elements have meets, then the meet is an operation on A. By definition, a binary operation ∧ on a set A is a meet, the pair then is a meet-semilattice. Moreover, we then may define a binary relation ≤ on A, by stating that x ≤ y if, in fact, this relation is a partial order on A. Note that both meets and joins equally satisfy this definition, a couple of associated meet and join operations yield partial orders which are the reverse of each other. When choosing one of these orders as the ones, one also fixes which operation is considered a meet. Thus, the partial order defined by the meet in the universal algebra approach coincides with the partial order
