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Wikipedia:A Course of Modern Analysis#0 | A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1915. The first edition was Whittaker's alone, but later editions were co-authored with Watson. == History == Its first, second, third, and the fourth edition were published in 1902, 1915, 1920, and 1927, respectively. Since then, it has continuously been reprinted and is still in print today. A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021. The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final honours on the advice of fellow student Vernon C. Morton, later Professor of Mathematics at Aberystwyth University. But its reach was much further than just the Cambridge school; André Weil in his obituary of the French mathematician Jean Delsarte noted that Delsarte always had a copy on his desk. In 1941, the book was included among a "selected list" of mathematical analysis books for use in universities in an article for that purpose published by American Mathematical Monthly. == Notable features == Some idiosyncratic but interesting problems from an older era of the Cambridge Mathematical Tripos are in the exercises. The book was one of the earliest to use decimal numbering for its sections, an innovation the authors attribute to Giuseppe Peano. == Contents == Below are the contents of the fourth edition: Part I. The Process of Analysis Part II. The Transcendental Functions == Reception == === Reviews of the first edition === George B. Mathews, in a 1903 review article published in The Mathematical Gazette opens by saying the book is "sure of a favorable reception" because of its "attractive account of some of the most valuable and interesting results of recent analysis". He notes that Part I deals mainly with infinite series, focusing on power series and Fourier expansions while including the "elements of" complex integration and the theory of residues. Part II, in contrast, has chapters on the gamma function, Legendre functions, the hypergeometric series, Bessel functions, elliptic functions, and mathematical physics. Arthur S. Hathaway, in another 1903 review published in the Journal of the American Chemical Society, notes that the book centers around complex analysis, but that topics such as infinite series are "considered in all their phases" along with "all those important series and functions" developed by mathematicians such as Joseph Fourier, Friedrich Bessel, Joseph-Louis Lagrange, Adrien-Marie Legendre, Pierre-Simon Laplace, Carl Friedrich Gauss, Niels Henrik Abel, and others in their respective studies of "practice problems". He goes on to say it "is a useful book for those who wish to make use of the most advanced developments of mathematical analysis in theoretical investigations of physical and chemical questions." In a third review of the first edition, Maxime Bôcher, in a 1904 review published in the Bulletin of the American Mathematical Society notes that while the book falls short of the "rigor" of French, German, and Italian writers, it is a "gratifying sign of progress to find in an English book such an attempt at rigorous treatment as is here made". He notes that important parts of the book were otherwise non-existent in the English language. == See also == Bateman Manuscript Project == References == == Further reading == Jourdain, Philip E. B. (1916-01-01). "(1) A Course of Pure Mathematics. By G. H. Hardy. Cambridge University Press, 1908. Pp. xvi, 428. Cloth, 12s. net. (2) A Course of Pure Mathematics. By G. H. Hardy. Second edition. Cambridge University Press, 1914. Pp. xii, 443. Cloth, 12s. net. (3) A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. By E. T. Whittaker. Cambridge University Press, 1902. Pp. xvi, 378. Cloth, 12s. 6d. net. (4) A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Second edition, completely revised. By E. T. Whittaker and G. N. Watson. Cambridge University Press, 1915. Pp. viii, 560. Cloth, 18s. net". VI. Critical Notices. Mind (review). XXV (4): 525–533. doi:10.1093/mind/XXV.4.525. ISSN 0026-4423. JSTOR 2248860. (9 pages) Neville, Eric Harold (1921). "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 10 (152): 283. doi:10.2307/3604927. ISSN 0025-5572. JSTOR 3604927. (1 page) Wrinch, Dorothy Maud (1921). "Review of A Course of Modern Analysis. Third Edition". Science Progress in the Twentieth Century (1919-1933) (review). 15 (60). Sage Publications, Inc.: 658. ISSN 2059-4941. JSTOR 43769035. (1 page) "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 14 (196): 245. 1928. doi:10.2307/3606904. ISSN 0025-5572. JSTOR 3606904. S2CID 3980161. (1 page) "Review of A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". The American Mathematical Monthly (review). 28 (4): 176. 1921. doi:10.2307/2972291. hdl:2027/coo1.ark:/13960/t17m0tq6p. ISSN 0002-9890. JSTOR 2972291. Φ (1916). "Review of A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Second edition, completely revised". The Monist (review). 26 (4): 639–640. ISSN 0026-9662. JSTOR 27900617. (2 pages) "Review of A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytical Functions, with an Account of the Principal Transcendental Functions. Second Edition". Science Progress (1916–1919) (review). 11 (41). Sage Publications, Inc.: 160–161. 1916. ISSN 2059-495X. JSTOR 43426733. (2 pages) "Review of A Course of Modern Analysis: An introduction to the General Theory of Infinite Processes and of Analytical Functions; With an Account of the Principal Transcendental Functions". The Mathematical Gazette (review). 8 (124): 306–307. 1916. doi:10.2307/3604810. ISSN 0025-5572. JSTOR 3604810. S2CID 40238008. (2 pages) Schubert, A. (1963). "E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Fourth Edition. 608 S. Cambridge 1962. Cambridge University Press. Preis brosch. 27/6 net". ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (review). 43 (9): 435. Bibcode:1963ZaMM...43R.435S. doi:10.1002/zamm.19630430916. ISSN 1521-4001. (1 page) "Modern Analysis. By E. T. Whittaker and G. N. Watson Pp. 608. 27s. 6d. 1962. (Cambridge University Press)". The Mathematical Gazette (review). 47 (359): 88. February 1963. doi:10.1017/S0025557200049032. ISSN 0025-5572. "A Course of Modern Analysis". Nature (review). 97 (2432): 298–299. 1916-06-08. Bibcode:1916Natur..97..298.. doi:10.1038/097298a0. ISSN 1476-4687. S2CID 3980161. (1 page) "A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". Nature (review). 106 (2669): 531. 1920-12-23. Bibcode:1920Natur.106R.531.. doi:10.1038/106531c0. hdl:2027/coo1.ark:/13960/t17m0tq6p. ISSN 1476-4687. S2CID 40238008. (1 page) M.-T., L. M. (1928-03-17). "A Course of Modern Analysis: an Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". Nature (review). 121 (3046): 417. Bibcode:1928Natur.121..417M. doi:10.1038/121417a0. ISSN 1476-4687. (1 page) Stuart, S. N. (1981). "Table errata: A course of modern analysis [fourth edition, Cambridge Univ. Press, Cambridge, 1927; Jbuch 53, 180] by E. T. Whittaker and G. N. Watson". Mathematics of Computation (errata). 36 (153). American Mathematical Society: 315–320 [319]. doi:10.1090/S0025-5718-1981-0595076-1. ISSN 0025-5718. JSTOR 2007758. (1 of 6 pages) |
Wikipedia:A Course of Pure Mathematics#0 | A Course of Pure Mathematics is a classic textbook on introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions (up to 1952) and several reprints. It is now out of copyright in UK and is downloadable from various internet web sites. It remains one of the most popular books on pure mathematics. == Contents == The book contains a large number of descriptive and study materials together with a number of difficult problems with regards to number theory analysis. The book is organized into the following chapters. I. REAL VARIABLES II. FUNCTIONS OF REAL VARIABLES III. COMPLEX NUMBERS IV. LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE V. LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS VI. DERIVATIVES AND INTEGRALS VII. ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS VIII. THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS IX. THE LOGARITHMIC, EXPONENTIAL AND CIRCULAR FUNCTIONS OF A REAL VARIABLE X. THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL AND CIRCULAR FUNCTIONS == Review == The book was intended to help reform mathematics teaching in the UK, and more specifically in the University of Cambridge and in schools preparing to study higher mathematics. It was aimed directly at "scholarship level" students – the top 10% to 20% by ability. Hardy himself did not originally find a passion for mathematics, only seeing it as a way to beat other students, which he did decisively, and gain scholarships. == References == == External links == === Online copies === Third edition (1921) at Internet Archive Third edition (1921) at Project Gutenberg First edition (1908) at University of Michigan Historical Math Collection === Other === A Course of Pure Mathematics at Cambridge University Press (10 e. 1952, reissued 2008) |
Wikipedia:A History of Greek Mathematics#0 | A History of Greek Mathematics is a book by English historian of mathematics Thomas Heath about history of Greek mathematics. It was published in Oxford in 1921, in two volumes titled Volume I, From Thales to Euclid and Volume II, From Aristarchus to Diophantus. It got positive reviews and is still used today. Ten years later, in 1931, Heath published A Manual of Greek Mathematics, a concise version of the two-volume History. == Background == Thomas Heath was a British civil servant, whose hobby was Greek mathematics (he called it a "hobby" himself). He published a number of translations of major works of Euclid, Archimedes, Apollonius of Perga and others; most are still used today. Heath wrote in the preface to the book: The work was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely, that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passion being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.' Ten years later, in 1931, Heath published A Manual of Greek Mathematics, a concise version of the two-volume History. In a preface Heath wrote that the Manual is for "the general reader who has not lost interest in the studies of his youth", while History was written for scholars. The Manual contains some discoveries made in ten years after the publication of History, for example the new edition of Rhind Papyrus (published in 1923), some parts of then unpublished Moscow Papyrus, and decipherment of Babylonian tablets and "the newest studies" of Babylonian astronomy. == Contents == I. Introductory II. Greek numerical notation and arithmetical operations (logistiké) III. Pythagorean arithmetic (arithmetiké) IV. The earliest Greek geometry (Thales) V. Pythagorean geometry (Pythagoras) VI. Progress in the Elements down to Plato's time ("the formative stage in which proofs were discovered and the logical bases of the science were beginning to be sought") VII. Special problems ("three famous problems" of antiquity) VIII. Zeno of Elea IX. Plato X. From Plato to Euclid (Eudoxus and Aristotle) XI. Euclid XII. Aristarchus of Samos XIII. Archimedes XIV. Conic Sections: Apollonius of Perga XV. The successors of the great geometers (Nicomedes, Diocles, Perseus, Zenodorus, Hypsicles, Dionysodorus, Posidonius, Geminus) XVI. Some handbooks (Cleomedes, Nicomachus, and Theon of Smyrna) XVII. Trigonometry: Hipparchus, Menelaus, Ptolemy XVIII. Mensuration: Heron of Alexandria XIX. Pappus of Alexandria XX. Algebra: Diophantus of Alexandria XXI. Commentators and Byzantines (Serenus, Theon of Alexandria, Proclus, Hypatia, Porphyry, Iamblichus, Marinus of Neapolis, Domninus of Larissa, Simplicius, Eutocius, Anthemius of Tralles, Hero the Younger, Michael Psellus, Georgius Pachymeres, Maximus Planudes, Manuel Moschopoulos, Nicholas Rhabdas, John Pediasimos, Barlaam of Seminara, Isaac Argyrus) == Reception == The book got positive reviews. Mathematician David Eugene Smith praised the book, writing in 1923 that "no man now living is more capable than he of interpreting the Greek mathematical mind to the scholar of today; indeed, there is no one who ranks even in the same class with Sir Thomas Heath in this particular". He also noted that Heath wrote in length about "five of the greatest names in the field of ancient mathematical research" (Euclid, Archimedes, Apollonius, Pappus, and Diophantus), given "each approximately a hundred pages". He called the book "destined to be the standard work". Philosopher John Alexander Smith wrote in 1923 that the book "has the eminent merit of being readable", and that "for most scholars the work is full and detailed enough to form almost a library of reference". Another reviewer from 1923 wrote that "covering as it does so much ground, it is not surprising that the book shows signs of ruthless compression". The author was praised for the book, with one reviewer writing "In Sir Thomas Heath we have, as Erasmus said of Tunstall, a scholar who is dictus ad unguem". Historian of science George Sarton also praised the book in his 1922 review, writing that "it seems hardly necessary to speak at great length of a book of which most scholars knew long before it appeared, for few books have been awaited with greater impatience". He also noted careful explanation of solutions written in modern language, and "perfect clearness of the exposition, its excellent order, its thoroughness". The Manual, concise version of History, also received positive reviews. It was called a "fascinating little book", "a mine of information, a delight to read". Sarton criticized the book because of the absence of chapters devoted to Egyptian and Mesopotamian mathematics. Herbert Turnbull praised the book, especially its treatment of new discoveries of Egyptian and Babylonian mathematics. Mathematician Howard Eves praised the book in his 1984 review, writing that "the English-speaking population is particularly fortunate in having available the extraordinary treatise ... one finds one of the most scholarly, most complete, and most charmingly written treatments of the subject, a treatment certain to kindle a deep appreciation of that early period of mathematical development and a genuine admiration of those who played leading roles in it." Fernando Q. Gouvêa, writing in 2006, criticizes Heath's books as outdated and old-fashioned. Benjamin Wardhaugh, writing in 2016, finds that Heath's approach to Greek mathematics is to "made them look like works of classic literature", and that "what Heath constructed might be characterized today as a history of the contents of Greek theoretical mathematics." Reviel Netz in his 2022 book calls Heath's History "a reliable guide to many generations of scholars and curious readers". He writes that "Historiographies went in and out of fashion, but Heath still stands, providing a clear and readable survey of the contents of most of the works of pure mathematics attested from Greek antiquity." He has also noted that there was no other book on the subject written in a hundred years. == Publication history == A History of Greek Mathematics, Oxford, Clarendon Press. 1921. Volume I, From Thales to Euclid, Volume II, From Aristarchus to Diophantus A History of Greek Mathematics. New York: Dover Publications. 1981. ISBN 978-0-486-24073-2. Volume I, From Thales to Euclid, Volume II, From Aristarchus to Diophantus Heath, T. L. (2013). A History of Greek Mathematics. Cambridge University Press. doi:10.1017/CBO9781139600576. ISBN 978-1-108-06306-7. A Manual of Greek Mathematics, Oxford, Clarendon Press. 1931. A Manual of Greek Mathematics. Mineola, NY: Dover Publications. 2003. ISBN 978-0486432311. == References == |
Wikipedia:A History of the Kerala School of Hindu Astronomy#0 | A History of the Kerala School of Hindu Astronomy (in perspective) is the first definitive book giving a comprehensive description of the contribution of Kerala to astronomy and mathematics. The book was authored by K. V. Sarma who was a Reader in Sanskrit at Vishveshvaranand Institute of Sanskrit and Indological Studies, Panjab University, Hoshiarpur, at the time of publication of the book (1972). The book, among other things, contains details of the lives and works of about 80 astronomers and mathematicians belonging to the Kerala School. It has also identified 752 works belonging to the Kerala school. Even though C. M. Whish, an officer of East India Company, had presented a paper on the achievements of the mathematicians of Kerala School as early as 1842, western scholars had hardly taken note of these contributions. Much later in the 1940s, C. T. Rajagopal and his associates made some efforts to study and popularize the discoveries of Whish. Their work was lying scattered in several journals and as parts of books. Even after these efforts by C. T. Rajagopal and others, the view that Bhaskara II was the last significant mathematician pre-modern India had produced had prevailed among scholars, and surprisingly, even among Indian scholars. It was in this context K. V. Sarma published his book as an attempt to present in a succinct form the results of the investigations of C. T. Rajagopal and others and also the findings of his own investigations into the history of the Kerala school of astronomy and mathematics. == Summary of the book == The book is divided into six chapters. Chapter 1 gives an outline of the salient features of Kerala astronomy. Sarma emphasizes the spirit of inquiry, stress on observation and experimentation, concern for accuracy, and continuity of tradition as the important features of Kerala astronomy. Adherence to the Aryabhatan system, use of the katapayadi system for expressing numbers, the use of the Parahita and Drik systems for astronomical computations are some other important aspects of Kerala astronomy. Chapter 2 gives a brief account of the mathematical discoveries of Kerala mathematicians which anticipate many modern day discoveries in mathematics and astronomy. Among other topics, Sarma specifically mentions the following: Tycho Brahe's reduction to the ecliptic, Newton-Gauss interpolation formula, Taylor series for sine and cosine functions, power series for sine and cosine functions, Lhuier's formula for the circum-radius of a cyclic quadrilateral, Gregory's series for the inverse tangent, and approximations to the value of pi. Chapter 3 contains a discussion on the major trends in the Kerala literature on Jyotisha. This gives an indication of the range and depth of the topics discussed in the Kerala literature on Jyotisha. Chapter 4 is devoted to providing brief accounts of the Kerala authors of mathematical and astronomical works. There are accounts of as many as 80 authors beginning with the legendary Vararuchi I who is believed to have flourished in the 4th century CE and ending with Rama Varma Koittampuran (1853–1910). Chapter 5 is a bibliography of Kerala Jyotisha literature. This chapter contains essential information about as many as 752 works produced by Kerala astronomers and mathematicians. Chapter 6, the last one of the book, discusses works produced in regions outside Kerala, based on Kerala jyotisha. == See also == Kerala school of astronomy and mathematics List of astronomers and mathematicians of the Kerala school == Notes == The full text of the book can be accessed from Internet Archive: A History of the Kerala School of Hindu Astronomy (in perspective). The full text of a review of the book appeared in the Indian Journal of History of Science: Sen, S. N.; Bag, A. K. (1973). "Review of A History of the Kerala School of Hindu Astronomy" (PDF). Indian Journal of History of Science. 8 (1 & 2): 117–118. Retrieved 17 February 2016. == References == |
Wikipedia:A Passage to Infinity#0 | A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact is a 2009 book by George Gheverghese Joseph chronicling the social and mathematical origins of the Kerala school of astronomy and mathematics. The book discusses the highlights of the achievements of Kerala school and also analyses the hypotheses and conjectures on the possible transmission of Kerala mathematics to Europe. == An outline of the contents == Introduction The Social Origins of the Kerala School The Mathematical Origins of the Kerala School The Highlights of Kerala Mathematics and Astronomy Indian Trigonometry: From Ancient Beginnings to Nilakantha Squaring the Circle: The Kerala Answer Reaching for the Stars: The Power Series for Sines and Cosines Changing Perspectives on Indian Mathematics Exploring Transmissions: A Case Study of Kerala Mathematics A Final Assessment == See also == Indian astronomy Indian mathematics History of mathematics == References == == Further references == In association with the Royal Society's 350th anniversary celebrations in 2010, Asia House presented a talk based on A Passage to Infinity. See : "A Passage to Infinity: Indian Mathematics in World Mathematics". Retrieved 3 May 2010. For an audio-visual presentation of George Gheverghese Joseph's views on the ideas presented in the book, see : Joseph, George Gheverghese (16 September 2008). "George Gheverghese Joseph on the Transmission to Europe of Non-European Mathematics". The Mathematical Association of America. Archived from the original on 15 April 2010. Retrieved 3 May 2010. The Economic Times talks to George Gheverghese Joseph on The Passage to Infinity. See : Lal, Amrith (23 April 2010). "Indian mathematics loved numbers". The Economic Times. Review of "A PASSAGE TO INFINITY: Medieval Indian Mathematics from Kerala and its impact" by M. Ram Murty in Hardy-Ramanujan Journal, 36 (2013), 43–46. Nair, R. Madhavan (3 February 2011). "In search of the roots of mathematics". The Hindu. Retrieved 15 October 2014. |
Wikipedia:A Primer of Real Functions#0 | A Primer of Real Functions is a revised edition of a classic Carus Monograph on the theory of functions of a real variable. It is authored by R. P. Boas, Jr and updated by his son Harold P. Boas. == References == |
Wikipedia:A-equivalence#0 | The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same trajectories and landing at identical times. The extended form by Albert Einstein requires special relativity to also hold in free fall and requires the weak equivalence to be valid everywhere. This form was a critical input for the development of the theory of general relativity. The strong form requires Einstein's form to work for stellar objects. Highly precise experimental tests of the principle limit possible deviations from equivalence to be very small. == Concept == In classical mechanics, Newton's equation of motion in a gravitational field, written out in full, is: inertial mass × acceleration = gravitational mass × gravitational acceleration Careful experiments have shown that the inertial mass on the left side and gravitational mass on the right side are numerically equal and independent of the material composing the masses. The equivalence principle is the hypothesis that this numerical equality of inertial and gravitational mass is a consequence of their fundamental identity.: 32 The equivalence principle can be considered an extension of the principle of relativity, the principle that the laws of physics are invariant under uniform motion. An observer in a windowless room cannot distinguish between being on the surface of the Earth and being in a spaceship in deep space accelerating at 1g and the laws of physics are unable to distinguish these cases.: 33 == History == By experimenting with the acceleration of different materials, Galileo Galilei determined that gravitation is independent of the amount of mass being accelerated. Isaac Newton, just 50 years after Galileo, investigated whether gravitational and inertial mass might be different concepts. He compared the periods of pendulums composed of different materials and found them to be identical. From this, he inferred that gravitational and inertial mass are the same thing. The form of this assertion, where the equivalence principle is taken to follow from empirical consistency, later became known as "weak equivalence". A version of the equivalence principle consistent with special relativity was introduced by Albert Einstein in 1907, when he observed that identical physical laws are observed in two systems, one subject to a constant gravitational field causing acceleration and the other subject to constant acceleration, like a rocket far from any gravitational field.: 152 Since the physical laws are the same, Einstein assumed the gravitational field and the acceleration were "physically equivalent". Einstein stated this hypothesis by saying he would: ...assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. In 1911 Einstein demonstrated the power of the equivalence principle by using it to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field.: 153 He connected the equivalence principle to his earlier principle of special relativity: This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course. Soon after completing work on his theory of gravity (known as general relativity): 111 and then also in later years, Einstein recalled the importance of the equivalence principle to his work: The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me: If a man falls freely, he would not feel his weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity. Einstein's development of general relativity necessitated some means of empirically discriminating the theory from other theories of gravity compatible with special relativity. Accordingly, Robert Dicke developed a test program incorporating two new principles – the § Einstein equivalence principle, and the § Strong equivalence principle – each of which assumes the weak equivalence principle as a starting point. == Definitions == Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.: 6 Some proposals also suggest finer divisions or minor alterations. === Weak equivalence principle === The weak equivalence principle, also known as the universality of free fall or the Galilean equivalence principle can be stated in many ways. The strong equivalence principle, a generalization of the weak equivalence principle, includes astronomic bodies with gravitational self-binding energy. Instead, the weak equivalence principle assumes falling bodies are self-bound by non-gravitational forces only (e.g. a stone). Either way: "All uncharged, freely falling test particles follow the same trajectories, once an initial position and velocity have been prescribed".: 6 "... in a uniform gravitational field all objects, regardless of their composition, fall with precisely the same acceleration." "The weak equivalence principle implicitly assumes that the falling objects are bound by non-gravitational forces." "... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass." Mass (measured with a balance) and weight (measured with a scale) are locally in identical ratio for all bodies (the opening page to Newton's Philosophiæ Naturalis Principia Mathematica, 1687). Uniformity of the gravitational field eliminates measurable tidal forces originating from a radial divergent gravitational field (e.g., the Earth) upon finite sized physical bodies. === Einstein equivalence principle === What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that: Here local means that experimental setup must be small compared to variations in the gravitational field, called tidal forces. The test experiment must be small enough so that its gravitational potential does not alter the result. The two additional constraints added to the weak principle to get the Einstein form − (1) the independence of the outcome on relative velocity (local Lorentz invariance) and (2) independence of "where" (known as local positional invariance) − have far reaching consequences. With these constraints alone Einstein was able to predict the gravitational redshift. Theories of gravity that obey the Einstein equivalence principle must be "metric theories", meaning that trajectories of freely falling bodies are geodesics of symmetric metric.: 9 Around 1960 Leonard I. Schiff conjectured that any complete and consistent theory of gravity that embodies the weak equivalence principle implies the Einstein equivalence principle; the conjecture can't be proven but has several plausibility arguments in its favor.: 20 Nonetheless, the two principles are tested with very different kinds of experiments. The Einstein equivalence principle has been criticized as imprecise, because there is no universally accepted way to distinguish gravitational from non-gravitational experiments (see for instance Hadley and Durand). === Strong equivalence principle === The strong equivalence principle applies the same constraints as the Einstein equivalence principle, but allows the freely falling bodies to be massive gravitating objects as well as test particles. Thus this is a version of the equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments. It requires that the gravitational constant be the same everywhere in the universe: 49 and is incompatible with a fifth force. It is much more restrictive than the Einstein equivalence principle. Like the Einstein equivalence principle, the strong equivalence principle requires gravity to be geometrical by nature, but in addition it forbids any extra fields, so the metric alone determines all of the effects of gravity. If an observer measures a patch of space to be flat, then the strong equivalence principle suggests that it is absolutely equivalent to any other patch of flat space elsewhere in the universe. Einstein's theory of general relativity (including the cosmological constant) is thought to be the only theory of gravity that satisfies the strong equivalence principle. A number of alternative theories, such as Brans–Dicke theory and the Einstein-aether theory add additional fields. === Active, passive, and inertial masses === Some of the tests of the equivalence principle use names for the different ways mass appears in physical formulae. In nonrelativistic physics three kinds of mass can be distinguished: Inertial mass intrinsic to an object, the sum of all of its mass–energy. Passive mass, the response to gravity, the object's weight. Active mass, the mass that determines the objects gravitational effect. By definition of active and passive gravitational mass, the force on M 1 {\displaystyle M_{1}} due to the gravitational field of M 0 {\displaystyle M_{0}} is: F 1 = M 0 a c t M 1 p a s s r 2 {\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}} Likewise the force on a second object of arbitrary mass2 due to the gravitational field of mass0 is: F 2 = M 0 a c t M 2 p a s s r 2 {\displaystyle F_{2}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}}}} By definition of inertial mass: F = m i n e r t a {\displaystyle F=m^{\mathrm {inert} }a} if m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} are the same distance r {\displaystyle r} from m 0 {\displaystyle m_{0}} then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same). a 1 = F 1 m 1 i n e r t = a 2 = F 2 m 2 i n e r t {\displaystyle a_{1}={\frac {F_{1}}{m_{1}^{\mathrm {inert} }}}=a_{2}={\frac {F_{2}}{m_{2}^{\mathrm {inert} }}}} Hence: M 0 a c t M 1 p a s s r 2 m 1 i n e r t = M 0 a c t M 2 p a s s r 2 m 2 i n e r t {\displaystyle {\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}m_{1}^{\mathrm {inert} }}}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}m_{2}^{\mathrm {inert} }}}} Therefore: M 1 p a s s m 1 i n e r t = M 2 p a s s m 2 i n e r t {\displaystyle {\frac {M_{1}^{\mathrm {pass} }}{m_{1}^{\mathrm {inert} }}}={\frac {M_{2}^{\mathrm {pass} }}{m_{2}^{\mathrm {inert} }}}} In other words, passive gravitational mass must be proportional to inertial mass for objects, independent of their material composition if the weak equivalence principle is obeyed. The dimensionless Eötvös-parameter or Eötvös ratio η ( A , B ) {\displaystyle \eta (A,B)} is the difference of the ratios of gravitational and inertial masses divided by their average for the two sets of test masses "A" and "B". η ( A , B ) = 2 ( m p a s s m i n e r t ) A − ( m p a s s m i n e r t ) B ( m p a s s m i n e r t ) A + ( m p a s s m i n e r t ) B . {\displaystyle \eta (A,B)=2{\frac {\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{A}-\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{B}}{\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{A}+\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{B}}}.} Values of this parameter are used to compare tests of the equivalence principle.: 10 A similar parameter can be used to compare passive and active mass. By Newton's third law of motion: F 1 = M 0 a c t M 1 p a s s r 2 {\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}} must be equal and opposite to F 0 = M 1 a c t M 0 p a s s r 2 {\displaystyle F_{0}={\frac {M_{1}^{\mathrm {act} }M_{0}^{\mathrm {pass} }}{r^{2}}}} It follows that: M 0 a c t M 0 p a s s = M 1 a c t M 1 p a s s {\displaystyle {\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}={\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}} In words, passive gravitational mass must be proportional to active gravitational mass for all objects. The difference, S 0 , 1 = M 0 a c t M 0 p a s s − M 1 a c t M 1 p a s s {\displaystyle S_{0,1}={\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}-{\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}} is used to quantify differences between passive and active mass. == Experimental tests == === Tests of the weak equivalence principle === Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. An obvious test is dropping different objects and verifying that they land at the same time. Historically this was the first approach – though probably not by Galileo's Leaning Tower of Pisa experiment: 19–21 but instead earlier by Simon Stevin, who dropped lead balls of different masses off the Delft churchtower and listened for the sound of them hitting a wooden plank. Newton measured the period of pendulums made with different materials as an alternative test giving the first precision measurements. Loránd Eötvös's approach in 1908 used a very sensitive torsion balance to give precision approaching 1 in a billion. Modern experiments have improved this by another factor of a million. A popular exposition of this measurement was done on the Moon by David Scott in 1971. He dropped a falcon feather and a hammer at the same time, showing on video that they landed at the same time. Experiments are still being performed at the University of Washington which have placed limits on the differential acceleration of objects towards the Earth, the Sun and towards dark matter in the Galactic Center. Future satellite experiments – Satellite Test of the Equivalence Principle and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy. With the first successful production of antimatter, in particular anti-hydrogen, a new approach to test the weak equivalence principle has been proposed. Experiments to compare the gravitational behavior of matter and antimatter are currently being developed. Proposals that may lead to a quantum theory of gravity such as string theory and loop quantum gravity predict violations of the weak equivalence principle because they contain many light scalar fields with long Compton wavelengths, which should generate fifth forces and variation of the fundamental constants. Heuristic arguments suggest that the magnitude of these equivalence principle violations could be in the 10−13 to 10−18 range. Currently envisioned tests of the weak equivalence principle are approaching a degree of sensitivity such that non-discovery of a violation would be just as profound a result as discovery of a violation. Non-discovery of equivalence principle violation in this range would suggest that gravity is so fundamentally different from other forces as to require a major reevaluation of current attempts to unify gravity with the other forces of nature. A positive detection, on the other hand, would provide a major guidepost towards unification. === Tests of the Einstein equivalence principle === In addition to the tests of the weak equivalence principle, the Einstein equivalence principle requires testing the local Lorentz invariance and local positional invariance conditions. Testing local Lorentz invariance amounts to testing special relativity, a theory with vast number of existing tests.: 12 Nevertheless, attempts to look for quantum gravity require even more precise tests. The modern tests include looking for directional variations in the speed of light (called "clock anisotropy tests") and new forms of the Michelson–Morley experiment. The anisotropy measures less than one part in 10−20.: 14 Testing local positional invariance divides in to tests in space and in time.: 17 Space-based tests use measurements of the gravitational redshift, the classic is the Pound–Rebka experiment in the 1960s. The most precise measurement was done in 1976 by flying a hydrogen maser and comparing it to one on the ground. The Global Positioning System requires compensation for this redshift to give accurate position values. Time-based tests search for variation of dimensionless constants and mass ratios. For example, Webb et al. reported detection of variation (at the 10−5 level) of the fine-structure constant from measurements of distant quasars. Other researchers dispute these findings. The present best limits on the variation of the fundamental constants have mainly been set by studying the naturally occurring Oklo natural nuclear fission reactor, where nuclear reactions similar to ones we observe today have been shown to have occurred underground approximately two billion years ago. These reactions are extremely sensitive to the values of the fundamental constants. === Tests of the strong equivalence principle === The strong equivalence principle can be tested by 1) finding orbital variations in massive bodies (Sun-Earth-Moon), 2) variations in the gravitational constant (G) depending on nearby sources of gravity or on motion, or 3) searching for a variation of Newton's gravitational constant over the life of the universe: 47 Orbital variations due to gravitational self-energy should cause a "polarization" of solar system orbits called the Nordtvedt effect. This effect has been sensitively tested by Lunar Laser Ranging experiments. Up to the limit of one part in 1013 there is no Nordtvedt effect. A tight bound on the effect of nearby gravitational fields on the strong equivalence principle comes from modeling the orbits of binary stars and comparing the results to pulsar timing data.: 49 In 2014, astronomers discovered a stellar triple system containing a millisecond pulsar PSR J0337+1715 and two white dwarfs orbiting it. The system provided them a chance to test the strong equivalence principle in a strong gravitational field with high accuracy. If there is any departure from the strong equivalence principle, it is no more than two parts per million. Most alternative theories of gravity predict a change in the gravity constant over time. Studies of Big Bang nucleosynthesis, analysis of pulsars, and the lunar laser ranging data have shown that G cannot have varied by more than 10% since the creation of the universe. The best data comes from studies of the ephemeris of Mars, based on three successive NASA missions, Mars Global Surveyor, Mars Odyssey, and Mars Reconnaissance Orbiter.: 50 == See also == == References == == Further reading == == External links == Gravity and the principle of equivalence – The Feynman Lectures on Physics Introducing The Einstein Principle of Equivalence from Syracuse University The Equivalence Principle at MathPages The Einstein Equivalence Principle at Living Reviews on General Relativity "...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..." |
Wikipedia:A. Edward Nussbaum#0 | Adolf Edward Nussbaum (10 January 1925 – 31 October 2009) was a German-born American theoretical mathematician who was a professor of mathematics in Arts and Sciences at Washington University in St. Louis for nearly 40 years. He worked with others in 20th-century theoretical physics and mathematics such as J. Robert Oppenheimer and John von Neumann, and was acquainted with Albert Einstein. == Early years == Nussbaum was born to a Jewish family in Rheydt, a borough of the German city Mönchengladbach in northwestern Germany, in 1925. The youngest of three children, he was a Holocaust survivor and was orphaned after the Nazi takeover of Germany. Both his father, Karl Nussbaum, a wounded veteran of World War I during which he had been awarded the Iron Cross, and his mother, Franziska, was murdered at Auschwitz. His brother, Erwin Nussbaum, was also captured and killed. Nussbaum and his sister, Lieselotte, were separated and sent on a Kindertransport to Belgium in 1939. When Belgium was invaded by Germany, Nussbaum escaped to southern France, then under the Vichy regime. He lived there at an orphanage known as Château de la Hille. He began his teaching career there, while still a teenager, teaching mathematics to the younger children. After being captured twice, and jailed once by the Nazis, he escaped on foot to Switzerland, where he attended the University of Zurich, studying both mathematics and physics. In 1947, he was sponsored by relatives in New Jersey to emigrate to the United States. == Career == Shortly after emigrating to the United States, he studied mathematics at Brooklyn College before transferring to Columbia University in New York where he earned his Master of Arts degree in 1950 and his Ph.D. in 1957. While writing his thesis for Columbia, he worked in the academic year 1952–1953 at the Institute for Advanced Study in Princeton with John von Neumann, a mathematician who used Hilbert spaces in his development of the mathematical basis of quantum mechanics. Hilbert spaces eventually became Nussbaum's area of expertise and he wrote several papers with von Neumann on this topic. During this period, Nussbaum also became acquainted with Albert Einstein, another of the original group at the Institute for Advanced Study. Nussbaum's thesis was accepted with no revisions and he received his doctorate shortly thereafter. In the meantime he had worked at the University of Connecticut in Storrs, where he co-authored papers with Allen Devinatz, and at the Rensselaer Polytechnic Institute in Troy, New York. He followed Devinatz to St. Louis to teach at Washington University in 1958. In 1962, he was a visiting scholar at the Institute for Advanced Studies working with Robert Oppenheimer; in 1967–68 he was a visiting scholar at Stanford University in Palo Alto, California. He joined Washington University's mathematics faculty as an assistant professor in 1958. He became a full professor in 1966 and taught until 1995, when he was named an emeritus professor. == Personal life == Nussbaum married his cousin's sister-in-law, Anne Ebbin, on September 1, 1957. They had a son, Karl Erich Nussbaum and a daughter, Franziska Suzanne Nussbaum. He died in St. Louis, Missouri, in 2009. == Selected publications == Devinatz, A.; ——; Neumann, J. Von (1955). "On the Permutability of Self-Adjoint Operators". The Annals of Mathematics. 62 (2): 199–203. doi:10.2307/1969674. ISSN 0003-486X. JSTOR 1969674. —— (1955). "The Hausdorff-Bernstein-Widder theorem for semi-groups in locally compact Abelian groups". Duke Mathematical Journal. 22 (4): 573–582. doi:10.1215/S0012-7094-55-02263-8. ISSN 0012-7094. Devinatz, A.; —— (1957). "On the Permutability of Normal Operators". Annals of Mathematics. 65 (1): 144–152. doi:10.2307/1969669. JSTOR 1969669. —— (1959). "Integral Representation of Semi-Groups of Unbounded Self-Adjoint Operators". Annals of Mathematics. 69 (1): 133–141. doi:10.2307/1970098. JSTOR 1970098. Devinatz, Allen; —— (1960). On Real Characters of Certain Semi-groups with Applications. Office of Scientific Research, US Air Force; 54 pages{{cite book}}: CS1 maint: postscript (link) —— (1962). "On a theorem by I. Glicksberg". Proceedings of the American Mathematical Society. 13 (4): 645–646. doi:10.1090/S0002-9939-1962-0138721-7. —— (1964). "On the reduction of C {\displaystyle C} *-algebras". Proceedings of the American Mathematical Society. 15 (4): 567–573. doi:10.1090/S0002-9939-1964-0165383-7. —— (1964). "Reduction theory for unbounded closed operators in Hilbert space". Duke Mathematical Journal. 31: 33–44. doi:10.1215/S0012-7094-64-03103-5. —— (1965). "Quasi-analytic vectors" (PDF). Arkiv för Matematik. 6 (10): 179–191. Bibcode:1965ArM.....6..179N. doi:10.1007/BF02591357. S2CID 122725979. —— (1967). "On the integral representation of positive linear functionals". Transactions of the American Mathematical Society. 128 (3): 460–473. doi:10.1090/S0002-9947-1967-0215108-9. —— (1969). "A commutativity theorem for unbounded operators in Hilbert space". Transactions of the American Mathematical Society. 140: 485–491. doi:10.1090/S0002-9947-1969-0242010-0. ISSN 0002-9947. —— (1970). "Spectral representation of certain one-parametric families of symmetric operators in Hilbert space". Transactions of the American Mathematical Society. 152 (2): 419–429. doi:10.1090/S0002-9947-1970-0268719-9. —— (1972). "Radial exponentially convex functions". Journal d'Analyse Mathématique. 25 (1): 277–288. doi:10.1007/BF02790041. ISSN 0021-7670. S2CID 122743981. —— (1973). "Integral representation of functions and distributions positive definite relative to the orthogonal group". Transactions of the American Mathematical Society. 175: 355–387. doi:10.1090/S0002-9947-1973-0333600-6. —— (1973). "On functions positive definite relative to the orthogonal group and the representation of functions as Hankel-Stieltjes transforms". Transactions of the American Mathematical Society. 175: 389–408. doi:10.1090/S0002-9947-1973-0333601-8. —— (1976). "Semi-Groups of Subnormal Operators". Journal of the London Mathematical Society. s2-14 (2): 340–344. doi:10.1112/jlms/s2-14.2.340.} —— (1982). "Multi-parameter local semi-groups of Hermetian operators". Journal of Functional Analysis. 48 (2): 213–223. doi:10.1016/0022-1236(82)90067-2. —— (1997). "A commutativity theorem for semibounded operators in Hilbert space". Proceedings of the American Mathematical Society. 125 (12): 3541–3545. doi:10.1090/S0002-9939-97-03977-4. JSTOR 2162252. == Notes == |
Wikipedia:A. F. Mujibur Rahman#0 | Abul Faiz Mujibur Rahman (born 23 September 1897, in Faridpur district of Bangladesh), was a jurist and the first Bengali Muslim Indian Civil Service (ICS) officer. == Early life == Rahman attended school in Faridpur Zilla School and graduated from Dhaka College. He moved to Calcutta University and in 1920 achieved master's degree in pure mathematics with the highest score in the history of Calcutta University beating the previous record mark achieved by Sir Ashutosh Mukherjee. He then later applied to join the Indian Civil Service. He attended Balliol College, University of Oxford for probationary studies after which he joined in the judicial branch and also served as district judge in Dhaka for sometimes. == Career == Rahman opposed the death penalty of revolutionary Ambika Chakrabarty for raiding Chittagong Armoury and the decision saved Chakrabarti's life. On request by Sher E Bangla A. K. Fazlul Huq, he took the responsibility of setting up the Land Acquisition Collectorate to ensure plots for hundreds and thousands of destitute Muslims living in Calcutta's slum area. == Death and legacy == Rahman died of heart failure on 12 May 1945 at the age of 48. In 1985 in remembrance of his father, Rezaur Rahman established a charitable trust called AF Mujibur Rahman Foundation. This foundation supports a number of institutions, especially the department of mathematics of Dhaka, Jagannath, Chittagong, Khulna and Rajshahi universities and awards the meritorious students of the department. It provide scholarships at the Institute of Business Administration of University of Dhaka, the Institute of Chartered Accountants of Bangladesh and Gono Bishawabidyalay. The foundation also supports the Bangladesh Mathematical Society to organise the National Mathematics Undergraduate Olympiad every year. Funded by the AF Mujibur Rahman Foundation, the new eight storey mathematics building of University of Dhaka called AF Mujibur Rahman Ganit Bhaban was inaugurated in 2014. == References == |
Wikipedia:A. K. Dewdney#0 | Alexander Keewatin Dewdney (August 5, 1941 – March 9, 2024) was a Canadian mathematician, computer scientist, author, filmmaker, and conspiracy theorist. Dewdney was the son of Canadian artist and author Selwyn Dewdney and art therapist Irene Dewdney, and brother of poet Christopher Dewdney. == Personal life == Dewdney was born in London, Ontario on August 5, 1941, and died there on March 9, 2024, at the age of 82. == Art and fiction == In his student days, Dewdney made a number of influential experimental films, including Malanga, on the poet Gerald Malanga, Four Girls, Scissors, and his most ambitious film, the pre-structural Maltese Cross Movement. Margaret Atwood wrote that a poetry scrapbook by Dewdney, based on the Maltese Cross Movement film, "raises scrapbooking to an art". The Academy Film Archive has preserved two of Dewdney's films: The Maltese Cross Movement in 2009 and Wildwood Flower in 2011. Dewdney wrote two novels, The Planiverse (about an imaginary two-dimensional world) and Hungry Hollow: The Story of a Natural Place. Dewdney lived in London, Ontario where he held the position of Professor Emeritus at the University of Western Ontario. == Computing, mathematics, and science == Dewdney wrote a number of books on mathematics, computing, and bad science. He also founded and edited a magazine on recreational programming called Algorithm between 1989 and 1993. Dewdney followed Martin Gardner and Douglas Hofstadter in authoring Scientific American magazine's recreational mathematics column, renamed to "Computer Recreations", then "Mathematical Recreations", from 1984 to 1991. He published more than 10 books on scientific possibilities and puzzles. Dewdney was a co-inventor of programming game Core War. Beginning in the nineties, Dewdney worked on biology, both as a field ecologist and as a mathematical biologist, contributing a solution to the problem of determining the underlying dynamics of species abundance in natural communities. == Conspiracy theories == Dewdney was a member of the 9/11 truth movement, and theorized that the planes used in the September 11 attacks had been emptied of passengers and were flown by remote control. He based these claims in part on a series of experiments (one with funding from Japan's TV Asahi) that, he claimed, showed that cell phones do not work on airplanes, from which he concluded that the phone calls received from hijacked passengers during the attacks must have been faked. == Works == The Planiverse: Computer Contact with a Two-Dimensional World (1984). ISBN 0-387-98916-1. The Armchair Universe: An Exploration of Computer Worlds (1988). ISBN 0-7167-1939-8. (collection of "Mathematical Recreations" columns) The Magic Machine: A Handbook of Computer Sorcery (1990). ISBN 0-7167-2144-9. (collection of "Mathematical Recreations" columns) The New Turing Omnibus: Sixty-Six Excursions in Computer Science (1993). ISBN 0-8050-7166-0. The Tinkertoy Computer and Other Machinations (1993). ISBN 0-7167-2491-X. (collection of "Mathematical Recreations" columns) Introductory Computer Science: Bits of Theory, Bytes of Practice (1996). ISBN 0-7167-8286-3. 200% of Nothing: An Eye Opening Tour Through the Twists and Turns of Math Abuse and Innumeracy (1996). ISBN 0-471-14574-2. Yes, We Have No Neutrons: An Eye-Opening Tour through the Twists and Turns of Bad Science (1997). ISBN 0-471-29586-8. Hungry Hollow: The Story of a Natural Place (1998). ISBN 0-387-98415-1. A Mathematical Mystery Tour: Discovering the Truth and Beauty of the Cosmos (2001). ISBN 0-471-40734-8. Beyond Reason: Eight Great Problems that Reveal the Limits of Science (2004). ISBN 0-471-01398-6. == References == == External links == Alexander Dewdney homepage Keewatin Dewdney at IMDb |
Wikipedia:A. Rod Gover#0 | Ashwin Rod Gover is a New Zealand mathematician and a Fellow of the Royal Society of New Zealand. He is currently employed as a Professor of Pure Mathematics at the University of Auckland in New Zealand. == Education and career == Gover received his secondary education at Tauranga Boys' College, where he was Head Boy and Dux. He earned a Bachelor of Science with Honours and Master of Science in physics at Canterbury University and a Doctor of Philosophy (DPhil) in Mathematics in 1989 at Oxford. He joined the University of Auckland as a lecturer in 1999, before being promoted to Senior Lecturer in 2001, Associate Professor in 2005, and Professor in 2008. == Research areas == His current main research areas are Differential geometry and its relationship to representation theory Applications to analysis on manifolds, PDE theory and Mathematical Physics Conformal, CR and related structures He has published work on a range of topics including integral transforms and their applications to representation theory and quantum groups. His main area of specialisation is the class of parabolic differential geometries. Tractor calculus is important for treating geometries in this class, and a current theme of his work is the further development of this calculus, its relationship to other geometric constructions and tools, as well as its applications to the construction and understanding of local and global geometric invariants and natural differential equations. A list of his publications can be found here. == References == |
Wikipedia:ABACABA pattern#0 | The ABACABA pattern is a recursive fractal pattern that shows up in many places in the real world (such as in geometry, art, music, poetry, number systems, literature and higher dimensions). Patterns often show a DABACABA type subset. AA, ABBA, and ABAABA type forms are also considered. == Generating the pattern == In order to generate the next sequence, first take the previous pattern, add the next letter from the alphabet, and then repeat the previous pattern. The first few steps are listed here. ABACABA is a "quickly growing word", often described as chiastic or "symmetrically organized around a central axis" (see: Chiastic structure and Χ). The number of members in each iteration is a(n) = 2n − 1, the Mersenne numbers (OEIS: A000225). == Gallery == == See also == Arch form Farey sequence Rondo Sesquipower == Notes == == References == == External links == Naylor, Mike: abacaba.org |
Wikipedia:ATS theorem#0 | In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful. == History of the problem == In some fields of mathematics and mathematical physics, sums of the form S = ∑ a < k ≤ b φ ( k ) e 2 π i f ( k ) ( 1 ) {\displaystyle S=\sum _{a<k\leq b}\varphi (k)e^{2\pi if(k)}\qquad (1)} are under study. Here φ ( x ) {\displaystyle \varphi (x)} and f ( x ) {\displaystyle f(x)} are real valued functions of a real argument, and i 2 = − 1. {\displaystyle i^{2}=-1.} Such sums appear, for example, in number theory in the analysis of the Riemann zeta function, in the solution of problems connected with integer points in the domains on plane and in space, in the study of the Fourier series, and in the solution of such differential equations as the wave equation, the potential equation, the heat conductivity equation. The problem of approximation of the series (1) by a suitable function was studied already by Euler and Poisson. We shall define the length of the sum S {\displaystyle S} to be the number b − a {\displaystyle b-a} (for the integers a {\displaystyle a} and b , {\displaystyle b,} this is the number of the summands in S {\displaystyle S} ). Under certain conditions on φ ( x ) {\displaystyle \varphi (x)} and f ( x ) {\displaystyle f(x)} the sum S {\displaystyle S} can be substituted with good accuracy by another sum S 1 , {\displaystyle S_{1},} S 1 = ∑ α < k ≤ β Φ ( k ) e 2 π i F ( k ) , ( 2 ) {\displaystyle S_{1}=\sum _{\alpha <k\leq \beta }\Phi (k)e^{2\pi iF(k)},\ \ \ (2)} where the length β − α {\displaystyle \beta -\alpha } is far less than b − a . {\displaystyle b-a.} First relations of the form S = S 1 + R , ( 3 ) {\displaystyle S=S_{1}+R,\qquad (3)} where S , {\displaystyle S,} S 1 {\displaystyle S_{1}} are the sums (1) and (2) respectively, R {\displaystyle R} is a remainder term, with concrete functions φ ( x ) {\displaystyle \varphi (x)} and f ( x ) , {\displaystyle f(x),} were obtained by G. H. Hardy and J. E. Littlewood, when they deduced approximate functional equation for the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} and by I. M. Vinogradov, in the study of the amounts of integer points in the domains on plane. In general form the theorem was proved by J. Van der Corput, (on the recent results connected with the Van der Corput theorem one can read at ). In every one of the above-mentioned works, some restrictions on the functions φ ( x ) {\displaystyle \varphi (x)} and f ( x ) {\displaystyle f(x)} were imposed. With convenient (for applications) restrictions on φ ( x ) {\displaystyle \varphi (x)} and f ( x ) , {\displaystyle f(x),} the theorem was proved by A. A. Karatsuba in (see also,). == Certain notations == [1]. For B > 0 , B → + ∞ , {\displaystyle B>0,B\to +\infty ,} or B → 0 , {\displaystyle B\to 0,} the record 1 ≪ A B ≪ 1 {\displaystyle 1\ll {\frac {A}{B}}\ll 1} means that there are the constants C 1 > 0 {\displaystyle C_{1}>0} and C 2 > 0 , {\displaystyle C_{2}>0,} such that C 1 ≤ | A | B ≤ C 2 . {\displaystyle C_{1}\leq {\frac {|A|}{B}}\leq C_{2}.} [2]. For a real number α , {\displaystyle \alpha ,} the record ‖ α ‖ {\displaystyle \|\alpha \|} means that ‖ α ‖ = min ( { α } , 1 − { α } ) , {\displaystyle \|\alpha \|=\min(\{\alpha \},1-\{\alpha \}),} where { α } {\displaystyle \{\alpha \}} is the fractional part of α . {\displaystyle \alpha .} == ATS theorem == Let the real functions ƒ(x) and φ ( x ) {\displaystyle \varphi (x)} satisfy on the segment [a, b] the following conditions: 1) f ⁗ ( x ) {\displaystyle f''''(x)} and φ ″ ( x ) {\displaystyle \varphi ''(x)} are continuous; 2) there exist numbers H , {\displaystyle H,} U {\displaystyle U} and V {\displaystyle V} such that H > 0 , 1 ≪ U ≪ V , 0 < b − a ≤ V {\displaystyle H>0,\qquad 1\ll U\ll V,\qquad 0<b-a\leq V} and 1 U ≪ f ″ ( x ) ≪ 1 U , φ ( x ) ≪ H , f ‴ ( x ) ≪ 1 U V , φ ′ ( x ) ≪ H V , f ⁗ ( x ) ≪ 1 U V 2 , φ ″ ( x ) ≪ H V 2 . {\displaystyle {\begin{array}{rc}{\frac {1}{U}}\ll f''(x)\ll {\frac {1}{U}}\ ,&\varphi (x)\ll H,\\\\f'''(x)\ll {\frac {1}{UV}}\ ,&\varphi '(x)\ll {\frac {H}{V}},\\\\f''''(x)\ll {\frac {1}{UV^{2}}}\ ,&\varphi ''(x)\ll {\frac {H}{V^{2}}}.\\\\\end{array}}} Then, if we define the numbers x μ {\displaystyle x_{\mu }} from the equation f ′ ( x μ ) = μ , {\displaystyle f'(x_{\mu })=\mu ,} we have ∑ a < μ ≤ b φ ( μ ) e 2 π i f ( μ ) = ∑ f ′ ( a ) ≤ μ ≤ f ′ ( b ) C ( μ ) Z ( μ ) + R , {\displaystyle \sum _{a<\mu \leq b}\varphi (\mu )e^{2\pi if(\mu )}=\sum _{f'(a)\leq \mu \leq f'(b)}C(\mu )Z(\mu )+R,} where R = O ( H U b − a + H T a + H T b + H log ( f ′ ( b ) − f ′ ( a ) + 2 ) ) ; {\displaystyle R=O\left({\frac {HU}{b-a}}+HT_{a}+HT_{b}+H\log \left(f'(b)-f'(a)+2\right)\right);} T j = { 0 , if f ′ ( j ) is an integer ; min ( 1 ‖ f ′ ( j ) ‖ , U ) , if ‖ f ′ ( j ) ‖ ≠ 0 ; {\displaystyle T_{j}={\begin{cases}0,&{\text{if }}f'(j){\text{ is an integer}};\\\min \left({\frac {1}{\|f'(j)\|}},{\sqrt {U}}\right),&{\text{if }}\|f'(j)\|\neq 0;\\\end{cases}}} j = a , b ; {\displaystyle j=a,b;} C ( μ ) = { 1 , if f ′ ( a ) < μ < f ′ ( b ) ; 1 2 , if μ = f ′ ( a ) or μ = f ′ ( b ) ; {\displaystyle C(\mu )={\begin{cases}1,&{\text{if }}f'(a)<\mu <f'(b);\\{\frac {1}{2}},&{\text{if }}\mu =f'(a){\text{ or }}\mu =f'(b);\\\end{cases}}} Z ( μ ) = 1 + i 2 φ ( x μ ) f ″ ( x μ ) e 2 π i ( f ( x μ ) − μ x μ ) . {\displaystyle Z(\mu )={\frac {1+i}{\sqrt {2}}}{\frac {\varphi (x_{\mu })}{\sqrt {f''(x_{\mu })}}}e^{2\pi i(f(x_{\mu })-\mu x_{\mu })}\ .} The most simple variant of the formulated theorem is the statement, which is called in the literature the Van der Corput lemma. == Van der Corput lemma == Let f {\displaystyle f} be a real differentiable function in the interval ] a , b ] , {\displaystyle ]a,b],} moreover, inside of this interval, its derivative f ′ {\displaystyle f'} is a monotonic and a sign-preserving function, and for the constant δ {\displaystyle \delta } such that 0 < δ < 1 {\displaystyle 0<\delta <1} satisfies the inequality | f ′ | ≤ δ . {\displaystyle |f'|\leq \delta .} Then ∑ a < k ≤ b e 2 π i f ( k ) = ∫ a b e 2 π i f ( x ) d x + θ ( 3 + 2 δ 1 − δ ) , {\displaystyle \sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}dx+\theta \left(3+{\frac {2\delta }{1-\delta }}\right),} where | θ | ≤ 1. {\displaystyle |\theta |\leq 1.} == Remark == If the parameters a {\displaystyle a} and b {\displaystyle b} are integers, then it is possible to substitute the last relation by the following ones: ∑ a < k ≤ b e 2 π i f ( k ) = ∫ a b e 2 π i f ( x ) d x + 1 2 e 2 π i f ( b ) − 1 2 e 2 π i f ( a ) + θ 2 δ 1 − δ , {\displaystyle \sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}\,dx+{\frac {1}{2}}e^{2\pi if(b)}-{\frac {1}{2}}e^{2\pi if(a)}+\theta {\frac {2\delta }{1-\delta }},} where | θ | ≤ 1. {\displaystyle |\theta |\leq 1.} == Additional sources == On the applications of ATS to the problems of physics see: Karatsuba, Ekatherina A. (2004). "Approximation of sums of oscillating summands in certain physical problems". Journal of Mathematical Physics. 45 (11). AIP Publishing: 4310–4321. doi:10.1063/1.1797552. ISSN 0022-2488. Karatsuba, Ekatherina A. (2007-07-20). "On an approach to the study of the Jaynes–Cummings sum in quantum optics". Numerical Algorithms. 45 (1–4). Springer Science and Business Media LLC: 127–137. doi:10.1007/s11075-007-9070-x. ISSN 1017-1398. S2CID 13485016. Chassande-Mottin, Éric; Pai, Archana (2006-02-27). "Best chirplet chain: Near-optimal detection of gravitational wave chirps". Physical Review D. 73 (4). American Physical Society (APS): 042003. arXiv:gr-qc/0512137. doi:10.1103/physrevd.73.042003. hdl:11858/00-001M-0000-0013-4BBD-B. ISSN 1550-7998. S2CID 56344234. Fleischhauer, M.; Schleich, W. P. (1993-05-01). "Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model". Physical Review A. 47 (5). American Physical Society (APS): 4258–4269. doi:10.1103/physreva.47.4258. ISSN 1050-2947. PMID 9909432. == Notes == |
Wikipedia:AWM–Microsoft Research Prize in Algebra and Number Theory#0 | The AWM–Microsoft Research Prize in Algebra and Number Theory and is a prize given every other year by the Association for Women in Mathematics to an outstanding young female researcher in algebra or number theory. It was funded in 2012 by Microsoft Research and first issued in 2014. == Winners == Sophie Morel (2014), for her research in number theory, particularly her contributions to the Langlands program, an application of her results on weighted cohomology, and a new proof of Brenti's combinatorial formula for Kazhdan-Lusztig polynomials. Lauren Williams (2016), for her research in algebraic combinatorics, particularly her contributions on the totally nonnegative Grassmannian, her work on cluster algebras, and her proof (with Musiker and Schiffler) of the famous Laurent positivity conjecture. Melanie Wood (2018), for her research in number theory and algebraic geometry, particularly her contributions in arithmetic statistics and tropical geometry, as well as her work with Ravi Vakil on the limiting behavior of natural families of varieties. Melody Chan (2020), in recognition of her advances at the interface between algebraic geometry and combinatorics. Jennifer Balakrishnan (2022), in recognition of her advances in computing rational points on algebraic curves over number fields. Yunqing Tang (2024), for "work in arithmetic geometry, including results on the Grothendieck–Katz p {\displaystyle p} -curvature conjecture, a conjecture of Ogus on algebraicity of cycles, arithmetic intersection theory, and the unbounded denominators conjecture of Atkin and Swinnerton-Dyer" == See also == List of awards honoring women List of mathematics awards == References == == External links == AWM–Microsoft Research Prize, Association for Women in Mathematics |
Wikipedia:Abacus#0 | An abacus (pl. abaci or abacuses), also called a counting frame, is a hand-operated calculating tool which was used from ancient times in the ancient Near East, Europe, China, and Russia, until the adoption of the Hindu–Arabic numeral system. An abacus consists of a two-dimensional array of slidable beads (or similar objects). In their earliest designs, the beads could be loose on a flat surface or sliding in grooves. Later the beads were made to slide on rods and built into a frame, allowing faster manipulation. Each rod typically represents one digit of a multi-digit number laid out using a positional numeral system such as base ten (though some cultures used different numerical bases). Roman and East Asian abacuses use a system resembling bi-quinary coded decimal, with a top deck (containing one or two beads) representing fives and a bottom deck (containing four or five beads) representing ones. Natural numbers are normally used, but some allow simple fractional components (e.g. 1⁄2, 1⁄4, and 1⁄12 in Roman abacus), and a decimal point can be imagined for fixed-point arithmetic. Any particular abacus design supports multiple methods to perform calculations, including addition, subtraction, multiplication, division, and square and cube roots. The beads are first arranged to represent a number, then are manipulated to perform a mathematical operation with another number, and their final position can be read as the result (or can be used as the starting number for subsequent operations). In the ancient world, abacuses were a practical calculating tool. It was widely used in Europe as late as the 17th century, but fell out of use with the rise of decimal notation and algorismic methods. Although calculators and computers are commonly used today instead of abacuses, abacuses remain in everyday use in some countries. The abacus has an advantage of not requiring a writing implement and paper (needed for algorism) or an electric power source. Merchants, traders, and clerks in some parts of Eastern Europe, Russia, China, and Africa use abacuses. The abacus remains in common use as a scoring system in non-electronic table games. Others may use an abacus due to visual impairment that prevents the use of a calculator. The abacus is still used to teach the fundamentals of mathematics to children in many countries such as Japan and China. == Etymology == The word abacus dates to at least 1387 AD when a Middle English work borrowed the word from Latin that described a sandboard abacus. The Latin word is derived from ancient Greek ἄβαξ (abax) which means something without a base, and colloquially, any piece of rectangular material. Alternatively, without reference to ancient texts on etymology, it has been suggested that it means "a square tablet strewn with dust", or "drawing-board covered with dust (for the use of mathematics)" (the exact shape of the Latin perhaps reflects the genitive form of the Greek word, ἄβακoς (abakos)). While the table strewn with dust definition is popular, some argue evidence is insufficient for that conclusion. Greek ἄβαξ probably borrowed from a Northwest Semitic language like Phoenician, evidenced by a cognate with the Hebrew word ʾābāq (אבק), or "dust" (in the post-Biblical sense "sand used as a writing surface"). Both abacuses and abaci are used as plurals. The user of an abacus is called an abacist. == History == === Mesopotamia === The Sumerian abacus appeared between 2700 and 2300 BC. It held a table of successive columns which delimited the successive orders of magnitude of their sexagesimal (base 60) number system. Some scholars point to a character in Babylonian cuneiform that may have been derived from a representation of the abacus. It is the belief of Old Babylonian scholars, such as Ettore Carruccio, that Old Babylonians "seem to have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations". === Egypt === Greek historian Herodotus mentioned the abacus in Ancient Egypt. He wrote that the Egyptians manipulated the pebbles from right to left, opposite in direction to the Greek left-to-right method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument are yet to be discovered. === Persia === At around 600 BC, Persians first began to use the abacus, during the Achaemenid Empire. Under the Parthian, Sassanian, and Iranian empires, scholars concentrated on exchanging knowledge and inventions with the countries around them – India, China, and the Roman Empire – which is how the abacus may have been exported to other countries. === Greece === The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC. Demosthenes (384–322 BC) complained that the need to use pebbles for calculations was too difficult. A play by Alexis from the 4th century BC mentions an abacus and pebbles for accounting, and both Diogenes and Polybius use the abacus as a metaphor for human behavior, stating "that men that sometimes stood for more and sometimes for less" like the pebbles on an abacus. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations. This Greek abacus was used in Achaemenid Persia, the Etruscan civilization, Ancient Rome, and the Western Christian world until the French Revolution. The Salamis Tablet, found on the Greek island Salamis in 1846 AD, dates to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble 149 cm (59 in) in length, 75 cm (30 in) wide, and 4.5 cm (2 in) thick, on which are 5 groups of markings. In the tablet's center is a set of 5 parallel lines equally divided by a vertical line, capped with a semicircle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these lines are marked with a cross where they intersect with the vertical line. Also from this time frame, the Darius Vase was unearthed in 1851. It was covered with pictures, including a "treasurer" holding a wax tablet in one hand while manipulating counters on a table with the other. === Rome === The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles (Latin: calculi) were used. Marked lines indicated units, fives, tens, etc. as in the Roman numeral system. Writing in the 1st century BC, Horace refers to the wax abacus, a board covered with a thin layer of black wax on which columns and figures were inscribed using a stylus. One example of archaeological evidence of the Roman abacus, shown nearby in reconstruction, dates to the 1st century AD. It has eight long grooves containing up to five beads in each and eight shorter grooves having either one or no beads in each. The groove marked I indicates units, X tens, and so on up to millions. The beads in the shorter grooves denote fives (five units, five tens, etc.) resembling a bi-quinary coded decimal system related to the Roman numerals. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions). === Medieval Europe === The Roman system of 'counter casting' was used widely in medieval Europe, and persisted in limited use into the nineteenth century. Wealthy abacists used decorative minted counters, called jetons. Due to Pope Sylvester II's reintroduction of the abacus with modifications, it became widely used in Europe again during the 11th century It used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster and was more easily moved. === China === The earliest known written documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, also known as the suanpan (算盤/算盘, lit. "calculating tray"), comes in various lengths and widths, depending on the operator. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads each in the bottom one, to represent numbers in a bi-quinary coded decimal-like system. The beads are usually rounded and made of hardwood. The beads are counted by moving them up or down towards the beam; beads moved toward the beam are counted, while those moved away from it are not. One of the top beads is 5, while one of the bottom beads is 1. Each rod has a number under it, showing the place value. The suanpan can be reset to the starting position instantly by a quick movement along the horizontal axis to spin all the beads away from the horizontal beam at the center. The prototype of the Chinese abacus appeared during the Han dynasty, and the beads are oval. The Song dynasty and earlier used the 1:4 type or four-beads abacus similar to the modern abacus including the shape of the beads commonly known as Japanese-style abacus. In the early Ming dynasty, the abacus began to appear in a 1:5 ratio. The upper deck had one bead and the bottom had five beads. In the late Ming dynasty, the abacus styles appeared in a 2:5 ratio. The upper deck had two beads, and the bottom had five. Various calculation techniques were devised for Suanpan enabling efficient calculations. Some schools teach students how to use it. In the long scroll Along the River During the Qingming Festival painted by Zhang Zeduan during the Song dynasty (960–1297), a suanpan is clearly visible beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao). The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, given evidence of a trade relationship between the Roman Empire and China. However, no direct connection has been demonstrated, and the similarity of the abacuses may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Korean and Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. Incidentally, this ancient Chinese calculation system 市用制 (Shì yòng zhì) allows use with a hexadecimal numeral system (or any base up to 18) which is used for traditional Chinese measures of weight [(jīn (斤) and liǎng (兩)]. (Instead of running on wires as in the Chinese, Korean, and Japanese models, the Roman model used grooves, presumably making arithmetic calculations much slower). Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of zero as a placeholder. The zero was probably introduced to the Chinese in the Tang dynasty (618–907) when travel in the Indian Ocean and the Middle East would have provided direct contact with India, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians. === India === The Abhidharmakośabhāṣya of Vasubandhu (316–396), a Sanskrit work on Buddhist philosophy, says that the second-century CE philosopher Vasumitra said that "placing a wick (Sanskrit vartikā) on the number one (ekāṅka) means it is a one while placing the wick on the number hundred means it is called a hundred, and on the number one thousand means it is a thousand". It is unclear exactly what this arrangement may have been. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the abacus. Hindu texts used the term śūnya (zero) to indicate the empty column on the abacus. === Japan === In Japan, the abacus is called soroban (算盤, そろばん, lit. "counting tray"). It was imported from China in the 14th century. It was probably in use by the working class a century or more before the ruling class adopted it, as the class structure obstructed such changes. The 1:4 abacus, which removes the seldom-used second and fifth bead, became popular in the 1940s. Today's Japanese abacus is a 1:4 type, four-bead abacus, introduced from China in the Muromachi era. It adopts the form of the upper deck one bead and the bottom four beads. The top bead on the upper deck was equal to five and the bottom one is similar to the Chinese or Korean abacus, and the decimal number can be expressed, so the abacus is designed as a 1:4 device. The beads are always in the shape of a diamond. The quotient division is generally used instead of the division method; at the same time, in order to make the multiplication and division digits consistently use the division multiplication. Later, Japan had a 3:5 abacus called 天三算盤, which is now in the Ize Rongji collection of Shansi Village in Yamagata City. Japan also used a 2:5 type abacus. The four-bead abacus spread, and became common around the world. Improvements to the Japanese abacus arose in various places. In China, an abacus with an aluminium frame and plastic beads has been used. The file is next to the four beads, and pressing the "clearing" button puts the upper bead in the upper position, and the lower bead in the lower position. The abacus is still manufactured in Japan, despite the proliferation, practicality, and affordability of pocket electronic calculators. The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation. Using visual imagery, one can complete a calculation as quickly as with a physical instrument. === Korea === The Chinese abacus migrated from China to Korea around 1400 AD. Koreans call it jupan (주판), supan (수판) or jusan (주산). The four-beads abacus (1:4) was introduced during the Goryeo Dynasty. The 5:1 abacus was introduced to Korea from China during the Ming Dynasty. === Native America === Some sources mention the use of an abacus called a nepohualtzintzin in ancient Aztec culture. This Mesoamerican abacus used a 5-digit base-20 system. The word Nepōhualtzintzin Nahuatl pronunciation: [nepoːwaɬˈt͡sint͡sin] comes from Nahuatl, formed by the roots; Ne – personal -; pōhual or pōhualli Nahuatl pronunciation: [ˈpoːwalːi] – the account -; and tzintzin Nahuatl pronunciation: [ˈt͡sint͡sin] – small similar elements. Its complete meaning was taken as: counting with small similar elements. Its use was taught in the Calmecac to the temalpouhqueh Nahuatl pronunciation: [temaɬˈpoʍkeʔ], who were students dedicated to taking the accounts of skies, from childhood. The Nepōhualtzintzin was divided into two main parts separated by a bar or intermediate cord. In the left part were four beads. Beads in the first row have unitary values (1, 2, 3, and 4), and on the right side, three beads had values of 5, 10, and 15, respectively. In order to know the value of the respective beads of the upper rows, it is enough to multiply by 20 (by each row), the value of the corresponding count in the first row. The device featured 13 rows with 7 beads, 91 in total. This was a basic number for this culture. It had a close relation to natural phenomena, the underworld, and the cycles of the heavens. One Nepōhualtzintzin (91) represented the number of days that a season of the year lasts, two Nepōhualtzitzin (182) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin (273) is the number of days of a baby's gestation, and four Nepōhualtzintzin (364) completed a cycle and approximated one year. When translated into modern computer arithmetic, the Nepōhualtzintzin amounted to the rank from 10 to 18 in floating point, which precisely calculated large and small amounts, although round off was not allowed. The rediscovery of the Nepōhualtzintzin was due to the Mexican engineer David Esparza Hidalgo, who in his travels throughout Mexico found diverse engravings and paintings of this instrument and reconstructed several of them in gold, jade, encrustations of shell, etc. Very old Nepōhualtzintzin are attributed to the Olmec culture, and some bracelets of Mayan origin, as well as a diversity of forms and materials in other cultures. Sanchez wrote in Arithmetic in Maya that another base 5, base 4 abacus had been found in the Yucatán Peninsula that also computed calendar data. This was a finger abacus, on one hand, 0, 1, 2, 3, and 4 were used; and on the other hand 0, 1, 2, and 3 were used. Note the use of zero at the beginning and end of the two cycles. The quipu of the Incas was a system of colored knotted cords used to record numerical data, like advanced tally sticks – but not used to perform calculations. Calculations were carried out using a yupana (Quechua for "counting tool"; see figure) which was still in use after the conquest of Peru. The working principle of a yupana is unknown, but in 2001 Italian mathematician De Pasquale proposed an explanation. By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1, 1, 2, 3, 5 and powers of 10, 20, and 40 as place values for the different fields in the instrument. Using the Fibonacci sequence would keep the number of grains within any one field at a minimum. === Russia === The Russian abacus, the schoty (Russian: счёты, plural from Russian: счёт, counting), usually has a single slanted deck, with ten beads on each wire (except one wire with four beads for quarter-ruble fractions). 4-bead wire was introduced for quarter-kopeks, which were minted until 1916. The Russian abacus is used vertically, with each wire running horizontally. The wires are usually bowed upward in the center, to keep the beads pinned to either side. It is cleared when all the beads are moved to the right. During manipulation, beads are moved to the left. For easy viewing, the middle 2 beads on each wire (the 5th and 6th bead) usually are of a different color from the other eight. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color. The Russian abacus was in use in shops and markets throughout the former Soviet Union, and its usage was taught in most schools until the 1990s. Even the 1874 invention of mechanical calculator, Odhner arithmometer, had not replaced them in Russia. According to Yakov Perelman, some businessmen attempting to import calculators into the Russian Empire were known to leave in despair after watching a skilled abacus operator. Likewise, the mass production of Felix arithmometers since 1924 did not significantly reduce abacus use in the Soviet Union. The Russian abacus began to lose popularity only after the mass production of domestic microcalculators in 1974. The Russian abacus was brought to France around 1820 by mathematician Jean-Victor Poncelet, who had served in Napoleon's army and had been a prisoner of war in Russia. To Poncelet's French contemporaries, it was something new. Poncelet used it, not for any applied purpose, but as a teaching and demonstration aid. The Turks and the Armenian people used abacuses similar to the Russian schoty. It was named a coulba by the Turks and a choreb by the Armenians. == School abacus == Around the world, abacuses have been used in pre-schools and elementary schools as an aid in teaching the numeral system and arithmetic. In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame is common (see image). Each bead represents one unit (e.g. 74 can be represented by shifting all beads on 7 wires and 4 beads on the 8th wire, so numbers up to 100 may be represented). In the bead frame shown, the gap between the 5th and 6th wire, corresponding to the color change between the 5th and the 6th bead on each wire, suggests the latter use. Teaching multiplication, e.g. 6 times 7, may be represented by shifting 7 beads on 6 wires. The red-and-white abacus is used in contemporary primary schools for a wide range of number-related lessons. The twenty bead version, referred to by its Dutch name rekenrek ("calculating frame"), is often used, either on a string of beads or on a rigid framework. == Neurological analysis == Learning how to calculate with the abacus may improve capacity for mental calculation. Abacus-based mental calculation (AMC), which was derived from the abacus, is the act of performing calculations, including addition, subtraction, multiplication, and division, in the mind by manipulating an imagined abacus. It is a high-level cognitive skill that runs calculations with an effective algorithm. People doing long-term AMC training show higher numerical memory capacity and experience more effectively connected neural pathways. They are able to retrieve memory to deal with complex processes. AMC involves both visuospatial and visuomotor processing that generate the visual abacus and move the imaginary beads. Since it only requires that the final position of beads be remembered, it takes less memory and less computation time. == Renaissance abacuses == == Binary abacus == The binary abacus is used to explain how computers manipulate numbers. The abacus shows how numbers, letters, and signs can be stored in a binary system on a computer, or via ASCII. The device consists of beads on parallel wires arranged in three rows; each bead represents a switch which can be either "on" or "off". == Visually impaired users == An adapted abacus, invented by Tim Cranmer, and called a Cranmer abacus is commonly used by visually impaired users. A piece of soft fabric or rubber is placed behind the beads, keeping them in place while the users manipulate them. The device is then used to perform the mathematical functions of multiplication, division, addition, subtraction, square root, and cube root. Although blind students have benefited from talking calculators, the abacus is often taught to these students in early grades. Blind students can also complete mathematical assignments using a braille-writer and Nemeth code (a type of braille code for mathematics) but large multiplication and long division problems are tedious. The abacus gives these students a tool to compute mathematical problems that equals the speed and mathematical knowledge required by their sighted peers using pencil and paper. Many blind people find this number machine a useful tool throughout life. == See also == Chinese Zhusuan Chisanbop Logical abacus Napier's bones Sand table Slide rule == Notes == == Footnotes == == References == == Further reading == == External links == Texts on Wikisource: "Abacus", from A Dictionary of Greek and Roman Antiquities, 3rd ed., 1890. "Abacus" . Encyclopædia Britannica. Vol. I (9th ed.). 1878. p. 4. "Abacus". Encyclopædia Britannica (11th ed.). 1911. === Tutorials === Heffelfinger, Totton & Gary Flom, Abacus: Mystery of the Bead - an Abacus Manual Min Multimedia Stephenson, Stephen Kent (2009), How to use a Counting Board Abacus === History === Esaulov, Vladimir (2019), History of Abacus and Ancient Computing The Abacus: a Brief History === Curiosities === Schreiber, Michael (2007), Abacus, The Wolfram Demonstrations Project Abacus in Various Number Systems at cut-the-knot Java applet of Chinese, Japanese and Russian abaci An atomic-scale abacus Examples of Abaci Aztex Abacus Indian Abacus Abacus Course |
Wikipedia:Abbas Bahri#0 | Abbas Bahri (1 January 1955 – 10 January 2016) was a Tunisian mathematician. He was the winner of the Fermat Prize and the Langevin Prize in mathematics. He was a professor of mathematics at Rutgers University. He mainly studied the calculus of variations, partial differential equations, and differential geometry. He introduced the method of the critical points at infinity, which is a fundamental step in the calculus of variations. == Biography == Bahri received his secondary education in Tunisia and higher education in France. He attended the École Normale Superieure in Paris in 1974, the first Tunisian to do so. In 1981, he completed his PhD from Pierre-and-Marie-Curie University. His dissertation advisor was the French mathematician Haïm Brezis. Afterwards, he was a visiting scientist at the University of Chicago. In October 1981, Bahri became a lecturer in mathematics at the University of Tunis. He taught as a lecturer at the École Polytechnique from 1984 to 1993. In 1988, he became a tenured professor at Rutgers University. At Rutgers, he was director of the Center for Nonlinear Analysis from 1988 to 2002. === Personal life === He married Diana Nunziante on 20 June 1991. His wife is from Italy and they had four children. On 10 January 2016, he died following a long illness at the age of 61. == Awards == In 1989, Bahri won the Fermat Prize for Mathematics, jointly with Kenneth Alan Ribet, for his introduction of new methods in the calculus of variations. == Works == Pseudo-orbits of contact forms (1988) Critical Points at Infinity in Some Variational Problems (1989) Classical and Quantic Periodic Motions of Multiply Polarized Spin-Particles (1998) Flow lines and algebraic invariants in contact form geometry (2003) Recent progress in conformal geometry with Yongzhong Xu (2007) === Selected publications === Bahri, Abbas (August 2009). "Variations at infinity in contact form geometry". Journal of Fixed Point Theory & Its Applications. 5 (2): 265–289. doi:10.1007/s11784-009-0102-0. S2CID 120456000. Bahri, Abbas; Taimonov, Iskander A. (July 1998). "Periodic Orbits in Magnetic Fields and Ricci Curvature of Lagrangian Systems" (PDF). Transactions of the American Mathematical Society. 350 (7): 2697–2717. arXiv:dg-ga/9511016. doi:10.1090/s0002-9947-98-02108-4. S2CID 15064503. Retrieved 11 July 2014. == References == == External links == Official Rutgers website |
Wikipedia:Abd as-Salam al-Alami#0 | Abd as-Salam ibn Mohammed ibn Ahmed al-Hasani al-Alami al-Fasi (Arabic: عبدالسلام العلمي) (1834-1895) was a scientist from Fes. He was an expert in the field of astronomy, mathematics and medicine. Al-Alami was the author of several books in these fields and the designer of solar instruments. == References == == External links == Clifford Edmund Bosworth, The Encyclopaedia of Islam: Supplement, Volume 12, p. 10 [1] (Retrieved August 2, 2010) |
Wikipedia:Abdel Fattah al-Maghrabi#0 | Abdel Fattah al-Maghrabi (Arabic: عبد الفتاح المغربي ; 1898 – 1985) was a Sudanese official and statesman. He served as a member of the collective body at the helm of the Sudanese state, the First Sudanese Sovereignty Council, from 1955 to 1958. == Biography == === Early life and education === Abdel Fattah Muhammad al-Maghrabi was born in 1898. Abdelfattah el Maghrabi's father, Mohamed Mustafa, held the position of Chief Clerk of Dongola Province, which was bestowed upon him by the Mahdi. In 1889, following his forces' defeat by General Grenfell at Battle of Toski, the Khalifa, who succeeded the Mahdi, sought out individuals to blame and subsequently imprisoned Mohamed Mustafa. However, Mohamed Mustafa managed to secure his release by composing a poem consisting of 40 verses that praised the Khalifa. al-Maghrabi studied at American University in Beirut as part of the first student delegation to study outside Sudan, and graduated with a PhD in mathematics. He then worked as a mathematics lecturer at Gordon Memorial College after his graduation. === Political career === al-Maghrabi was appointed in 1951 as the only member of the opposition in the Legislative Assembly that was discussing the matter of the country’s constitution. After independence, al-Maghrabi became a member of the first Sudanese Sovereignty Council from 26 December 1955 to 17 November 1958, the head of the state's five-man supreme council. The prime minister was Ismail al-Azhari until 5 July 1956 followed by Abdallah Khalil until 17 November 1958. The First Sudanese Sovereignty Council ended on 17 November 1958 when General Ibrahim Abboud seized power in a military coup. Ibrahim Abboud assumed the presidency, and the council was dissolved, leading to a change in Sudan's governance structure from a parliamentary system to military rule. === Personal life and death === al-Maghrabi married the Phillippa Maghrabi in 1937 who was a British nurse working for the Sudanese Health Services Authority. Between 1937 and 1948, He and Phillippa resided in Gereif, which was located five miles outside of Khartoum along the banks of the Blue Nile. During their time there, Phillippa undertook the role of an informal district nurse in the neighboring villages. She wrote a memoirs documenting the history of Khartoum in a book called Early days at Gereif. al-Maghribi died in 1985 in Newton Abbot in Devon, England. == References == |
Wikipedia:Abduhamid Juraev#0 | Abduhamid Juraev (10 October 1932 – 5 June 2005) Isfara, Tajikistan was a Tajik mathematician. He published many articles and books. == References == Iraj Bashiri, Prominent Tajik Figures of the Twentieth Century, International Borbad Foundation, Academy of Sciences of Tajikistan, Dushanbe, 2003. == External links == Books by Abduhamid Juraev |
Wikipedia:Abdul Hakim (writer)#0 | Abdul Hakim Haqqani (Pashto: عبد الحكيم حقاني Pashto pronunciation: [ˈabdʊl haˈkim haqɑˈni]; born 1967), also known as Abdul Hakim Ishaqzai (Pashto: عبد الحكيم اسحاقزى Pashto pronunciation: [ˈabdʊl haˈkim ɪshaqˈzai]), is an Afghan Islamic scholar and writer who has been the chief justice of Afghanistan since 2021 in the internationally unrecognized Taliban regime. He has also served as chief justice of the Supreme Court in the 1996–2001 Islamic Emirate of Afghanistan. He was the chairman of the Taliban negotiation team in the Qatar office. He is one of the founding members of the Taliban and was a close associate of the late leader Mullah Mohammed Omar. == Early life == He was born to Mawlawi Khudaidad in 1967 in the Panjwayi District of Kandahar Province, Afghanistan. He graduated from Darul Uloom Haqqania, a Deobandi Islamic seminary (darul uloom), in Pakistan, and taught there. == Career == === Teaching === Apart from teaching at the Darul Uloom Haqqania, until recently he also ran his own Islamic seminary or madrasa in the Ishaqabad area of Quetta, in Pakistan’s Balochistan province. === Judiciary === During the rule of the first Islamic Emirate, in addition to teaching, he also served in the Appellate Court and at the Central Dar ul-Ifta. Following the appointment of Hibatullah Akhundzada as Supreme Leader, Ishaqzai was appointed Chief Justice. === Diplomacy === In September 2020, he was appointed the Taliban's chief negotiator for peace talks in Qatar with the government of Afghanistan, replacing Sher Mohammad Abbas Stanikzai, who became his deputy in the 21-member negotiating team. == Controversies == === International sanctions due to alleged violation of women's rights === On 20 July 2023, Hakim Haqqani was sanctioned by the EU due to his instrumental role as Chief Justice of the Supreme Court in implementing policies and spreading ideological teachings aimed at creating and justifying gender-based repressions against women in Afghanistan. On January 23, 2025, the International Criminal Court's chief prosecutor, Karim Khan, announced the submission of arrest warrant applications for Taliban leaders, including supreme leader Haibatullah Akhundzada and Chief Justice Abdul Hakim Haqqani. He is accused of crimes against humanity, specifically the persecution of women and girls, since the Taliban's return to power in August 2021. The charges highlight severe restrictions imposed on Afghan females, encompassing bans on education, employment, and public participation. == Books == Specializing in Islamic jurisprudence, especially its justice system, Hakim Haqqani has written books on various subjects which have been translated into many languages. == References == |
Wikipedia:Abdul Jerri#0 | Abdul Jabbar Hassoon Jerri (Arabic: عبد الجبار حسون جري) is an Iraqi American mathematician, most recognized for his contributions to Shannon Sampling Theory, Its Generalizations, Error Analysis, and Historical Reviews, and in particular his establishment in 2002 of the journal Sampling Theory in Signal and Image Processing (STSIP-ISSN 1530-6429) with over thirty top international experts as its editors, besides establishing its Sampling Publishing, also his contribution to the general understanding of the Gibbs Phenomenon, where he wrote the first book ever on the subject, published by Springer - Verlag, then he followed it by editing another book on Advances in Gibbs Phenomenon published by Sampling Publishing. == Academic life == Jerri earned a B.Sc. in physics with honors at the University of Baghdad (1955) and M.S. in physics from Illinois Institute of Technology (1960) in Chicago where he continued to work within the research group (1960–63) in Reactor Physics and Radiation streaming in Shelter Entrance ways. He also earned a Ph.D. in mathematics from Oregon State University in 1967 with the dissertation title On Extensions of the Generalized Sampling Theorem. Jerri commenced his tenure with the faculty of the Department of Mathematics and Computer Science at Clarkson University in Potsdam, NY (1967), where he worked from 1967 until his retirement as professor emeritus in 2002. Jerri's career includes visiting positions at the American University in Cairo, where he established the Study Programs in Mathematics and Computer Science (1972–74). He was also the Director of the Graduate Mathematics Study Program at Kuwait University (1978–80). === Awards === Jerri is a double-awarded Fulbright Scholar at the Sultan Qaboos University in Muscat, Oman(1997), and a second time, at the Yarmouk University in Irbid, Jordan (2001). He is the Founding Executive Editor of Sampling Theory In Signal And Image Processing (STSIP) - An International Journal., and the owner of Sampling Publishing. In 1995, Jerri was one of the few researchers who helped establish the SAMPTA (Sampling Theory and Applications) Workshops, that holds a workshop in different country every two years (Starting in Latvia in 1995, then Portugal, Norway, the US, Austria, Turkey, Greece, France, Singapore, Germany, the US, and Estonia in 2017.) === Research === He is the author of several other popular books: Introduction to Integral Equations with Applications, accompanied by a Students Solution Manual: Sampling Publishing, Introduction to Wavelets accompanied by a Students Solution Manual( The latter Manual was co-authored with Prof Masaru Kamada); Sampling Publishing. Other books include Integral and Discrete Transforms with Applications and Error Analysis: Marcel Dekker, and Linear Difference Equations with Discrete Transform Methods:Sp ringer-Verlag. He had published over forty papers, with numerous lectures on his areas of research interest nationally and internationally. Jerri's main research interests include the areas of Integral and Discrete Transforms, Sampling Expansion and its Applications, History and Error Analysis, the Gibbs Phenomenon, Transform-Iterative Methods for Nonlinear Problems, and Operational Sum Methods for Difference Equations. In his first workshop, he introduced the subject of Shannon Sampling Theory in four-one hour lectures. In 1997 workshop in Aveiro, Portugal, the Proceedings of the workshop was dedicated to jerri 65th birthday. Presently, he is working on writing a Tutorial review paper on the subject "Multidimensional sampling in Signal Processing. He is dedicating this paper for the occasion of the CENTENNIAL of the American Scientist of the century, the father of Information Theory, Claude Elwood Shannon. == References == |
Wikipedia:Abdulla A'zamov#0 | Abdulla Aʼzamovich Aʼzamov (born April 21, 1947) is a Soviet and Uzbek doctor of physical and mathematical sciences (1987), professor, member of the Academy of Sciences of Uzbekistan (2013), president of the Uzbek Mathematical Society (2013), vice-president of TWMS (Turkic World Mathematical Society) (2017). He is the recipient of the "People's Education Hero" medal, the International Babur Award (2015), the "Order of Labor Glory" (2016) and the honorary title of "Scientist of the Republic of Uzbekistan". He was a member of the jury of the republic mathematics olympiad since 1973, and a member of the jury of the former union mathematics olympiad from 1989 to 1991. He defended his candidate dissertation at the Council of the Faculty of Mathematics of Tashkent State University in 1974, and his doctoral dissertation at the Special Council of the Faculty of Mathematics and Mechanics of St. Petersburg State University in 1987. Aʼzamov was awarded the title of associate professor in 1981 and professor in 1989. He has been a member of the Uzbekistan Writers' Union since 2013. He became an academician of the Academy of Sciences of the Republic of Uzbekistan in 2017. He worked as a senior editor at the "Uzbek Soviet Encyclopedia" main editorial office from 1972 to 1975 (on a part-time basis), as a deputy editor-in-chief of the "Tafakkur" journal from 1999 to 2000, and as a responsible employee at the Office of the President of the Republic of Uzbekistan from 2000 to 2011. == Biography == Abdulla Aʼzamov was born on April 21, 1947, in Baliqchi District of Andijan Region. He is an Uzbek by nationality. He studied at the 1st secondary school of his native district from 1953 to 1964. He participated in the external olympiad organized by the "Young Leninist" newspaper in 1963. He graduated from secondary school with a gold medal in 1964 and entered the Faculty of Mechanics and Mathematics of Tashkent State University (now Mirzo Ulugbek National University of Uzbekistan). He was sent to study at the Moscow State University named after M. V. Lomonosov with a group of students in 1966. He graduated in 1970. == Activities == Abdulla Aʼzamov started lecturing on differential equations in 1971–1972 academic year. He worked as a senior editor at the "Uzbek Soviet Encyclopedia" main editorial office from 1972 to 1975 (on a part-time basis). He was a member of the jury of the Uzbek SSR mathematics olympiad since 1973, and a member of the jury of the former union mathematics olympiad from 1989 to 1991. He was admitted to the position of assistant at the Department of Mathematical Analysis of Tashkent State University, and worked as an associate professor, professor and head of the Department of Differential Equations from 1980 to 1993. He was appointed as the dean of the Faculty of Mathematics in 1992–1993. Aʼzamov was appointed as the rector of Namangan State University in 1993. He worked as the head of the department of secondary and vocational education at the Ministry of Higher and Secondary Specialized Education from 1998 to 1999, as the deputy editor-in-chief of the "Tafakkur" journal from 1999 to 2000, and as a responsible employee at the Office of the President of the Republic of Uzbekistan from 2000 to 2011. He defended his candidate dissertation at the Council of the Faculty of Mathematics of Tashkent State University in 1974, and his doctoral dissertation at the Special Council of the Faculty of Mathematics and Mechanics of St. Petersburg State University in 1987. He was awarded the title of associate professor in 1981 and professor in 1989. His work “Munojatnoma” is devoted to the scientific-historical analysis of Alisher Navoi's ghazals. He wrote the play “Where is Usmon Nosir?” during his tenure as the rector of Namangan State University and the work was staged at the regional theater. He led the Namangan branch of the Academy of Sciences of Uzbekistan from 1993 to 1997. Aʼzamov lectured at the National University of Uzbekistan and the Tashkent branch of Moscow State University. He became a member of the Uzbekistan Writers’ Union in 2013 and was elected as the president of the Uzbek Mathematical Society. He is the vice-president of the Turkic World Mathematical Society (2017) and an academician of the Academy of Sciences of the Republic of Uzbekistan since 2017. == Family == Aʼzamov has two sons and a daughter. == Books == Abdulla Aʼzamov (2018). Taqvim jadvali. Hijriy va milodiy taqvimlarda sanalarni kunma-kun o‘girish jadvallari. Toshkent: G‘afur G‘ulom. p. 436. ISBN 978-9943-03-992-6. Abdulla Aʼzamov (2017). Differential Equations and Dynamical Systems. Urganch.{{cite book}}: CS1 maint: location missing publisher (link) === Scientific articles === Abdulla Aʼzamov (2021). Existence and uniqueness theorems for the Pfaff equation with continuous coefficients. Abdulla Aʼzamov (2021). An existence theorem and an approximate solution method for the Pfaff equation with continuous coefficients. Abdulla Aʼzamov (2020). A pursuit-evasion differential game with slow pursuers on the edge graph of simplexes. Abdulla Aʼzamov (2020). Four-dimensional brusselator model with periodical solution. Abdulla Aʼzamov (2019). The pursuit-evasion game on the 1-skeleton graph of the regular polyhedron. Abdulla Aʼzamov (2019). On the Сhernous’ko time-optimal problem for the equation of heat conductivity in a rod. Abdulla Aʼzamov (2018). On generators of a matrix algebra and some of its subalgebras. == References == == Bibliography == Ummatov R (2022). Gʻaznai Namangon/ Encyclopedia (3 volume) 1st book/ Oltin odamlar. Toshkent: „Sharq“. p. 448. |
Wikipedia:Abe Sklar#0 | Abe Sklar (November 25, 1925 – October 30, 2020) was an American mathematician and a professor of applied mathematics at the Illinois Institute of Technology (Illinois Tech) and the inventor of copulas in probability theory. == Education and career == Sklar was born in Chicago to Jewish parents who immigrated to the United States from Ukraine. He attended Von Steuben High School and later enrolled at the University of Chicago in 1942, when he was only 16. Sklar went on to become a student of Tom M. Apostol at the California Institute of Technology, where he earned his Ph.D. in 1956. His students at IIT have included geometers Clark Kimberling and Marjorie Senechal. In 1959, Sklar introduced the notion of and the name of "copulas" into probability theory and proved the theorem that bears his name, Sklar's theorem. That is, that multivariate cumulative distribution functions can be expressed in terms of copulas. This representation of distribution functions, which is valid in any dimension and unique when the margins are continuous, is the basis of copula modeling, a widespread data analytical technique used in statistics; this representation is often termed Sklar's representation. Schweizer–Sklar t-norms are also named after Sklar and Berthold Schweizer, who studied them together in the early 1960s. == Bibliography == Golland, Louise; McGuinness, Brian; Sklar, Abe, eds. (1994). Karl Menger - Reminiscences of the Vienna Circle and the Mathematical Colloquium. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-2711-X. OCLC 30026523. Menger, Karl; Schweizer, Berthold; Sklar, Abe; Sigmund; Schmetterer, Leopold; Gruber, Peter M.; Hlawka, Edmund, eds. (2002). Selecta Mathematica. Volume 1. Vienna. ISBN 978-3-7091-6110-4. OCLC 1149922502.{{cite book}}: CS1 maint: location missing publisher (link) Menger, Karl; Schweizer, Berthold; Sklar, Abe; Sigmund, Karl, eds. (2003). Selecta mathematica. Volume 2. Wien; New York: Springer-Verlag. ISBN 978-3-211-83834-1. OCLC 492462474. Schweizer, Berthold; Sklar, Abe (2005). Probabilistic metric spaces. Mineola, N.Y. ISBN 978-0-486-14375-0. OCLC 873840651.{{cite book}}: CS1 maint: location missing publisher (link) == References == |
Wikipedia:Abel equation#0 | The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form f ( h ( x ) ) = h ( x + 1 ) {\displaystyle f(h(x))=h(x+1)} or α ( f ( x ) ) = α ( x ) + 1 {\displaystyle \alpha (f(x))=\alpha (x)+1} . The forms are equivalent when α is invertible. h or α control the iteration of f. == Equivalence == The second equation can be written α − 1 ( α ( f ( x ) ) ) = α − 1 ( α ( x ) + 1 ) . {\displaystyle \alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.} Taking x = α−1(y), the equation can be written f ( α − 1 ( y ) ) = α − 1 ( y + 1 ) . {\displaystyle f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.} For a known function f(x) , a problem is to solve the functional equation for the function α−1 ≡ h, possibly satisfying additional requirements, such as α−1(0) = 1. The change of variables sα(x) = Ψ(x), for a real parameter s, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) . The further change F(x) = exp(sα(x)) into Böttcher's equation, F(f(x)) = F(x)s. The Abel equation is a special case of (and easily generalizes to) the translation equation, ω ( ω ( x , u ) , v ) = ω ( x , u + v ) , {\displaystyle \omega (\omega (x,u),v)=\omega (x,u+v)~,} e.g., for ω ( x , 1 ) = f ( x ) {\displaystyle \omega (x,1)=f(x)} , ω ( x , u ) = α − 1 ( α ( x ) + u ) {\displaystyle \omega (x,u)=\alpha ^{-1}(\alpha (x)+u)} . (Observe ω(x,0) = x.) The Abel function α(x) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups). == History == Initially, the equation in the more general form was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis. In the case of a linear transfer function, the solution is expressible compactly. == Special cases == The equation of tetration is a special case of Abel's equation, with f = exp. In the case of an integer argument, the equation encodes a recurrent procedure, e.g., α ( f ( f ( x ) ) ) = α ( x ) + 2 , {\displaystyle \alpha (f(f(x)))=\alpha (x)+2~,} and so on, α ( f n ( x ) ) = α ( x ) + n . {\displaystyle \alpha (f_{n}(x))=\alpha (x)+n~.} == Solutions == The Abel equation has at least one solution on E {\displaystyle E} if and only if for all x ∈ E {\displaystyle x\in E} and all n ∈ N {\displaystyle n\in \mathbb {N} } , f n ( x ) ≠ x {\displaystyle f^{n}(x)\neq x} , where f n = f ∘ f ∘ . . . ∘ f {\displaystyle f^{n}=f\circ f\circ ...\circ f} , is the function f iterated n times. We have the following existence and uniqueness theorem: Theorem B Let h : R → R {\displaystyle h:\mathbb {R} \to \mathbb {R} } be analytic, meaning it has a Taylor expansion. To find: real analytic solutions α : R → C {\displaystyle \alpha :\mathbb {R} \to \mathbb {C} } of the Abel equation α ∘ h = α + 1 {\textstyle \alpha \circ h=\alpha +1} . === Existence === A real analytic solution α {\displaystyle \alpha } exists if and only if both of the following conditions hold: h {\displaystyle h} has no fixed points, meaning there is no y ∈ R {\displaystyle y\in \mathbb {R} } such that h ( y ) = y {\displaystyle h(y)=y} . The set of critical points of h {\displaystyle h} , where h ′ ( y ) = 0 {\displaystyle h'(y)=0} , is bounded above if h ( y ) > y {\displaystyle h(y)>y} for all y {\displaystyle y} , or bounded below if h ( y ) < y {\displaystyle h(y)<y} for all y {\displaystyle y} . === Uniqueness === The solution is essentially unique in the sense that there exists a canonical solution α 0 {\displaystyle \alpha _{0}} with the following properties: The set of critical points of α 0 {\displaystyle \alpha _{0}} is bounded above if h ( y ) > y {\displaystyle h(y)>y} for all y {\displaystyle y} , or bounded below if h ( y ) < y {\displaystyle h(y)<y} for all y {\displaystyle y} . This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by { α 0 + β ∘ α 0 | β : R → R is analytic, with period 1 } . {\displaystyle \{\alpha _{0}+\beta \circ \alpha _{0}|\beta :\mathbb {R} \to \mathbb {R} {\text{ is analytic, with period 1}}\}.} === Approximate solution === Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point. The analytic solution is unique up to a constant. == See also == Functional equation Schröder's equation Böttcher's equation Infinite compositions of analytic functions Iterated function Shift operator Superfunction == References == M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968). M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990. |
Wikipedia:Abel's binomial theorem#0 | Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑ k = 0 m ( m k ) ( w + m − k ) m − k − 1 ( z + k ) k = w − 1 ( z + w + m ) m . {\displaystyle \sum _{k=0}^{m}{\binom {m}{k}}(w+m-k)^{m-k-1}(z+k)^{k}=w^{-1}(z+w+m)^{m}.} == Example == === The case m = 2 === ( 2 0 ) ( w + 2 ) 1 ( z + 0 ) 0 + ( 2 1 ) ( w + 1 ) 0 ( z + 1 ) 1 + ( 2 2 ) ( w + 0 ) − 1 ( z + 2 ) 2 = ( w + 2 ) + 2 ( z + 1 ) + ( z + 2 ) 2 w = ( z + w + 2 ) 2 w . {\displaystyle {\begin{aligned}&{}\quad {\binom {2}{0}}(w+2)^{1}(z+0)^{0}+{\binom {2}{1}}(w+1)^{0}(z+1)^{1}+{\binom {2}{2}}(w+0)^{-1}(z+2)^{2}\\&=(w+2)+2(z+1)+{\frac {(z+2)^{2}}{w}}\\&={\frac {(z+w+2)^{2}}{w}}.\end{aligned}}} == See also == Binomial theorem Binomial type == References == Weisstein, Eric W. "Abel's binomial theorem". MathWorld. |
Wikipedia:Abel's identity#0 | In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be generalised to nth-order linear ordinary differential equations. The identity is named after the Norwegian mathematician Niels Henrik Abel. Since Abel's identity relates to the different linearly independent solutions of the differential equation, it can be used to find one solution from the other. It provides useful identities relating the solutions, and is also useful as a part of other techniques such as the method of variation of parameters. It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly. A generalisation of first-order systems of homogeneous linear differential equations is given by Liouville's formula. == Statement == Consider a homogeneous linear second-order ordinary differential equation y ″ + p ( x ) y ′ + q ( x ) y = 0 {\displaystyle y''+p(x)y'+q(x)\,y=0} on an interval I of the real line with real- or complex-valued continuous functions p and q. Abel's identity states that the Wronskian W = ( y 1 , y 2 ) {\displaystyle W=(y_{1},y_{2})} of two real- or complex-valued solutions y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} of this differential equation, that is the function defined by the determinant W ( y 1 , y 2 ) ( x ) = | y 1 ( x ) y 2 ( x ) y 1 ′ ( x ) y 2 ′ ( x ) | = y 1 ( x ) y 2 ′ ( x ) − y 1 ′ ( x ) y 2 ( x ) , x ∈ I , {\displaystyle W(y_{1},y_{2})(x)={\begin{vmatrix}y_{1}(x)&y_{2}(x)\\y'_{1}(x)&y'_{2}(x)\end{vmatrix}}=y_{1}(x)\,y'_{2}(x)-y'_{1}(x)\,y_{2}(x),\quad x\in I,} satisfies the relation W ( y 1 , y 2 ) ( x ) = W ( y 1 , y 2 ) ( x 0 ) ⋅ exp ( − ∫ x 0 x p ( t ) d t ) , x ∈ I , {\displaystyle W(y_{1},y_{2})(x)=W(y_{1},y_{2})(x_{0})\cdot \exp \left(-\int _{x_{0}}^{x}p(t)\,dt\right),\quad x\in I,} for each point x 0 ∈ I {\displaystyle x_{0}\in I} . === Remarks === When the differential equation is real-valued, since exp ( − ∫ x 0 x p ( t ) d t ) {\displaystyle \exp \left(-\int _{x_{0}}^{x}p(t)\,dt\right)} is strictly positive, the Wronskian W ( y 1 , y 2 ) {\displaystyle W(y_{1},y_{2})} is always either identically zero, always positive, or always negative at every point x {\displaystyle x} in I {\displaystyle I} . If the two solutions y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are linearly dependent, then the Wronskian is identically zero. Conversely, if the Wronskian is not zero at any point on the interval, then they are linearly independent. It is not necessary to assume that the second derivatives of the solutions y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are continuous. If p ( x ) = 0 {\displaystyle p(x)=0} then W {\displaystyle W} is constant. === Proof === == Generalization == The Wronskian W ( y 1 , … , y n ) {\displaystyle W(y_{1},\ldots ,y_{n})} of n {\displaystyle n} functions y 1 , … , y n {\displaystyle y_{1},\ldots ,y_{n}} on an interval I {\displaystyle I} is the function defined by the determinant W ( y 1 , … , y n ) ( x ) = | y 1 ( x ) y 2 ( x ) ⋯ y n ( x ) y 1 ′ ( x ) y 2 ′ ( x ) ⋯ y n ′ ( x ) ⋮ ⋮ ⋱ ⋮ y 1 ( n − 1 ) ( x ) y 2 ( n − 1 ) ( x ) ⋯ y n ( n − 1 ) ( x ) | , x ∈ I , {\displaystyle W(y_{1},\ldots ,y_{n})(x)={\begin{vmatrix}y_{1}(x)&y_{2}(x)&\cdots &y_{n}(x)\\y'_{1}(x)&y'_{2}(x)&\cdots &y'_{n}(x)\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-1)}(x)&y_{2}^{(n-1)}(x)&\cdots &y_{n}^{(n-1)}(x)\end{vmatrix}},\qquad x\in I,} Consider a homogeneous linear ordinary differential equation of order n ≥ 1 {\displaystyle n\geq 1} : y ( n ) + p n − 1 ( x ) y ( n − 1 ) + ⋯ + p 1 ( x ) y ′ + p 0 ( x ) y = 0 , {\displaystyle y^{(n)}+p_{n-1}(x)\,y^{(n-1)}+\cdots +p_{1}(x)\,y'+p_{0}(x)\,y=0,} on an interval I {\displaystyle I} of the real line with a real- or complex-valued continuous function p n − 1 {\displaystyle p_{n-1}} . Let y 1 , … , y n {\displaystyle y_{1},\ldots ,y_{n}} by solutions of this nth order differential equation. Then the generalisation of Abel's identity states that this Wronskian satisfies the relation: W ( y 1 , … , y n ) ( x ) = W ( y 1 , … , y n ) ( x 0 ) exp ( − ∫ x 0 x p n − 1 ( ξ ) d ξ ) , x ∈ I , {\displaystyle W(y_{1},\ldots ,y_{n})(x)=W(y_{1},\ldots ,y_{n})(x_{0})\exp {\biggl (}-\int _{x_{0}}^{x}p_{n-1}(\xi )\,{\textrm {d}}\xi {\biggr )},\qquad x\in I,} for each point x 0 ∈ I {\displaystyle x_{0}\in I} . === Direct proof === For brevity, we write W {\displaystyle W} for W ( y 1 , … , y n ) {\displaystyle W(y_{1},\ldots ,y_{n})} and omit the argument x {\displaystyle x} . It suffices to show that the Wronskian solves the first-order linear differential equation W ′ = − p n − 1 W , {\displaystyle W'=-p_{n-1}\,W,} because the remaining part of the proof then coincides with the one for the case n = 2 {\displaystyle n=2} . In the case n = 1 {\displaystyle n=1} we have W = y 1 {\displaystyle W=y_{1}} and the differential equation for W {\displaystyle W} coincides with the one for y 1 {\displaystyle y_{1}} . Therefore, assume n ≥ 2 {\displaystyle n\geq 2} in the following. The derivative of the Wronskian W {\displaystyle W} is the derivative of the defining determinant. It follows from the Leibniz formula for determinants that this derivative can be calculated by differentiating every row separately, hence W ′ = | y 1 ′ y 2 ′ ⋯ y n ′ y 1 ′ y 2 ′ ⋯ y n ′ y 1 ″ y 2 ″ ⋯ y n ″ y 1 ‴ y 2 ‴ ⋯ y n ‴ ⋮ ⋮ ⋱ ⋮ y 1 ( n − 1 ) y 2 ( n − 1 ) ⋯ y n ( n − 1 ) | + | y 1 y 2 ⋯ y n y 1 ″ y 2 ″ ⋯ y n ″ y 1 ″ y 2 ″ ⋯ y n ″ y 1 ‴ y 2 ‴ ⋯ y n ‴ ⋮ ⋮ ⋱ ⋮ y 1 ( n − 1 ) y 2 ( n − 1 ) ⋯ y n ( n − 1 ) | + ⋯ + | y 1 y 2 ⋯ y n y 1 ′ y 2 ′ ⋯ y n ′ ⋮ ⋮ ⋱ ⋮ y 1 ( n − 3 ) y 2 ( n − 3 ) ⋯ y n ( n − 3 ) y 1 ( n − 2 ) y 2 ( n − 2 ) ⋯ y n ( n − 2 ) y 1 ( n ) y 2 ( n ) ⋯ y n ( n ) | . {\displaystyle {\begin{aligned}W'&={\begin{vmatrix}y'_{1}&y'_{2}&\cdots &y'_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\y''_{1}&y''_{2}&\cdots &y''_{n}\\y'''_{1}&y'''_{2}&\cdots &y'''_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-1)}&y_{2}^{(n-1)}&\cdots &y_{n}^{(n-1)}\end{vmatrix}}+{\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y''_{1}&y''_{2}&\cdots &y''_{n}\\y''_{1}&y''_{2}&\cdots &y''_{n}\\y'''_{1}&y'''_{2}&\cdots &y'''_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-1)}&y_{2}^{(n-1)}&\cdots &y_{n}^{(n-1)}\end{vmatrix}}\\&\qquad +\ \cdots \ +{\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-3)}&y_{2}^{(n-3)}&\cdots &y_{n}^{(n-3)}\\y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots &y_{n}^{(n-2)}\\y_{1}^{(n)}&y_{2}^{(n)}&\cdots &y_{n}^{(n)}\end{vmatrix}}.\end{aligned}}} However, note that every determinant from the expansion contains a pair of identical rows, except the last one. Since determinants with linearly dependent rows are equal to 0, one is only left with the last one: W ′ = | y 1 y 2 ⋯ y n y 1 ′ y 2 ′ ⋯ y n ′ ⋮ ⋮ ⋱ ⋮ y 1 ( n − 2 ) y 2 ( n − 2 ) ⋯ y n ( n − 2 ) y 1 ( n ) y 2 ( n ) ⋯ y n ( n ) | . {\displaystyle W'={\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots &y_{n}^{(n-2)}\\y_{1}^{(n)}&y_{2}^{(n)}&\cdots &y_{n}^{(n)}\end{vmatrix}}.} Since every y i {\displaystyle y_{i}} solves the ordinary differential equation, we have y i ( n ) + p n − 2 y i ( n − 2 ) + ⋯ + p 1 y i ′ + p 0 y i = − p n − 1 y i ( n − 1 ) {\displaystyle y_{i}^{(n)}+p_{n-2}\,y_{i}^{(n-2)}+\cdots +p_{1}\,y'_{i}+p_{0}\,y_{i}=-p_{n-1}\,y_{i}^{(n-1)}} for every i ∈ { 1 , … , n } {\displaystyle i\in \lbrace 1,\ldots ,n\rbrace } . Hence, adding to the last row of the above determinant p 0 {\displaystyle p_{0}} times its first row, p 1 {\displaystyle p_{1}} times its second row, and so on until p n − 2 {\displaystyle p_{n-2}} times its next to last row, the value of the determinant for the derivative of W {\displaystyle W} is unchanged and we get W ′ = | y 1 y 2 ⋯ y n y 1 ′ y 2 ′ ⋯ y n ′ ⋮ ⋮ ⋱ ⋮ y 1 ( n − 2 ) y 2 ( n − 2 ) ⋯ y n ( n − 2 ) − p n − 1 y 1 ( n − 1 ) − p n − 1 y 2 ( n − 1 ) ⋯ − p n − 1 y n ( n − 1 ) | = − p n − 1 W . {\displaystyle W'={\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots &y_{n}^{(n-2)}\\-p_{n-1}\,y_{1}^{(n-1)}&-p_{n-1}\,y_{2}^{(n-1)}&\cdots &-p_{n-1}\,y_{n}^{(n-1)}\end{vmatrix}}=-p_{n-1}W.} === Proof using Liouville's formula === The solutions y 1 , … , y n {\displaystyle y_{1},\ldots ,y_{n}} form the square-matrix valued solution Φ ( x ) = ( y 1 ( x ) y 2 ( x ) ⋯ y n ( x ) y 1 ′ ( x ) y 2 ′ ( x ) ⋯ y n ′ ( x ) ⋮ ⋮ ⋱ ⋮ y 1 ( n − 2 ) ( x ) y 2 ( n − 2 ) ( x ) ⋯ y n ( n − 2 ) ( x ) y 1 ( n − 1 ) ( x ) y 2 ( n − 1 ) ( x ) ⋯ y n ( n − 1 ) ( x ) ) , x ∈ I , {\displaystyle \Phi (x)={\begin{pmatrix}y_{1}(x)&y_{2}(x)&\cdots &y_{n}(x)\\y'_{1}(x)&y'_{2}(x)&\cdots &y'_{n}(x)\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-2)}(x)&y_{2}^{(n-2)}(x)&\cdots &y_{n}^{(n-2)}(x)\\y_{1}^{(n-1)}(x)&y_{2}^{(n-1)}(x)&\cdots &y_{n}^{(n-1)}(x)\end{pmatrix}},\qquad x\in I,} of the n {\displaystyle n} -dimensional first-order system of homogeneous linear differential equations ( y ′ y ″ ⋮ y ( n − 1 ) y ( n ) ) = ( 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 − p 0 ( x ) − p 1 ( x ) − p 2 ( x ) ⋯ − p n − 1 ( x ) ) ( y y ′ ⋮ y ( n − 2 ) y ( n − 1 ) ) . {\displaystyle {\begin{pmatrix}y'\\y''\\\vdots \\y^{(n-1)}\\y^{(n)}\end{pmatrix}}={\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\\-p_{0}(x)&-p_{1}(x)&-p_{2}(x)&\cdots &-p_{n-1}(x)\end{pmatrix}}{\begin{pmatrix}y\\y'\\\vdots \\y^{(n-2)}\\y^{(n-1)}\end{pmatrix}}.} The trace of this matrix is − p n − 1 ( x ) {\displaystyle -p_{n-1}(x)} , hence Abel's identity follows directly from Liouville's formula. == References == Abel, N. H., "Précis d'une théorie des fonctions elliptiques" J. Reine Angew. Math., 4 (1829) pp. 309–348. Boyce, W. E. and DiPrima, R. C. (1986). Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. Weisstein, Eric W. "Abel's Differential Equation Identity". MathWorld. |
Wikipedia:Abel's irreducibility theorem#0 | In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if f(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of f(x). Equivalently, if f(x) shares at least one root with g(x) then f is divisible evenly by g(x), meaning that f(x) can be factored as g(x)h(x) with h(x) also having coefficients in F. Corollaries of the theorem include: If f(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has 2 {\displaystyle {\sqrt {2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2 {\displaystyle {\sqrt {2}}} as a root. Furthermore, there is no same-degree polynomial that shares any roots with f(x), other than constant multiples of f(x). If f(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots. == References == == External links == Larry Freeman. Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility. September 4, 2008. Weisstein, Eric W. "Abel's Irreducibility Theorem". MathWorld. |
Wikipedia:Abraham Neyman#0 | Abraham Neyman (Hebrew: אברהם ניימן; born June 14, 1949) is an Israeli mathematician and game theorist, Professor of Mathematics at the Federmann Center for the Study of Rationality and the Einstein Institute of Mathematics at the Hebrew University of Jerusalem. He served as president of the Israeli Chapter of the Game Theory Society (2014–2018). == Biography == Neyman received his BSc in mathematics in 1970 and his MSc in mathematics in 1972 from the Hebrew University. His MSc thesis was on the subject of “The Range of a Vector Measure” and was supervised by Joram Lindenstrauss. His PhD thesis, "Values of Games with a Continuum of Players," was completed under Robert Aumann in 1977. Neyman has been professor of mathematics at the Hebrew University since 1982, including serving as the chairman of the institute of mathematics 1992–1994, as well as holding a professorship in economics, 1982–1990. He has been a member of the Center for the Study of Rationality at the Hebrew University since its inception in 1991. He held various positions at Stony Brook University of New York, 1985–2001. He has also held positions and has been visiting scholar at Cornell University, University of California at Berkeley, Stanford University, the Graduate School of Business Administration at Harvard University, and Ohio State University. Neyman has had 12 graduate students complete Ph.D. theses under his supervision, five at Stony Brook University and seven at the Hebrew University. Neyman has also served as the Game Theory Area Editor for the journal Mathematics of Operations Research (1987–1993) and on the editorial board for Games and Economic Behavior (1993–2001) and the International Journal of Game Theory (2001–2007). == Awards and honors == Neyman has been a fellow of the Econometric Society since 1989. The Game Theory Society released, in March 2016, a special issue of the International Journal of Game Theory in honour of Neyman, "in recognition of his important contributions to game theory". A Festschrift conference in Neyman's honour was held at Hebrew University in June 2015, on the occasion of Neyman's 66th birthday. He gave the inaugural von-Neumann lecture at the 2008 Congress of the Game Theory Society as well as delivering it at the 2012 World Congress on behalf of the recently deceased Jean-Francois Mertens. His Ph.D. thesis won two prizes from the Hebrew University: the 1977 Abraham Urbach prize for distinguished thesis in mathematics and the 1979 Aharon Katzir prize (for the best Ph. D. thesis in the Faculties of Exact Science, Mathematics, Agriculture and Medicine). In addition, Neyman won the Israeli under 20 chess championship in 1966. == Research contributions == Neyman has made numerous contributions to game theory, including to stochastic games, the Shapley value, and repeated games. === Stochastic games === Together with Jean-Francois Mertens, he proved the existence of the uniform value of zero-sum undiscounted stochastic games. This work is considered one of the most important works in the theory of stochastic games, solving a problem that had been open for over 20 years. Together with Elon Kohlberg, he applied operator techniques to study convergence properties of the discounted and finite stage values. Recently, he has pioneered a model of stochastic games in continuous time and derived uniform equilibrium existence results. He also co-edited, together with Sylvain Sorin, a comprehensive collection of works in the field of stochastic games. === Repeated games === Neyman has made many contributions to the theory of repeated games. One idea that appears, in different contexts, in some of his papers, is that the model of an infinitely repeated game serves also as a powerful paradigm for a long finitely repeated game. A related insight appears in a 1999 paper, where he showed that in a long finitely repeated game, an exponentially small deviation from common knowledge of the number of repetitions is enough to dramatically alter the equilibrium analysis, producing a folk-theorem-like result. Neyman is one of the pioneers and a most notable leader of the study of repeated games under complexity constraints. In his seminal paper he showed that bounded memory can justify cooperation in a finitely repeated prisoner's dilemma game. His paper was followed by many others who started working on bounded memory games. Most notable was Neyman's M.Sc. student Elchanan Ben-Porath who was the first to shed light on the strategic value of bounded complexity. The two main models of bounded complexity, automaton size and recall capacity, continued to pose intriguing open problems in the following decades. A major breakthrough was achieved when Neyman and his Ph.D. student Daijiro Okada proposed a new approach to these problems, based on information theoretic techniques, introducing the notion of strategic entropy. His students continued to employ Neyman's entropy technique to achieve a better understanding of repeated games under complexity constraints. Neyman's information theoretic approach opened new research areas beyond bounded complexity. A classic example is the communication game he introduced jointly with Olivier Gossner and Penelope Hernandez. === The Shapley value === Neyman has made numerous fundamental contributions to the theory of the value. In a "remarkable tour-de-force of combinatorial reasoning", he proved the existence of an asymptotic value for weighted majority games. The proof was facilitated by his fundamental contribution to renewal theory. In subsequent work Neyman proved that many of the assumptions made in these works can be relaxed, while showing that others are essential. Neyman proved the diagonality of continuous values, which had many implications on further developments of the theory. Together with Pradeep Dubey and Robert James Weber he studied the theory of semivalues, and separately demonstrated its importance in political economy. Together with Pradeep Dubey he characterized the well-known phenomenon of value correspondence, a fundamental notion in economics, originating already in Edgeworth's work and Adam Smith before him. In loose terms, it essentially states that in a large economy consisting of many economically insignificant agents, the core of the economy coincides with the perfectly competitive outcomes, which in the case of differentiable preferences is a unique element that is the Aumann–Shapley value. Another major contribution of Neyman was the introduction of the Neyman value, a far-reaching generalization of the Aumann–Shapley value to the case of non-differentiable vector measure games. === Other === Neyman has made contributions to other fields of mathematics, usually motivated by problems in game theory. Among these contributions are a renewal theorem for sampling without replacement (mentioned above as applied to the theory of the value), contributions to embeddings of Lp spaces, contributions to the theory of vector measures, and to the theory of non-expansive mappings. == Business involvements == Neyman previously served (2005–8) as director at Tradus (previously named QXL). He also held a directorship (2004–5) at Gilat Satellite Networks. In 1999, Neyman co-founded Bidorbuy, the first online auction company to operate in India and in South Africa, and serves as the chairman of the board. Since 2013, he has held a directorship at the Israeli bank Bank Mizrahi-Tefahot. == References == == External links == Neyman’s homepage Full publication list |
Wikipedia:Abramov's algorithm#0 | In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989. == Universal denominator == The main concept in Abramov's algorithm is a universal denominator. Let K {\textstyle \mathbb {K} } be a field of characteristic zero. The dispersion dis ( p , q ) {\textstyle \operatorname {dis} (p,q)} of two polynomials p , q ∈ K [ n ] {\textstyle p,q\in \mathbb {K} [n]} is defined as dis ( p , q ) = max { k ∈ N : deg ( gcd ( p ( n ) , q ( n + k ) ) ) ≥ 1 } ∪ { − 1 } , {\displaystyle \operatorname {dis} (p,q)=\max\{k\in \mathbb {N} \,:\,\deg(\gcd(p(n),q(n+k)))\geq 1\}\cup \{-1\},} where N {\textstyle \mathbb {N} } denotes the set of non-negative integers. Therefore the dispersion is the maximum k ∈ N {\textstyle k\in \mathbb {N} } such that the polynomial p {\textstyle p} and the k {\textstyle k} -times shifted polynomial q {\displaystyle q} have a common factor. It is − 1 {\textstyle -1} if such a k {\textstyle k} does not exist. The dispersion can be computed as the largest non-negative integer root of the resultant res n ( p ( n ) , q ( n + k ) ) ∈ K [ k ] {\textstyle \operatorname {res} _{n}(p(n),q(n+k))\in \mathbb {K} [k]} . Let ∑ k = 0 r p k ( n ) y ( n + k ) = f ( n ) {\textstyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=f(n)} be a recurrence equation of order r {\textstyle r} with polynomial coefficients p k ∈ K [ n ] {\displaystyle p_{k}\in \mathbb {K} [n]} , polynomial right-hand side f ∈ K [ n ] {\textstyle f\in \mathbb {K} [n]} and rational sequence solution y ( n ) ∈ K ( n ) {\textstyle y(n)\in \mathbb {K} (n)} . It is possible to write y ( n ) = p ( n ) / q ( n ) {\textstyle y(n)=p(n)/q(n)} for two relatively prime polynomials p , q ∈ K [ n ] {\textstyle p,q\in \mathbb {K} [n]} . Let D = dis ( p r ( n − r ) , p 0 ( n ) ) {\textstyle D=\operatorname {dis} (p_{r}(n-r),p_{0}(n))} and u ( n ) = gcd ( [ p 0 ( n + D ) ] D + 1 _ , [ p r ( n − r ) ] D + 1 _ ) {\displaystyle u(n)=\gcd([p_{0}(n+D)]^{\underline {D+1}},[p_{r}(n-r)]^{\underline {D+1}})} where [ p ( n ) ] k _ = p ( n ) p ( n − 1 ) ⋯ p ( n − k + 1 ) {\textstyle [p(n)]^{\underline {k}}=p(n)p(n-1)\cdots p(n-k+1)} denotes the falling factorial of a function. Then q ( n ) {\textstyle q(n)} divides u ( n ) {\textstyle u(n)} . So the polynomial u ( n ) {\textstyle u(n)} can be used as a denominator for all rational solutions y ( n ) {\textstyle y(n)} and hence it is called a universal denominator. == Algorithm == Let again ∑ k = 0 r p k ( n ) y ( n + k ) = f ( n ) {\textstyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=f(n)} be a recurrence equation with polynomial coefficients and u ( n ) {\textstyle u(n)} a universal denominator. After substituting y ( n ) = z ( n ) / u ( n ) {\textstyle y(n)=z(n)/u(n)} for an unknown polynomial z ( n ) ∈ K [ n ] {\textstyle z(n)\in \mathbb {K} [n]} and setting ℓ ( n ) = lcm ( u ( n ) , … , u ( n + r ) ) {\textstyle \ell (n)=\operatorname {lcm} (u(n),\dots ,u(n+r))} the recurrence equation is equivalent to ∑ k = 0 r p k ( n ) z ( n + k ) u ( n + k ) ℓ ( n ) = f ( n ) ℓ ( n ) . {\displaystyle \sum _{k=0}^{r}p_{k}(n){\frac {z(n+k)}{u(n+k)}}\ell (n)=f(n)\ell (n).} As the u ( n + k ) {\textstyle u(n+k)} cancel this is a linear recurrence equation with polynomial coefficients which can be solved for an unknown polynomial solution z ( n ) {\textstyle z(n)} . There are algorithms to find polynomial solutions. The solutions for z ( n ) {\textstyle z(n)} can then be used again to compute the rational solutions y ( n ) = z ( n ) / u ( n ) {\textstyle y(n)=z(n)/u(n)} . algorithm rational_solutions is input: Linear recurrence equation ∑ k = 0 r p k ( n ) y ( n + k ) = f ( n ) , p k , f ∈ K [ n ] , p 0 , p r ≠ 0 {\textstyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=f(n),p_{k},f\in \mathbb {K} [n],p_{0},p_{r}\neq 0} . output: The general rational solution y {\textstyle y} if there are any solutions, otherwise false. D = disp ( p r ( n − r ) , p 0 ( n ) ) {\textstyle D=\operatorname {disp} (p_{r}(n-r),p_{0}(n))} u ( n ) = gcd ( [ p 0 ( n + D ) ] D + 1 _ , [ p r ( n − r ) ] D + 1 _ ) {\textstyle u(n)=\gcd([p_{0}(n+D)]^{\underline {D+1}},[p_{r}(n-r)]^{\underline {D+1}})} ℓ ( n ) = lcm ( u ( n ) , … , u ( n + r ) ) {\textstyle \ell (n)=\operatorname {lcm} (u(n),\dots ,u(n+r))} Solve ∑ k = 0 r p k ( n ) z ( n + k ) u ( n + k ) ℓ ( n ) = f ( n ) ℓ ( n ) {\textstyle \sum _{k=0}^{r}p_{k}(n){\frac {z(n+k)}{u(n+k)}}\ell (n)=f(n)\ell (n)} for general polynomial solution z ( n ) {\textstyle z(n)} if solution z ( n ) {\textstyle z(n)} exists then return general solution y ( n ) = z ( n ) / u ( n ) {\textstyle y(n)=z(n)/u(n)} else return false end if == Example == The homogeneous recurrence equation of order 1 {\textstyle 1} ( n − 1 ) y ( n ) + ( − n − 1 ) y ( n + 1 ) = 0 {\displaystyle (n-1)\,y(n)+(-n-1)\,y(n+1)=0} over Q {\textstyle \mathbb {Q} } has a rational solution. It can be computed by considering the dispersion D = dis ( p 1 ( n − 1 ) , p 0 ( n ) ) = disp ( − n , n − 1 ) = 1. {\displaystyle D=\operatorname {dis} (p_{1}(n-1),p_{0}(n))=\operatorname {disp} (-n,n-1)=1.} This yields the following universal denominator: u ( n ) = gcd ( [ p 0 ( n + 1 ) ] 2 _ , [ p r ( n − 1 ) ] 2 _ ) = ( n − 1 ) n {\displaystyle u(n)=\gcd([p_{0}(n+1)]^{\underline {2}},[p_{r}(n-1)]^{\underline {2}})=(n-1)n} and ℓ ( n ) = lcm ( u ( n ) , u ( n + 1 ) ) = ( n − 1 ) n ( n + 1 ) . {\displaystyle \ell (n)=\operatorname {lcm} (u(n),u(n+1))=(n-1)n(n+1).} Multiplying the original recurrence equation with ℓ ( n ) {\textstyle \ell (n)} and substituting y ( n ) = z ( n ) / u ( n ) {\textstyle y(n)=z(n)/u(n)} leads to ( n − 1 ) ( n + 1 ) z ( n ) + ( − n − 1 ) ( n − 1 ) z ( n + 1 ) = 0. {\displaystyle (n-1)(n+1)\,z(n)+(-n-1)(n-1)\,z(n+1)=0.} This equation has the polynomial solution z ( n ) = c {\textstyle z(n)=c} for an arbitrary constant c ∈ Q {\textstyle c\in \mathbb {Q} } . Using y ( n ) = z ( n ) / u ( n ) {\textstyle y(n)=z(n)/u(n)} the general rational solution is y ( n ) = c ( n − 1 ) n {\displaystyle y(n)={\frac {c}{(n-1)n}}} for arbitrary c ∈ Q {\textstyle c\in \mathbb {Q} } . == References == |
Wikipedia:Absolute value (algebra)#0 | In algebra, an absolute value is a function that generalizes the usual absolute value. More precisely, if D is a field or (more generally) an integral domain, an absolute value on D is a function, commonly denoted | x | , {\displaystyle |x|,} from D to the real numbers satisfying: It follows from the axioms that | 1 | = 1 , {\displaystyle |1|=1,} | − 1 | = 1 , {\displaystyle |-1|=1,} and | − x | = | x | {\displaystyle |-x|=|x|} for every x {\displaystyle x} . Furthermore, for every positive integer n, | n | ≤ n , {\displaystyle |n|\leq n,} where the leftmost n denotes the sum of n summands equal to the identity element of D. The classical absolute value and its square root are examples of absolute values, but not the square of the classical absolute value, which does not fulfill the triangular inequality. An absolute value such that | x + y | ≤ max ( | x | , | y | ) {\displaystyle |x+y|\leq \max(|x|,|y|)} is an ultrametric absolute value. An absolute value induces a metric (and thus a topology) by d ( f , g ) = | f − g | . {\displaystyle d(f,g)=|f-g|.} == Examples == The standard absolute value on the integers. The standard absolute value on the complex numbers. The p-adic absolute value on the rational numbers. If F ( x ) {\displaystyle F(x)} is the field of rational fractions over a field F and P {\displaystyle P} is an irreducible polynomial over F, the P-adic absolute value on F ( x ) {\displaystyle F(x)} is defined as | f | P = 2 − n , {\displaystyle |f|_{P}=2^{-n},} where n is the unique integer such that f ( x ) = P n G H , {\textstyle f(x)=P^{n}{\frac {G}{H}},} where G and H are two polynomials, both coprime with P. == Types of absolute value == The trivial absolute value is the absolute value with |x| = 0 when x = 0 and |x| = 1 otherwise. Every integral domain can carry at least the trivial absolute value. The trivial value is the only possible absolute value on a finite field because any non-zero element can be raised to some power to yield 1. If an absolute value satisfies the stronger property |x + y| ≤ max(|x|, |y|) for all x and y, then |x| is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean absolute value. == Places == If |x|1 and |x|2 are two absolute values on the same integral domain D, then the two absolute values are equivalent if |x|1 < 1 if and only if |x|2 < 1 for all x. If two nontrivial absolute values are equivalent, then for some exponent e we have |x|1e = |x|2 for all x. Raising an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it violates the rule |x+y| ≤ |x|+|y|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place. Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p. For a given prime p, any rational number q can be written as pn(a/b), where a and b are integers not divisible by p and n is an integer. The p-adic absolute value of q is | p n a b | p = p − n . {\displaystyle \left|p^{n}{\frac {a}{b}}\right|_{p}=p^{-n}.} Since the ordinary absolute value and the p-adic absolute values are absolute values according to the definition above, these define places. == Valuations == If for some ultrametric absolute value and any base b > 1, we define ν(x) = −logb|x| for x ≠ 0 and ν(0) = ∞, where ∞ is ordered to be greater than all real numbers, then we obtain a function from D to R ∪ {∞}, with the following properties: ν(x) = ∞ ⇒ x = 0, ν(xy) = ν(x) + ν(y), ν(x + y) ≥ min(ν(x), ν(y)). Such a function is known as a valuation in the terminology of Bourbaki, but other authors use the term valuation for absolute value and then say exponential valuation instead of valuation. == Completions == Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every ε > 0 there is a positive integer N such that for all integers m, n > N one has |xm − xn| < ε. Cauchy sequences form a ring under pointwise addition and multiplication. One can also define null sequences as sequences (an) of elements of D such that |an| converges to zero. Null sequences are a prime ideal in the ring of Cauchy sequences, and the quotient ring is therefore an integral domain. The domain D is embedded in this quotient ring, called the completion of D with respect to the absolute value |x|. Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value. To show that the result is a field, and not just an integral domain, we can either show that null sequences form a maximal ideal, or else construct the inverse directly. The latter can be easily done by taking, for all nonzero elements of the quotient ring, a sequence starting from a point beyond the last zero element of the sequence. Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we can find a representative inverse element. Another theorem of Alexander Ostrowski has it that any field complete with respect to an Archimedean absolute value is isomorphic to either the real or the complex numbers, and the valuation is equivalent to the usual one. The Gelfand-Tornheim theorem states that any field with an Archimedean valuation is isomorphic to a subfield of C, the valuation being equivalent to the usual absolute value on C. == Fields and integral domains == If D is an integral domain with absolute value |x|, then we may extend the definition of the absolute value to the field of fractions of D by setting | x / y | = | x | / | y | . {\displaystyle |x/y|=|x|/|y|.\,} On the other hand, if F is a field with ultrametric absolute value |x|, then the set of elements of F such that |x| ≤ 1 defines a valuation ring, which is a subring D of F such that for every nonzero element x of F, at least one of x or x−1 belongs to D. Since F is a field, D has no zero divisors and is an integral domain. It has a unique maximal ideal consisting of all x such that |x| < 1, and is therefore a local ring. == Notes == == References == |
Wikipedia:Absolutely convex set#0 | In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set. == Definition == A subset S {\displaystyle S} of a real or complex vector space X {\displaystyle X} is called a disk and is said to be disked, absolutely convex, and convex balanced if any of the following equivalent conditions is satisfied: S {\displaystyle S} is a convex and balanced set. for any scalars a {\displaystyle a} and b , {\displaystyle b,} if | a | + | b | ≤ 1 {\displaystyle |a|+|b|\leq 1} then a S + b S ⊆ S . {\displaystyle aS+bS\subseteq S.} for all scalars a , b , {\displaystyle a,b,} and c , {\displaystyle c,} if | a | + | b | ≤ | c | , {\displaystyle |a|+|b|\leq |c|,} then a S + b S ⊆ c S . {\displaystyle aS+bS\subseteq cS.} for any scalars a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} and c , {\displaystyle c,} if | a 1 | + ⋯ + | a n | ≤ | c | {\displaystyle |a_{1}|+\cdots +|a_{n}|\leq |c|} then a 1 S + ⋯ + a n S ⊆ c S . {\displaystyle a_{1}S+\cdots +a_{n}S\subseteq cS.} for any scalars a 1 , … , a n , {\displaystyle a_{1},\ldots ,a_{n},} if | a 1 | + ⋯ + | a n | ≤ 1 {\displaystyle |a_{1}|+\cdots +|a_{n}|\leq 1} then a 1 S + ⋯ + a n S ⊆ S . {\displaystyle a_{1}S+\cdots +a_{n}S\subseteq S.} The smallest convex (respectively, balanced) subset of X {\displaystyle X} containing a given set is called the convex hull (respectively, the balanced hull) of that set and is denoted by co S {\displaystyle \operatorname {co} S} (respectively, bal S {\displaystyle \operatorname {bal} S} ). Similarly, the disked hull, the absolute convex hull, and the convex balanced hull of a set S {\displaystyle S} is defined to be the smallest disk (with respect to subset inclusion) containing S . {\displaystyle S.} The disked hull of S {\displaystyle S} will be denoted by disk S {\displaystyle \operatorname {disk} S} or cobal S {\displaystyle \operatorname {cobal} S} and it is equal to each of the following sets: co ( bal S ) , {\displaystyle \operatorname {co} (\operatorname {bal} S),} which is the convex hull of the balanced hull of S {\displaystyle S} ; thus, cobal S = co ( bal S ) . {\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} S).} In general, cobal S ≠ bal ( co S ) {\displaystyle \operatorname {cobal} S\neq \operatorname {bal} (\operatorname {co} S)} is possible, even in finite dimensional vector spaces. the intersection of all disks containing S . {\displaystyle S.} { a 1 s 1 + ⋯ a n s n : n ∈ N , s 1 , … , s n ∈ S , and a 1 , … , a n are scalars satisfying | a 1 | + ⋯ + | a n | ≤ 1 } . {\displaystyle \left\{a_{1}s_{1}+\cdots a_{n}s_{n}~:~n\in \mathbb {N} ,\,s_{1},\ldots ,s_{n}\in S,\,{\text{ and }}a_{1},\ldots ,a_{n}{\text{ are scalars satisfying }}|a_{1}|+\cdots +|a_{n}|\leq 1\right\}.} == Sufficient conditions == The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore. If D {\displaystyle D} is a disk in X , {\displaystyle X,} then D {\displaystyle D} is absorbing in X {\displaystyle X} if and only if span D = X . {\displaystyle \operatorname {span} D=X.} == Properties == If S {\displaystyle S} is an absorbing disk in a vector space X {\displaystyle X} then there exists an absorbing disk E {\displaystyle E} in X {\displaystyle X} such that E + E ⊆ S . {\displaystyle E+E\subseteq S.} If D {\displaystyle D} is a disk and r {\displaystyle r} and s {\displaystyle s} are scalars then s D = | s | D {\displaystyle sD=|s|D} and ( r D ) ∩ ( s D ) = ( min { | r | , | s | } ) D . {\displaystyle (rD)\cap (sD)=(\min _{}\{|r|,|s|\})D.} The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded. If D {\displaystyle D} is a bounded disk in a TVS X {\displaystyle X} and if x ∙ = ( x i ) i = 1 ∞ {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} is a sequence in D , {\displaystyle D,} then the partial sums s ∙ = ( s n ) n = 1 ∞ {\displaystyle s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }} are Cauchy, where for all n , {\displaystyle n,} s n := ∑ i = 1 n 2 − i x i . {\displaystyle s_{n}:=\sum _{i=1}^{n}2^{-i}x_{i}.} In particular, if in addition D {\displaystyle D} is a sequentially complete subset of X , {\displaystyle X,} then this series s ∙ {\displaystyle s_{\bullet }} converges in X {\displaystyle X} to some point of D . {\displaystyle D.} The convex balanced hull of S {\displaystyle S} contains both the convex hull of S {\displaystyle S} and the balanced hull of S . {\displaystyle S.} Furthermore, it contains the balanced hull of the convex hull of S ; {\displaystyle S;} thus bal ( co S ) ⊆ cobal S = co ( bal S ) , {\displaystyle \operatorname {bal} (\operatorname {co} S)~\subseteq ~\operatorname {cobal} S~=~\operatorname {co} (\operatorname {bal} S),} where the example below shows that this inclusion might be strict. However, for any subsets S , T ⊆ X , {\displaystyle S,T\subseteq X,} if S ⊆ T {\displaystyle S\subseteq T} then cobal S ⊆ cobal T {\displaystyle \operatorname {cobal} S\subseteq \operatorname {cobal} T} which implies cobal ( co S ) = cobal S = cobal ( bal S ) . {\displaystyle \operatorname {cobal} (\operatorname {co} S)=\operatorname {cobal} S=\operatorname {cobal} (\operatorname {bal} S).} === Examples === Although cobal S = co ( bal S ) , {\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} S),} the convex balanced hull of S {\displaystyle S} is not necessarily equal to the balanced hull of the convex hull of S . {\displaystyle S.} For an example where cobal S ≠ bal ( co S ) {\displaystyle \operatorname {cobal} S\neq \operatorname {bal} (\operatorname {co} S)} let X {\displaystyle X} be the real vector space R 2 {\displaystyle \mathbb {R} ^{2}} and let S := { ( − 1 , 1 ) , ( 1 , 1 ) } . {\displaystyle S:=\{(-1,1),(1,1)\}.} Then bal ( co S ) {\displaystyle \operatorname {bal} (\operatorname {co} S)} is a strict subset of cobal S {\displaystyle \operatorname {cobal} S} that is not even convex; in particular, this example also shows that the balanced hull of a convex set is not necessarily convex. The set cobal S {\displaystyle \operatorname {cobal} S} is equal to the closed and filled square in X {\displaystyle X} with vertices ( − 1 , 1 ) , ( 1 , 1 ) , ( − 1 , − 1 ) , {\displaystyle (-1,1),(1,1),(-1,-1),} and ( 1 , − 1 ) {\displaystyle (1,-1)} (this is because the balanced set cobal S {\displaystyle \operatorname {cobal} S} must contain both S {\displaystyle S} and − S = { ( − 1 , − 1 ) , ( 1 , − 1 ) } , {\displaystyle -S=\{(-1,-1),(1,-1)\},} where since cobal S {\displaystyle \operatorname {cobal} S} is also convex, it must consequently contain the solid square co ( ( − S ) ∪ S ) , {\displaystyle \operatorname {co} ((-S)\cup S),} which for this particular example happens to also be balanced so that cobal S = co ( ( − S ) ∪ S ) {\displaystyle \operatorname {cobal} S=\operatorname {co} ((-S)\cup S)} ). However, co ( S ) {\displaystyle \operatorname {co} (S)} is equal to the horizontal closed line segment between the two points in S {\displaystyle S} so that bal ( co S ) {\displaystyle \operatorname {bal} (\operatorname {co} S)} is instead a closed "hour glass shaped" subset that intersects the x {\displaystyle x} -axis at exactly the origin and is the union of two closed and filled isosceles triangles: one whose vertices are the origin together with S {\displaystyle S} and the other triangle whose vertices are the origin together with − S = { ( − 1 , − 1 ) , ( 1 , − 1 ) } . {\displaystyle -S=\{(-1,-1),(1,-1)\}.} This non-convex filled "hour-glass" bal ( co S ) {\displaystyle \operatorname {bal} (\operatorname {co} S)} is a proper subset of the filled square cobal S = co ( bal S ) . {\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} S).} == Generalizations == Given a fixed real number 0 < p ≤ 1 , {\displaystyle 0<p\leq 1,} a p {\displaystyle p} -convex set is any subset C {\displaystyle C} of a vector space X {\displaystyle X} with the property that r c + s d ∈ C {\displaystyle rc+sd\in C} whenever c , d ∈ C {\displaystyle c,d\in C} and r , s ≥ 0 {\displaystyle r,s\geq 0} are non-negative scalars satisfying r p + s p = 1. {\displaystyle r^{p}+s^{p}=1.} It is called an absolutely p {\displaystyle p} -convex set or a p {\displaystyle p} -disk if r c + s d ∈ C {\displaystyle rc+sd\in C} whenever c , d ∈ C {\displaystyle c,d\in C} and r , s {\displaystyle r,s} are scalars satisfying | r | p + | s | p ≤ 1. {\displaystyle |r|^{p}+|s|^{p}\leq 1.} A p {\displaystyle p} -seminorm is any non-negative function q : X → R {\displaystyle q:X\to \mathbb {R} } that satisfies the following conditions: Subadditivity/Triangle inequality: q ( x + y ) ≤ q ( x ) + q ( y ) {\displaystyle q(x+y)\leq q(x)+q(y)} for all x , y ∈ X . {\displaystyle x,y\in X.} Absolute homogeneity of degree p {\displaystyle p} : q ( s x ) = | s | p q ( x ) {\displaystyle q(sx)=|s|^{p}q(x)} for all x ∈ X {\displaystyle x\in X} and all scalars s . {\displaystyle s.} This generalizes the definition of seminorms since a map is a seminorm if and only if it is a 1 {\displaystyle 1} -seminorm (using p := 1 {\displaystyle p:=1} ). There exist p {\displaystyle p} -seminorms that are not seminorms. For example, whenever 0 < p < 1 {\displaystyle 0<p<1} then the map q ( f ) = ∫ R | f ( t ) | p d t {\displaystyle q(f)=\int _{\mathbb {R} }|f(t)|^{p}dt} used to define the Lp space L p ( R ) {\displaystyle L_{p}(\mathbb {R} )} is a p {\displaystyle p} -seminorm but not a seminorm. Given 0 < p ≤ 1 , {\displaystyle 0<p\leq 1,} a topological vector space is p {\displaystyle p} -seminormable (meaning that its topology is induced by some p {\displaystyle p} -seminorm) if and only if it has a bounded p {\displaystyle p} -convex neighborhood of the origin. == See also == Absorbing set – Set that can be "inflated" to reach any point Auxiliary normed space Balanced set – Construct in functional analysis Bounded set (topological vector space) – Generalization of boundedness Convex set – In geometry, set whose intersection with every line is a single line segment Star domain – Property of point sets in Euclidean spaces Symmetric set – Property of group subsets (mathematics) Vector (geometric) – Geometric object that has length and directionPages displaying short descriptions of redirect targets, for vectors in physics Vector field – Assignment of a vector to each point in a subset of Euclidean space == References == == Bibliography == Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 4–6. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. |
Wikipedia:Absorption law#0 | In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: a ¤ (a ⁂ b) = a ⁂ (a ¤ b) = a. A set equipped with two commutative and associative binary operations ∨ {\displaystyle \scriptstyle \lor } ("join") and ∧ {\displaystyle \scriptstyle \land } ("meet") that are connected by the absorption law is called a lattice; in this case, both operations are necessarily idempotent (i.e. a ∨ {\displaystyle \scriptstyle \lor } a = a and a ∧ {\displaystyle \scriptstyle \land } a = a). Examples of lattices include Heyting algebras and Boolean algebras, in particular sets of sets with union (∪) and intersection (∩) operators, and ordered sets with min and max operations. In classical logic, and in particular Boolean algebra, the operations OR and AND, which are also denoted by ∨ {\displaystyle \scriptstyle \lor } and ∧ {\displaystyle \scriptstyle \land } , satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic. The absorption law does not hold in many other algebraic structures, such as commutative rings, e.g. the field of real numbers, relevance logics, linear logics, and substructural logics. In the last case, there is no one-to-one correspondence between the free variables of the defining pair of identities. == See also == Absorption (logic) == References == Brian A. Davey; Hilary Ann Priestley (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. LCCN 2001043910. "Absorption laws", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Absorption Law". MathWorld. |
Wikipedia:Abstract algebra#0 | In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory gives a unified framework to study properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the variety of groups. == History == Before the nineteenth century, algebra was defined as the study of polynomials. Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of algebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal axiomatic definitions of various algebraic structures such as groups, rings, and fields. This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's Moderne Algebra, which start each chapter with a formal definition of a structure and then follow it with concrete examples. === Elementary algebra === The study of polynomial equations or algebraic equations has a long history. c. 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as rhetorical algebra and was the dominant approach up to the 16th century. Al-Khwarizmi originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until François Viète's 1591 New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637 La Géométrie. The formal study of solving symbolic equations led Leonhard Euler to accept what were then considered "nonsense" roots such as negative numbers and imaginary numbers, in the late 18th century. However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century. George Peacock's 1830 Treatise of Algebra was the first attempt to place algebra on a strictly symbolic basis. He distinguished a new symbolical algebra, distinct from the old arithmetical algebra. Whereas in arithmetical algebra a − b {\displaystyle a-b} is restricted to a ≥ b {\displaystyle a\geq b} , in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as ( − a ) ( − b ) = a b {\displaystyle (-a)(-b)=ab} , by letting a = 0 , c = 0 {\displaystyle a=0,c=0} in ( a − b ) ( c − d ) = a c + b d − a d − b c {\displaystyle (a-b)(c-d)=ac+bd-ad-bc} . Peacock used what he termed the principle of the permanence of equivalent forms to justify his argument, but his reasoning suffered from the problem of induction. For example, a b = a b {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}} holds for the nonnegative real numbers, but not for general complex numbers. === Early group theory === Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to the Galois group of a polynomial. Gauss's 1801 study of Fermat's little theorem led to the ring of integers modulo n, the multiplicative group of integers modulo n, and the more general concepts of cyclic groups and abelian groups. Klein's 1872 Erlangen program studied geometry and led to symmetry groups such as the Euclidean group and the group of projective transformations. In 1874 Lie introduced the theory of Lie groups, aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group of Möbius transformations, and its subgroups such as the modular group and Fuchsian group, based on work on automorphic functions in analysis. The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term "group", signifying a collection of permutations closed under composition. Arthur Cayley's 1854 paper On the theory of groups defined a group as a set with an associative composition operation and the identity 1, today called a monoid. In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left cancellation property b ≠ c → a ⋅ b ≠ a ⋅ c {\displaystyle b\neq c\to a\cdot b\neq a\cdot c} , similar to the modern laws for a finite abelian group. Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation. Walther von Dyck in 1882 was the first to require inverse elements as part of the definition of a group. Once this abstract group concept emerged, results were reformulated in this abstract setting. For example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group. Otto Hölder was particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed the Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator of two elements. Burnside, Frobenius, and Molien created the representation theory of finite groups at the end of the nineteenth century. J. A. de Séguier's 1905 monograph Elements of the Theory of Abstract Groups presented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916 Abstract Theory of Groups. === Early ring theory === Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented biquaternions, Cayley introduced octonions, and Grassman introduced exterior algebras. James Cockle presented tessarines in 1848 and coquaternions in 1849. William Kingdon Clifford introduced split-biquaternions in 1873. In addition Cayley introduced group algebras over the real and complex numbers in 1854 and square matrices in two papers of 1855 and 1858. Once there were sufficient examples, it remained to classify them. In an 1870 monograph, Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of an associative algebra. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the Peirce decomposition. Frobenius in 1878 and Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over R {\displaystyle \mathbb {R} } were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimple Lie algebras could be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } uniquely decomposes into the direct sums of a nilpotent algebra and a semisimple algebra that is the product of some number of simple algebras, square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called the Wedderburn principal theorem and Artin–Wedderburn theorem. For commutative rings, several areas together led to commutative ring theory. In two papers in 1828 and 1832, Gauss formulated the Gaussian integers and showed that they form a unique factorization domain (UFD) and proved the biquadratic reciprocity law. Jacobi and Eisenstein at around the same time proved a cubic reciprocity law for the Eisenstein integers. The study of Fermat's last theorem led to the algebraic integers. In 1847, Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the cyclotomic fields were UFDs, yet as Kummer pointed out, Q ( ζ 23 ) ) {\displaystyle \mathbb {Q} (\zeta _{23}))} was not a UFD. In 1846 and 1847 Kummer introduced ideal numbers and proved unique factorization into ideal primes for cyclotomic fields. Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of prime ideals, a precursor of the theory of Dedekind domains. Overall, Dedekind's work created the subject of algebraic number theory. In the 1850s, Riemann introduced the fundamental concept of a Riemann surface. Riemann's methods relied on an assumption he called Dirichlet's principle, which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the direct method in the calculus of variations. In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially M. Noether studied algebraic functions and curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring R [ x , y ] {\displaystyle \mathbb {R} [x,y]} , although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory of algebraic function fields which allowed the first rigorous definition of a Riemann surface and a rigorous proof of the Riemann–Roch theorem. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit in E. Noether's work. Lasker proved a special case of the Lasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection of primary ideals. Macauley proved the uniqueness of this decomposition. Overall, this work led to the development of algebraic geometry. In 1801 Gauss introduced binary quadratic forms over the integers and defined their equivalence. He further defined the discriminant of these forms, which is an invariant of a binary form. Between the 1860s and 1890s invariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the Jacobian and the Hessian for binary quartic forms and cubic forms. In 1868 Gordan proved that the graded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis. Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to Hilbert's basis theorem. Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given by Abraham Fraenkel in 1914. His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element. In addition, he had two axioms on "regular elements" inspired by work on the p-adic numbers, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative. Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one. In 1920, Emmy Noether, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called Idealtheorie in Ringbereichen (Ideal theory in rings'), analyzing ascending chain conditions with regard to (mathematical) ideals. The publication gave rise to the term "Noetherian ring", and several other mathematical objects being called Noetherian. Noted algebraist Irving Kaplansky called this work "revolutionary"; results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom. Artin, inspired by Noether's work, came up with the descending chain condition. These definitions marked the birth of abstract ring theory. === Early field theory === In 1801 Gauss introduced the integers mod p, where p is a prime number. Galois extended this in 1830 to finite fields with p n {\displaystyle p^{n}} elements. In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893. In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms. The first clear definition of an abstract field was due to Heinrich Martin Weber in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry. In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by their characteristic, and proved many theorems commonly seen today. === Other major areas === Solving of systems of linear equations, which led to linear algebra === Modern algebra === The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group. Questions of structure and classification of various mathematical objects came to the forefront. These processes were occurring throughout all of mathematics but became especially pronounced in algebra. Formal definitions through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. Hence such things as group theory and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of the 20th century were systematically exposed in Bartel van der Waerden's Moderne Algebra, the two-volume monograph published in 1930–1931 that reoriented the idea of algebra from the theory of equations to the theory of algebraic structures. == Basic concepts == By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer nontrivial theorems and fewer applications. Examples of algebraic structures with a single binary operation are: Magma Quasigroup Monoid Semigroup Group Examples involving several operations include: == Branches of abstract algebra == === Group theory === A group is a set G {\displaystyle G} together with a "group product", a binary operation ⋅ : G × G → G {\displaystyle \cdot :G\times G\rightarrow G} . The group satisfies the following defining axioms (cf. Group (mathematics) § Definition): Identity: there exists an element e {\displaystyle e} such that, for each element a {\displaystyle a} in G {\displaystyle G} , it holds that e ⋅ a = a ⋅ e = a {\displaystyle e\cdot a=a\cdot e=a} . Inverse: for each element a {\displaystyle a} of G {\displaystyle G} , there exists an element b {\displaystyle b} so that a ⋅ b = b ⋅ a = e {\displaystyle a\cdot b=b\cdot a=e} . Associativity: for each triplet of elements a , b , c {\displaystyle a,b,c} in G {\displaystyle G} , it holds that ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)} . === Ring theory === A ring is a set R {\displaystyle R} with two binary operations, addition: ( x , y ) ↦ x + y , {\displaystyle (x,y)\mapsto x+y,} and multiplication: ( x , y ) ↦ x y {\displaystyle (x,y)\mapsto xy} satisfying the following axioms. R {\displaystyle R} is a commutative group under addition. R {\displaystyle R} is a monoid under multiplication. Multiplication is distributive with respect to addition. == Applications == Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem. In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian. == See also == Coding theory Group theory List of publications in abstract algebra == References == === Bibliography === == Further reading == Allenby, R. B. J. T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2 Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra Gilbert, Jimmie; Gilbert, Linda (2005), Elements of Modern Algebra, Thomson Brooks/Cole, ISBN 978-0-534-40264-8 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 Sethuraman, B. A. (1996), Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94848-5 Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave, ISBN 978-0-333-79447-0 W. Keith Nicholson (2012) Introduction to Abstract Algebra, 4th edition, John Wiley & Sons ISBN 978-1-118-13535-8 . John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons == External links == Charles C. Pinter (1990) [1982] A Book of Abstract Algebra, second edition, from University of Maryland |
Wikipedia:Abstract and Applied Analysis#0 | Abstract and Applied Analysis is a peer-reviewed mathematics journal covering the fields of abstract and applied analysis and traditional forms of analysis such as linear and nonlinear ordinary and partial differential equations, optimization theory, and control theory. It is published by Hindawi Publishing Corporation. It was established by Athanassios G. Kartsatos (University of South Florida) in 1996, who was editor-in-chief until 2005. Martin Bohner (Missouri S&T) was editor-in-chief from 2006 until 2011 when the journal converted to a model shared by all Hindawi journals of not having an editor-in-chief, with editorial decisions made by editorial board members. The journal has faced delisting from the Journal Citation Reports (thus not receive an impact factor), for anomalous citation patterns. == References == == External links == Official website |
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