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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 |
Wikipedia:Abu Kamil#0 | Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel, Arabic: أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as Al-ḥāsib al-miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930) was a prominent Egyptian mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations. His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe. Abu Kamil made important contributions to algebra and geometry. He was the first Islamic mathematician to work easily with algebraic equations with powers higher than x 2 {\displaystyle x^{2}} (up to x 8 {\displaystyle x^{8}} ), and solved sets of non-linear simultaneous equations with three unknown variables. He illustrated the rules of signs for expanding the multiplication ( a ± b ) ( c ± d ) {\displaystyle (a\pm b)(c\pm d)} . He wrote all problems rhetorically, and some of his books lacked any mathematical notation beside those of integers. For example, he uses the Arabic expression "māl māl shayʾ" ("square-square-thing") for x 5 {\displaystyle x^{5}} (as x 5 = x 2 ⋅ x 2 ⋅ x {\displaystyle x^{5}=x^{2}\cdot x^{2}\cdot x} ). One notable feature of his works was enumerating all the possible solutions to a given equation. The Muslim encyclopedist Ibn Khaldūn classified Abū Kāmil as the second greatest algebraist chronologically after al-Khwarizmi. == Life == Almost nothing is known about the life and career of Abu Kamil except that he was a successor of al-Khwarizmi, whom he never personally met. == Works == === Book of Algebra (Kitāb fī al-jabr wa al-muqābala) === The Algebra is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of Al-Khwarizmi. Whereas the Algebra of al-Khwarizmi was geared towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's Elements. In this book Abu Kamil solves systems of equations whose solutions are whole numbers and fractions, and accepted irrational numbers (in the form of a square root or fourth root) as solutions and coefficients to quadratic equations. The first chapter teaches algebra by solving problems of application to geometry, often involving an unknown variable and square roots. The second chapter deals with the six types of problems found in Al-Khwarizmi's book, but some of which, especially those of x 2 {\displaystyle x^{2}} , were now worked out directly instead of first solving for x {\displaystyle x} and accompanied with geometrical illustrations and proofs. The third chapter contains examples of quadratic irrationalities as solutions and coefficients. The fourth chapter shows how these irrationalities are used to solve problems involving polygons. The rest of the book contains solutions for sets of indeterminate equations, problems of application in realistic situations, and problems involving unrealistic situations intended for recreational mathematics. A number of Islamic mathematicians wrote commentaries on this work, including al-Iṣṭakhrī al-Ḥāsib and ʿAli ibn Aḥmad al-ʿImrānī (d. 955-6), but both commentaries are now lost. In Europe, similar material to this book is found in the writings of Fibonacci, and some sections were incorporated and improved upon in the Latin work of John of Seville, Liber mahameleth. A partial translation to Latin was done in the 14th century by William of Luna, and in the 15th century the whole work also appeared in a Hebrew translation by Mordekhai Finzi. === Book of Rare Things in the Art of Calculation (Kitāb al-ṭarā’if fi’l-ḥisāb) === Abu Kamil describes a number of systematic procedures for finding integral solutions for indeterminate equations. It is also the earliest known Arabic work where solutions are sought to the type of indeterminate equations found in Diophantus's Arithmetica. However, Abu Kamil explains certain methods not found in any extant copy of the Arithmetica. He also describes one problem for which he found 2,678 solutions. === On the Pentagon and Decagon (Kitāb al-mukhammas wa’al-mu‘ashshar) === In this treatise algebraic methods are used to solve geometrical problems. Abu Kamil uses the equation x 4 + 3125 = 125 x 2 {\displaystyle x^{4}+3125=125x^{2}} to calculate a numerical approximation for the side of a regular pentagon in a circle of diameter 10. He also uses the golden ratio in some of his calculations. Fibonacci knew about this treatise and made extensive use of it in his Practica geometriae. === Book of Birds (Kitāb al-ṭair) === A small treatise teaching how to solve indeterminate linear systems with positive integral solutions. The title is derived from a type of problems known in the east which involve the purchase of different species of birds. Abu Kamil wrote in the introduction: I found myself before a problem that I solved and for which I discovered a great many solutions; looking deeper for its solutions, I obtained two thousand six hundred and seventy-six correct ones. My astonishment about that was great, but I found out that, when I recounted this discovery, those who did not know me were arrogant, shocked, and suspicious of me. I thus decided to write a book on this kind of calculations, with the purpose of facilitating its treatment and making it more accessible. According to Jacques Sesiano, Abu Kamil remained seemingly unparalleled throughout the Middle Ages in trying to find all the possible solutions to some of his problems. === On Measurement and Geometry (Kitāb al-misāḥa wa al-handasa) === A manual of geometry for non-mathematicians, like land surveyors and other government officials, which presents a set of rules for calculating the volume and surface area of solids (mainly rectangular parallelepipeds, right circular prisms, square pyramids, and circular cones). The first few chapters contain rules for determining the area, diagonal, perimeter, and other parameters for different types of triangles, rectangles and squares. === Lost works === Some of Abu Kamil's lost works include: A treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭaʾayn). Book on Augmentation and Diminution (Kitāb al-jamʿ wa al-tafrīq), which gained more attention after historian Franz Woepcke linked it with an anonymous Latin work, Liber augmenti et diminutionis. Book of Estate Sharing using Algebra (Kitāb al-waṣāyā bi al-jabr wa al-muqābala), which contains algebraic solutions for problems of Islamic inheritance and discusses the opinions of known jurists. Ibn al-Nadim in his Fihrist listed the following additional titles: Book of Fortune (Kitāb al-falāḥ), Book of the Key to Fortune (Kitāb miftāḥ al-falāḥ), Book of the Adequate (Kitāb al-kifāya), and Book of the Kernel (Kitāb al-ʿasīr). == Legacy == The works of Abu Kamil influenced other mathematicians, like al-Karaji and Fibonacci, and as such had a lasting impact on the development of algebra. Many of his examples and algebraic techniques were later copied by Fibonacci in his Practica geometriae and other works. Unmistakable borrowings, but without Abu Kamil being explicitly mentioned and perhaps mediated by lost treatises, are also found in Fibonacci's Liber Abaci. == On al-Khwarizmi == Abu Kamil was one of the earliest mathematicians to recognize al-Khwarizmi's contributions to algebra, defending him against Ibn Barza who attributed the authority and precedent in algebra to his grandfather, 'Abd al-Hamīd ibn Turk. Abu Kamil wrote in the introduction of his Algebra: I have studied with great attention the writings of the mathematicians, examined their assertions, and scrutinized what they explain in their works; I thus observed that the book by Muḥammad ibn Mūsā al-Khwārizmī known as Algebra is superior in the accuracy of its principle and the exactness of its argumentation. It thus behooves us, the community of mathematicians, to recognize his priority and to admit his knowledge and his superiority, as in writing his book on algebra he was an initiator and the discoverer of its principles, ... == Notes == == References == Sesiano, Jacques (2009-07-09). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore. ISBN 978-0-8218-4473-1. Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 30–32. ISBN 0-684-10114-9. O'Connor, John J.; Robertson, Edmund F., "Abu Kamil", MacTutor History of Mathematics Archive, University of St Andrews == Further reading == Yadegari, Mohammad (1978-06-01). "The Use of Mathematical Induction by Abū Kāmil Shujā' Ibn Aslam (850-930)". Isis. 69 (2): 259–262. doi:10.1086/352009. ISSN 0021-1753. JSTOR 230435. S2CID 144112534. Karpinski, L. C. (1914-02-01). "The Algebra of Abu Kamil". The American Mathematical Monthly. 21 (2): 37–48. doi:10.2307/2972073. ISSN 0002-9890. JSTOR 2972073. Herz-Fischler, Roger (June 1987). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier Univ Pr. ISBN 0-88920-152-8. Djebbar, Ahmed. Une histoire de la science arabe: Entretiens avec Jean Rosmorduc. Seuil (2001) |
Wikipedia:Action groupoid#0 | In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action X × G → X , {\displaystyle X\times G\to X,} we get the groupoid G {\displaystyle {\mathcal {G}}} (= a category whose morphisms are all invertible) where objects are elements of X {\displaystyle X} , morphisms from x {\displaystyle x} to y {\displaystyle y} are the actions of elements g {\displaystyle g} in G {\displaystyle G} such that y = x g {\displaystyle y=xg} , compositions for x → g y {\displaystyle x{\overset {g}{\to }}y} and y → h z {\displaystyle y{\overset {h}{\to }}z} is x → h g z {\displaystyle x{\overset {hg}{\to }}z} . A groupoid is often depicted using two arrows. Here the above can be written as: X × G ⇉ t s X {\displaystyle X\times G\,{\overset {s}{\underset {t}{\rightrightarrows }}}\,X} where s , t {\displaystyle s,t} denote the source and the target of a morphism in G {\displaystyle {\mathcal {G}}} ; thus, s ( x , g ) = x {\displaystyle s(x,g)=x} is the projection and t ( x , g ) = x g {\displaystyle t(x,g)=xg} is the given group action (here the set of morphisms in G {\displaystyle {\mathcal {G}}} is identified with X × G {\displaystyle X\times G} ). == In an ∞-category == Let C {\displaystyle C} be an ∞-category and G {\displaystyle G} a groupoid object in it. Then a group action or an action groupoid on an object X in C is the simplicial diagram ⋯ ⇉ ⇉ X × G × G ⇉ → X × G ⇉ X {\displaystyle \cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,X\times G\times G\,{\underset {\rightarrow }{\rightrightarrows }}\,X\times G\,\rightrightarrows \,X} that satisfies the axioms similar to an action groupoid in the usual case. == References == === Works cited === == Further reading == https://ncatlab.org/nlab/show/action+groupoid https://mathoverflow.net/questions/130950/groupoids-vs-action-groupoids https://www.math.sci.hokudai.ac.jp/~wakate/mcyr/2023/pdf/uchimura_tomoki.pdf in Japanese |
Wikipedia:Activation energy asymptotics#0 | Activation energy asymptotics (AEA), also known as large activation energy asymptotics, is an asymptotic analysis used in the combustion field utilizing the fact that the reaction rate is extremely sensitive to temperature changes due to the large activation energy of the chemical reaction. == History == The techniques were pioneered by the Russian scientists Yakov Borisovich Zel'dovich, David A. Frank-Kamenetskii and co-workers in the 30s, in their study on premixed flames and thermal explosions (Frank-Kamenetskii theory), but not popular to western scientists until the 70s. In the early 70s, due to the pioneering work of Williams B. Bush, Francis E. Fendell, Forman A. Williams, Amable Liñán and John F. Clarke, it became popular in western community and since then it was widely used to explain more complicated problems in combustion. == Method overview == In combustion processes, the reaction rate ω {\displaystyle \omega } is dependent on temperature T {\displaystyle T} in the following form (Arrhenius law), ω ( T ) ∝ e − E a / R T , {\displaystyle \omega (T)\propto \mathrm {e} ^{-E_{\rm {a}}/RT},} where E a {\displaystyle E_{\rm {a}}} is the activation energy, and R {\displaystyle R} is the universal gas constant. In general, the condition E a / R ≫ T b {\displaystyle E_{\rm {a}}/R\gg T_{b}} is satisfied, where T b {\displaystyle T_{\rm {b}}} is the burnt gas temperature. This condition forms the basis for activation energy asymptotics. Denoting T u {\displaystyle T_{\rm {u}}} for unburnt gas temperature, one can define the Zel'dovich number and heat release parameter as follows β = E a R T b T b − T u T b , q = T b − T u T u . {\displaystyle \beta ={\frac {E_{\rm {a}}}{RT_{\rm {b}}}}{\frac {T_{\rm {b}}-T_{\rm {u}}}{T_{\rm {b}}}},\quad q={\frac {T_{\rm {b}}-T_{\rm {u}}}{T_{\rm {u}}}}.} In addition, if we define a non-dimensional temperature θ = T − T u T b − T u , {\displaystyle \theta ={\frac {T-T_{\rm {u}}}{T_{\rm {b}}-T_{\rm {u}}}},} such that θ {\displaystyle \theta } approaching zero in the unburnt region and approaching unity in the burnt gas region (in other words, 0 ≤ θ ≤ 1 {\displaystyle 0\leq \theta \leq 1} ), then the ratio of reaction rate at any temperature to reaction rate at burnt gas temperature is given by ω ( T ) ω ( T b ) ∝ e − E a / R T e − E a / R T b = exp [ − β ( 1 − θ ) 1 + q 1 + q θ ] . {\displaystyle {\frac {\omega (T)}{\omega (T_{\rm {b}})}}\propto {\frac {\mathrm {e} ^{-E_{\rm {a}}/RT}}{\mathrm {e} ^{-E_{\rm {a}}/RT_{\rm {b}}}}}=\exp \left[-\beta (1-\theta ){\frac {1+q}{1+q\theta }}\right].} Now in the limit of β → ∞ {\displaystyle \beta \rightarrow \infty } (large activation energy) with q ∼ O ( 1 ) {\displaystyle q\sim O(1)} , the reaction rate is exponentially small i.e., O ( e − β ) {\displaystyle O(e^{-\beta })} and negligible everywhere, but non-negligible when β ( 1 − θ ) ∼ O ( 1 ) {\displaystyle \beta (1-\theta )\sim O(1)} . In other words, the reaction rate is negligible everywhere, except in a small region very close to burnt gas temperature, where 1 − θ ∼ O ( 1 / β ) {\displaystyle 1-\theta \sim O(1/\beta )} . Thus, in solving the conservation equations, one identifies two different regimes, at leading order, Outer convective-diffusive zone Inner reactive-diffusive layer where in the convective-diffusive zone, reaction term will be neglected and in the thin reactive-diffusive layer, convective terms can be neglected and the solutions in these two regions are stitched together by matching slopes using method of matched asymptotic expansions. The above mentioned two regime are true only at leading order since the next order corrections may involve all the three transport mechanisms. == See also == Zeldovich–Frank-Kamenetskii equation Burke–Schumann limit == References == |
Wikipedia:Acyuta Piṣāraṭi#0 | Acyuta Piṣāraṭi (also Achyuta Pisharati or Achyutha Pisharadi) (c. 1550 at Thrikkandiyur (aka Kundapura), Tirur, Kerala, India – 7 July 1621 in Kerala) was a Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. == Works == He discovered the techniques of 'the reduction of the ecliptic'. He authored Sphuta-nirnaya, Raasi-gola-sphuta-neeti (raasi meaning zodiac, gola meaning sphere and neeti roughly meaning rule), Karanottama (1593) and a four- chapter treatise Uparagakriyakrama on lunar and solar eclipses. Praveśaka: An introduction to Sanskrit grammar. Karaṇottama: Astronomical work dealing with the computation of the mean and true longitudes of the planets, with eclipses, and with the vyatipātas of the sun and moon. Uparāgakriyākrama (1593): Treatise on lunar and solar eclipses. Sphuṭanirṇaya: Astronomical text. Chāyāṣṭaka: Astronomical text. Uparāgaviṃśati: Manual on the computation of eclipses. Rāśigolasphuṭānīti: Work concerned with the reduction of the moon’s true longitude in its own orbit to the ecliptic. Veṇvārohavyākhyā: Malayalam commentary on the Veṇvāroha of Mādhava of Saṅgamagrāma (ca. 1340–1425) written at the request of the Azhvanchery Thambrakkal. Horāsāroccaya: An adaptation of the Jātakapaddhati of Śrīpati. == Narayaneeyam == Pisharati is known to have scolded and provoked an errant Narayana to take up the Brahmin's duties of prayer and religious practices. He accepted Narayana as his student. Later when Pisharati was struck with paralysis (or rheumatism by another account), Narayana, unable to bear the pain of his dear guru, by way of Gurudakshina took the disease upon himself. As a result, Pisharati is said to have been cured, but no medicine could cure Narayana. As a last resort, Narayana went to Guruvayur and requested Thunchaththu Ramanujan Ezhuthachan, a great devotee of Guruvayoorappan, to suggest a remedy for his disease. Ramajunan Ezhuthachan advised him to compose a poetical work on the Avatars (incarnations) of Lord Vishnu beginning with that of Matsya (Fish). Narayana composed beautiful slokas in praise of Lord Guruvayurappan and recited them before the deity. He was soon cured of his disease. The book of slokas written by Narayana were named Narayaneeyam. The day on which Narayana dedicated his Narayaneeyam to Sri Guruvayurappan is celebrated as "Narayaneeyam Dinam" every year at Guruvayur. == See also == Melpathur Narayana Bhattathiri Narayaneeyam Indian mathematics List of astronomers and mathematicians of the Kerala school == References == == Additional reading == David Pingree. "Acyuta Piṣāraṭi". Dictionary of Scientific Biography. S. Venkitasubramonia Iyer. "Acyuta Piṣāroṭi; His Date and Works" in JOR Madras', 22 (1952–1953), 40–46. K. V. Sarma (2008), "Acyuta Pisarati", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition) edited by Helaine Selin, p. 19, Springer, ISBN 978-1-4020-4559-2. K. Kunjunni Raja. The Contribution of Kerala to Sanskrit Literature (Madras, 1958), pp. 122–125. "Astronomy and Mathematics in Kerala" in Brahmavidyā, 27 (1963), 158–162. |
Wikipedia:Adalbold II of Utrecht#0 | Adalbold II of Utrecht (died 27 November 1026) was a bishop of Utrecht (1010–1026). == Biography == He was born in 975 probably in the Low Countries, and received his education partly from Notker of Liège. He became a canon of Laubach, and apparently was a teacher there. Henry II, Holy Roman Emperor, who had a great regard for him, invited him to the court, and nominated him as Bishop of Utrecht in 1010, and he is regarded as the principal founder of the territorial possessions of the diocese, especially by the acquisition in 1024 and 1026 of the counties of Drente and Teisterbant. He was obliged to defend the bishopric not only against frequent inroads by the Normans, but also against the aggressions of neighboring nobles. He was unsuccessful in the attempt to vindicate the possession of the district of Merwede, between the mouths of the Maas and the Waal, against Dirk III, Count of Holland, in the Battle of Vlaardingen in 1018. The imperial award required the restitution of this territory to the bishop and the destruction of a castle which Dirk had built to control the navigation of the Maas; but the expedition under Godfrey of Brabant which undertook to enforce this decision was defeated; and in the subsequent agreement the disputed land remained in Dirk's possession. Adalbold was active in promoting the building of churches and monasteries in his diocese. His principal achievement of this kind was the completion within a few years of the great romanesque Cathedral of Saint Martin at Utrecht. He restored the monastery of Tiel, and completed that of Hohorst, begun by his predecessor Ansfried. To the charge of the latter he appointed Poppo of Stablo, and thus introduced Cluniac monastic reform into the diocese. Adalbold is also mentioned as an author. A biography of Henry II, Vita Heinrici II imperatoris, has been ascribed to him; but the evidence for attributing this to him is not decisive. He wrote a mathematical treatise on establishing the volume of a sphere, Libellus de ratione inveniendi crassitudinem sphaerae, which he dedicated to Pope Sylvester II, who was himself a noted mathematician. He wrote a philosophical exposition of a passage of Boethius. A music theory discussion, Quemadmodum indubitanter musicae consonantiae judicari possint, seems, according to Hauck, to have been ascribed to him on insufficient grounds. == References == This article incorporates text from a publication in the public domain: Hauck, Albert (1908). "Adalbold". In Jackson, Samuel Macauley (ed.). New Schaff–Herzog Encyclopedia of Religious Knowledge. Vol. 1 (third ed.). London and New York: Funk and Wagnalls. p. 32. == External links == (in German) mittelalter-genealogie.de Opera Omnia by Migne Patrologia Latina with analytical indexes |
Wikipedia:Adam Kanigowski#0 | Adam Kanigowski (born 15 April 1989) is a Polish mathematician specializing in dynamical systems and ergodic theory. He is a professor at the University of Maryland. == Education == Kanigowski was born in Toruń. He earned his master's degree in mathematics from the Nicolaus Copernicus University in Toruń in 2012, and his Ph.D. in 2015 from the Institute of Mathematics of the Polish Academy of Sciences, under the supervision of Mariusz Lemańczyk and Joanna Kułaga-Przymus. His dissertation was entitled Własności ergodyczne gładkich potoków na powierzchniach (Ergodic properties of smooth flows on surfaces) and awarded the International Stefan Banach Prize in 2016. == Career and research == After graduating, Kanigowski joined Penn State University as an S. Chowla Research Assistant Professor in 2015 and then joined UMD as an assistant professor in 2018, where he was promoted to full professor in 2024. Since December 2022, Kanigowski has led a flagship project at Jagiellonian University that partly supports a research collaboration with UMD. Kanigowski's research interests include dynamical systems and ergodic theory as well as their interaction with number theory, geometry and probability theory. In particular, he is interested in randomness and chaos in smooth dynamical systems, classification problems in abstract ergodic theory, and non-standard ergodic theorems that find application in number theory. Together with collaborators, he solved several longstanding open problems and conjuctures, such as the Rokhlin problem, the Sarnak hypothesis, the Katok hypothesis and the Ratner problem. Kanigowski has published more than 30 papers in premier mathematical journals including the most prestigious ones, such as Annals of Mathematics, Journal of the American Mathematical Society, and Inventiones Mathematicae. Among his collaborators are Dmitry Dolgopyat, Bassam Fayad, Giovanni Forni, Mariusz Lemańczyk, Maksym Radziwiłł, Federico Rodriguez Hertz, and Corinna Ulcigrai. == Recognition == In 2015, the Polish Mathematical Society gave Kanigowski their Prize for Young Mathematicians (he was awarded for a series of six papers in the field of ergodic theory and operator theory). He was the 2016 winner of the International Stefan Banach Prize for a doctoral dissertation in the mathematical sciences. In 2017 he received the Kazimierz Kuratowski Award from the Institute of Mathematics of the Polish Academy of Sciences and the Polish Mathematical Society. In March 2024, the Simons Foundation named Kanigowski a 2024 Simons Fellow in Mathematics, in April he received the Institute of Mathematics of the Polish Academy of Sciences Prize for outstanding scientific achievements in mathematics for his "fundamental results in the field of dynamical systems and ergodic theory", and in July he was awarded the EMS Prize for "his outstanding contributions to the spectral classification and the mixing properties of slowly chaotic dynamical systems". == Personal life == Adam Kanigowski has two daughters. In June 2024 he finished his first Ironman triathlon. == References == |
Wikipedia:Adam Logan#0 | Adam Logan (born 1975) is a Canadian mathematician and Scrabble player. He won the World Scrabble Championship in 2005, beating Pakorn Nemitrmansuk of Thailand 3–0 in the final. He is the only player to have won the Canadian Scrabble Championship five times (1996, 2005, 2008, 2013 and 2016). He was also the winner of the 1996 National Scrabble Championship, North America's top rated player in 1997, and the winner of the Collins division of the 2014 North American Scrabble Championship. Since his competitive career began in 1985, Logan has played nearly 2200 tournament games, compiling a winning percentage of over 68%, and earning over $100,000 in prize money. He was a Putnam Fellow in 1992 and 1993. Logan completed his first degree, in mathematics, at Princeton University in 1995 and received a PhD from Harvard University in 1999. He completed his Post-doctoral work at McGill University between 2002 through 2003. From 2008 to 2009, he was employed as a Quantitative Analyst at D. E. Shaw & Co. in New York City. He works for the Tutte Institute for Mathematics and Computing in Ottawa, Ontario, Canada. == References == == External links == Adam Logan Scrabble tournament results at cross-tables.com Adam Logan's NSA player profile Adam Logan's professional home page (legacy; no longer in this position) Adam Logan at the Mathematics Genealogy Project |
Wikipedia:Adam Tanner (Jesuit theologian)#0 | Adam Tanner (in Latin, Tannerus; April 14, 1572 – May 25, 1632) was an Austrian Jesuit theologian. == Teaching career == He was born in Innsbruck, Austria. In 1589 he joined the Society of Jesus and became a teacher. By 1603 he was invited to join the Jesuit College of Ingolstadt and take the chair of theology at the University of Ingolstadt. Fifteen years later he was given a position at the University of Vienna by the Emperor Matthias. == Theological work == He was noted for his defense of the Catholic church and their practices against Lutheran reformers, as well as the Utraquists. His greatest work was the Universa theologia scholastica, published in 1626–1627. Tanner was also noted for his opposition to the witch hunts. During his time in Bavaria, he witnessed contemporary debates in which the skeptics had some success imposing limits on the witch trials. He included a number of these skeptics' arguments in his Universa theologia scholastica, for instance, "that the use of torture makes the death of innocent people inevitable, that several denunciations are not sufficient to warrant torture, that torture may not be repeated". These arguments were subsequently influential on his fellow Jesuit Friedrich Spee, another opponent of the witch hunts. == Death and controversy over his burial == He died at the village of Unken near Salzburg, and rests in an unmarked grave. Apparently, the parishioners refused to give him a Christian burial because a "hairy little imp" was found on a glass plate among his possessions. == Legacy == The crater Tannerus on the Moon is named after him. == Bibliography == Anatomiæ confessionis augustanæ, 1613, Ingolstadt. Astrologia sacra, 1615, Ingolstadt. Apologia pro Societate Iesu ex Boemiae regno: Ab eiusdem regni statibus religionis sub utraque publico decreto immerito proscripta, 1618, Vienna. Universa theologia scholastica, 1627, Ingolstadt. == References == == External links == Catholic encyclopedia Chapter in Paul Carus, History of the Devil |
Wikipedia:Adams operation#0 | In mathematics, an Adams operation, denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in symmetric function theory, at the level of vector bundles or other representing object in more abstract theories. Adams operations can be defined more generally in any λ-ring. == Adams operations in K-theory == Adams operations ψk on K theory (algebraic or topological) are characterized by the following properties. ψk are ring homomorphisms. ψk(l)= lk if l is the class of a line bundle. ψk are functorial. The fundamental idea is that for a vector bundle V on a topological space X, there is an analogy between Adams operators and exterior powers, in which ψk(V) is to Λk(V) as the power sum Σ αk is to the k-th elementary symmetric function σk of the roots α of a polynomial P(t). (Cf. Newton's identities.) Here Λk denotes the k-th exterior power. From classical algebra it is known that the power sums are certain integral polynomials Qk in the σk. The idea is to apply the same polynomials to the Λk(V), taking the place of σk. This calculation can be defined in a K-group, in which vector bundles may be formally combined by addition, subtraction and multiplication (tensor product). The polynomials here are called Newton polynomials (not, however, the Newton polynomials of interpolation theory). Justification of the expected properties comes from the line bundle case, where V is a Whitney sum of line bundles. In this special case the result of any Adams operation is naturally a vector bundle, not a linear combination of ones in K-theory. Treating the line bundle direct factors formally as roots is something rather standard in algebraic topology (cf. the Leray–Hirsch theorem). In general a mechanism for reducing to that case comes from the splitting principle for vector bundles. == Adams operations in group representation theory == The Adams operation has a simple expression in group representation theory. Let G be a group and ρ a representation of G with character χ. The representation ψk(ρ) has character χ ψ k ( ρ ) ( g ) = χ ρ ( g k ) . {\displaystyle \chi _{\psi ^{k}(\rho )}(g)=\chi _{\rho }(g^{k})\ .} == References == Adams, J.F. (May 1962). "Vector Fields on Spheres". Annals of Mathematics. Second Series. 75 (3): 603–632. doi:10.2307/1970213. JSTOR 1970213. Zbl 0112.38102. |
Wikipedia:Addition theorem#0 | In mathematics, an addition theorem is a formula such as that for the exponential function: ex + y = ex · ey, that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is more apparent than real, in that case, since there cos is an algebraic function of sin (in other words, we usually take their functions both as defined on the unit circle). The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions. To "classify" addition theorems it is necessary to put some restriction on the type of function G admitted, such that F(x + y) = G(F(x), F(y)). In this identity one can assume that F and G are vector-valued (have several components). An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution. In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups. The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law. The so-called quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of formal groups. == See also == Timeline of abelian varieties Addition theorem for spherical harmonics Mordell–Weil theorem == References == "Addition theorems in the theory of special functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994] |
Wikipedia:Additional Mathematics#0 | Additional Mathematics is a qualification in mathematics, commonly taken by students in high-school (or GCSE exam takers in the United Kingdom). It features a range of problems set out in a different format and wider content to the standard Mathematics at the same level. == Additional Mathematics in Singapore == In Singapore, Additional Mathematics is an elective subject offered to pupils in secondary school—specifically those who have an aptitude in Mathematics and are in the Normal (Academic) stream or Express stream. The syllabus covered is more in-depth as compared to Elementary Mathematics, with additional topics including Algebra binomial expansion, proofs in plane geometry, differential calculus and integral calculus. Additional Mathematics is also a prerequisite for students who are intending to offer H2 Mathematics and H2 Further Mathematics at A-level (if they choose to enter a Junior College after secondary school). Students without Additional Mathematics at the 'O' level will usually be offered H1 Mathematics instead. === Examination Format === The syllabus was updated starting with the 2021 batch of candidates. There are two written papers, each comprising half of the weightage towards the subject. Each paper is 2 hours 15 minutes long and worth 90 marks. Paper 1 has 12 to 14 questions, while Paper 2 has 9 to 11 questions. Generally, Paper 2 would have a graph plotting question based on linear law. It was originated in the year 2003 == GCSE Additional Mathematics in Northern Ireland == In Northern Ireland, Additional Mathematics was offered as a GCSE subject by the local examination board, CCEA. There were two examination papers: one which tested topics in Pure Mathematics, and one which tested topics in Mechanics and Statistics. It was discontinued in 2014 and replaced with GCSE Further Mathematics—a new qualification whose level exceeds both those offered by GCSE Mathematics, and the analogous qualifications offered in England. == Further Maths IGCSE and Additional Maths FSMQ in England == Starting from 2012, Edexcel and AQA have started a new course which is an IGCSE in Further Maths. Edexcel and AQA both offer completely different courses, with Edexcel including the calculation of solids formed through integration, and AQA not including integration. AQA's syllabus mainly offers further algebra, with the factor theorem and the more complex algebra such as algebraic fractions. It also offers differentiation up to—and including—the calculation of normals to a curve. AQA's syllabus also includes a wide selection of matrices work, which is an AS Further Mathematics topic. AQA's syllabus is much more famous than Edexcel's, mainly for its controversial decision to award an A* with Distinction (A^), a grade higher than the maximum possible grade in any Level 2 qualification; it is known colloquially as a Super A* or A**. A new Additional Maths course from 2018 is OCR Level 3 FSMQ: Additional Maths (6993). In addition to algebra, coordinate geometry, Pythagorean theorem, trigonometry and calculus, which were on the previous specification, this course also includes: 'Enumeration' content, which expands the topic of the binomial distribution to include permutations and combinations 'Numerical methods’ content, which expands upon the informal graphical approximations in GCSE 'Exponentials and Logarithms’ content, which develops the growth and decay content and the graphs section of GCSE 'Sequences' content, which uses subscript notation to support the iterative work on numerical methods. == Additional Mathematics in Malaysia == In Malaysia, Additional Mathematics is offered as an elective to upper secondary students within the public education system. This subject is included in the Sijil Pelajaran Malaysia examination. Science stream students are required to apply for Additional Mathematics as one of the subjects in the Sijil Pelajaran Malaysia examination, while Additional Mathematics is an optional subject for students who are from arts or commerce streams. Additional Mathematics in Malaysia—also commonly known as Add Maths—can be organized into two learning packages: the Core Package, which includes geometry, algebra, calculus, trigonometry and statistics, and the Elective Package, which includes science and technology application and social science application. It covers various topics including: Format for Additional Mathematics Exam based on the Malaysia Certificate of Education is as follows: Paper 1 (Duration: 2 Hours): Questions are categorised into Sections A and B and are tested based on the student's knowledge to grasp the concepts and formulae learned during their 2 years of learning. Section A consists of 12 questions in which all must all be answered, whereas Section B consists of 3 questions and students are given the choice to answer 2 of the three questions only. Each question may contain from zero to three subsets of questions with marks ranging from 2 to 8 marks. The total weighting of the paper is 80 marks and constitutes 44% of the grade. Paper 2 (Duration: 2 hours 30 minutes): Questions are categorised into 3 sections: A, B and C. Section A contains 7 questions which must all be answered. Section B contains 4 questions where students are given the choice to answer 3 out of 4 of them. Section C contains 4 questions where students are only required to answer 2 out of 4 of the given questions. All Section C questions are based on the same chapters every year and are thus predictable. A question in Section C carries 10 marks with at 3 to 4 subquestions per question. This paper tests the student's ability to apply various concepts and formulae in real-life situations. The total weighting of the paper is 100 marks and constitutes 56% of the grade. In 2020, the first batch of students learning the new syllabus, KSSM, will receive new Form 4 textbooks with new chapters which contain certain topics from A-levels. == Additional Mathematics in Mauritius == In Mauritius, Additional Mathematics, more commonly referred to as Add Maths, is offered in secondary school as an optional subject in the Arts Streams, and a compulsory subject in the Science, Technical and Economics Stream. This subject is included in the University of Cambridge International Examinations, with covered topics including functions, quadratic equations, differentiation and integration (calculus). == Additional Mathematics in Hong Kong == In Hong Kong, the syllabus of HKCEE additional mathematics covered three main topics, algebra, calculus and analytic geometry. In algebra, the topics covered include mathematical induction, binomial theorem, quadratic equations, trigonometry, inequalities, 2D-vectors and complex number, whereas in calculus, the topics covered include limit, differentiation and integration. In the HKDSE, additional mathematics has been replaced by two Mathematics Extend Modules, which include a majority of topics in the original additional mathematics, and a few topics, such as matrix and determinant, from the syllabus of HKALE pure mathematics and applied mathematics, while notably missing analytic geometry, inequalities involving absolute values and rational functions, and complex numbers (of which only the basic arithmetic is covered in the Mathematics Compulsory part). == See also == Advanced level mathematics Further mathematics == References == == External links == [Vault] (Online resources on higher mathematics) |
Wikipedia:Additive identity#0 | In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings. == Elementary examples == The additive identity familiar from elementary mathematics is zero, denoted 0. For example, 5 + 0 = 5 = 0 + 5. {\displaystyle 5+0=5=0+5.} In the natural numbers N {\displaystyle \mathbb {N} } (if 0 is included), the integers Z , {\displaystyle \mathbb {Z} ,} the rational numbers Q , {\displaystyle \mathbb {Q} ,} the real numbers R , {\displaystyle \mathbb {R} ,} and the complex numbers C , {\displaystyle \mathbb {C} ,} the additive identity is 0. This says that for a number n belonging to any of these sets, n + 0 = n = 0 + n . {\displaystyle n+0=n=0+n.} == Formal definition == Let N be a group that is closed under the operation of addition, denoted +. An additive identity for N, denoted e, is an element in N such that for any element n in N, e + n = n = n + e . {\displaystyle e+n=n=n+e.} == Further examples == In a group, the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof). A ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below). In the ring Mm × n(R) of m-by-n matrices over a ring R, the additive identity is the zero matrix, denoted O or 0, and is the m-by-n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2×2 matrices over the integers M 2 ( Z ) {\displaystyle \operatorname {M} _{2}(\mathbb {Z} )} the additive identity is 0 = [ 0 0 0 0 ] {\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}} In the quaternions, 0 is the additive identity. In the ring of functions from R → R {\displaystyle \mathbb {R} \to \mathbb {R} } , the function mapping every number to 0 is the additive identity. In the additive group of vectors in R n , {\displaystyle \mathbb {R} ^{n},} the origin or zero vector is the additive identity. == Properties == === The additive identity is unique in a group === Let (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G, 0 + g = g = g + 0 , 0 ′ + g = g = g + 0 ′ . {\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.} It then follows from the above that 0 ′ = 0 ′ + 0 = 0 ′ + 0 = 0 . {\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.} === The additive identity annihilates ring elements === In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any s in S, s · 0 = 0. This follows because: s ⋅ 0 = s ⋅ ( 0 + 0 ) = s ⋅ 0 + s ⋅ 0 ⇒ s ⋅ 0 = s ⋅ 0 − s ⋅ 0 ⇒ s ⋅ 0 = 0. {\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}} === The additive and multiplicative identities are different in a non-trivial ring === Let R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let r be any element of R. Then r = r × 1 = r × 0 = 0 {\displaystyle r=r\times 1=r\times 0=0} proving that R is trivial, i.e. R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown. == See also == 0 (number) Additive inverse Identity element Multiplicative identity == References == == Bibliography == David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, ISBN 0-471-43334-9. == External links == Uniqueness of additive identity in a ring at PlanetMath. |
Wikipedia:Additive inverse#0 | In mathematics, the additive inverse of an element x, denoted −x, is the element that when added to x, yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element. In elementary mathematics, the additive inverse is often referred to as the opposite number, or its negative. The unary operation of arithmetic negation is closely related to subtraction and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers. == Common examples == When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1. The concept can also be extended to algebraic expressions, which is often used when balancing equations. == Relation to subtraction == The additive inverse is closely related to subtraction, which can be viewed as an addition using the inverse: a − b = a + (−b). Conversely, the additive inverse can be thought of as subtraction from zero: −a = 0 − a. This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors. == Formal definition == Given an algebraic structure defined under addition ( S , + ) {\displaystyle (S,+)} with an additive identity e ∈ S {\displaystyle e\in S} , an element x ∈ S {\displaystyle x\in S} has an additive inverse y {\displaystyle y} if and only if y ∈ S {\displaystyle y\in S} , x + y = e {\displaystyle x+y=e} , and y + x = e {\displaystyle y+x=e} . Addition is typically only used to refer to a commutative operation, but it is not necessarily associative. When it is associative, so ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} , the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist. The definition requires closure, that the additive element y {\displaystyle y} be found in S {\displaystyle S} . This is why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be negative numbers, which is why the integers do have an additive inverse. == Further examples == In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). In a Boolean ring, which has elements { 0 , 1 } {\displaystyle \{0,1\}} addition is often defined as the symmetric difference. So 0 + 0 = 0 {\displaystyle 0+0=0} , 0 + 1 = 1 {\displaystyle 0+1=1} , 1 + 0 = 1 {\displaystyle 1+0=1} , and 1 + 1 = 0 {\displaystyle 1+1=0} . Our additive identity is 0, and both elements are their own additive inverse as 0 + 0 = 0 {\displaystyle 0+0=0} and 1 + 1 = 0 {\displaystyle 1+1=0} . == See also == Absolute value (related through the identity |−x| = |x|). Monoid Inverse function Involution (mathematics) Multiplicative inverse Reflection (mathematics) Reflection symmetry Semigroup == Notes and references == |
Wikipedia:Adela Ruiz de Royo#0 | Adela María Ruiz González, customary married name Ruiz de Royo (December 15, 1943 – June 19, 2019) was a Spanish-born Panamanian mathematics academic and educator. She served as the First Lady of Panama from 1978 until 1982 during the presidency of her husband, Aristides Royo. She also served President of the Panamanian Academy of Language (Academia Panameña de la Lengua). == Biography == Ruiz was born Adela María Ruiz González in a home in the municipality of Grado, Asturias, Spain to parents, José María and Rosalina. She was raised in the nearby city of Oviedo alongside her three sisters, Marta, Mabel, and María José. Ruiz was nicknamed Deli. By 1960, Ruiz had moved to Salamanca to study medicine. That same year, she met her future husband, a Panamanian national and fellow student at the University of Salamanca named Aristides Royo. The couple married in the early 1960s and eventually had three children - Marta Elena, Irma Natalia, and Aristides José. Ruiz, Royo and their oldest daughter, Marta, moved to Panama permanently on September 17, 1965. In addition to her own career, Ruiz held the role of the wife of a government minister and politician. She became First Lady of Panama from 1978 to 1982. During her tenure as first lady, Ruiz created the Asociación Pro Obras de Beneficencia. Ruiz was diagnosed with colon and liver cancer in 2017. She died from the disease on June 19, 2019, at the age of 75. Adela Ruiz was survived by her husband, Aristides Royo, and their three children, Marta Elena, Natalia, and Arístides José. Her funeral was held at the National Sanctuary in Bella Vista, Panama City on June 24, 2019. Ruiz's ashes were returned to her native Spain, where they were partially buried at the Praviano cemetery in Riberas, Asturias. A second funeral mass was held at the Carmelite Catholic Church of Oviedo on October 4, 2019. Shortly before the funeral, her remaining ashes were sprinkled into the Cantabrian Sea by her husband and children. In December 2019, Ruiz's daughter, Natalia Royo de Hagerman, was appointed as Panama's ambassador to the United Kingdom. == References == |
Wikipedia:Adela Świątek#0 | Adela Świątek (2 September 1945 – 27 September 2019) was a Polish mathematician and teacher at the Nicolaus Copernicus University Faculty of Mathematics and Computer Science, and a popularizer of mathematics. == Biography == Adela Świątek was born on 2 September 1945 in Człuchów, Poland. She graduated in mathematics from the Nicolaus Copernicus University in 1968 and since then, she was professionally associated with the University's Institute of Mathematics. In 1975, she defended her doctoral thesis entitled: Category of cube objects, supervised by Professor Stanisław Balcerzyk. In the years 1996–2001 she headed the postgraduate study Mathematics Teachers. In her professional work, she dealt with issues related to teaching mathematics and the education of students specializing in teaching. She was one of the originators of the Task League competition in the region, and she participated in the organization of the International Mathematical Kangaroo Competition in Poland. She was a co-author of popular science publications thematically related to the Mathematical Kangaroo competition. She was associated with the competition since 1993 and she participated in the work of international bodies setting competition tasks, prepared tasks for Polish environments (translations, etc.). She conducted workshops and lectures for competition winners as part of youth mathematics camps. An obituary notes: "Ada's knowledge and teaching skills did not result solely from her great natural abilities. She did not theorize and did not limit herself to repeating to students the content found in pedagogical publications. For many years she taught in the so-called university classes at the 4th Secondary School in Toruń, she lived school every day, understood its essence, difficulties and challenges. Many of her students took part in the finals of the Mathematical Olympiad." In 2009, Dr. Adela Świątek was awarded the Medal of the National Education Commission. She died on 27 September 2019 in Toruń and was buried there at the Central Municipal Cemetery. == Selected publications == Świątek was the author or co-author of more than 30 popular science publications about mathematics, including: Bobiński Zbigniew, Jarek Paweł, Nodzyński Piotr, Świątek Adela, Uscki Mirosław: Mathematics with a cheerful Kangaroo (grades III-V middle school). Toruń: Wydawnictwo Aksjomat, 1995. 89 pp. Bobiński Zbigniew, Jarek Paweł, Nodzyński Piotr, Świątek Adela, Uscki Mirosław: Mathematics with a cheerful Kangaroo (grades VII-VIII and I-II of secondary school). Toruń: Wydawnictwo Aksjomat, 1995. 181 pp. Bobiński Zbigniew, Burnicka Katarzyna, Jarek Paweł, Nodzyński Piotr, Świątek Adela, Uscki Mirosław: Mathematics with a cheerful Kangaroo, grades III-VI. Toruń: Wydawnictwo Aksjomat, 1995. 111 pp. Świątek Adela, Uscki Mirosław: Operative teaching of mathematics on selected examples of developed concepts. – Works of the Mathematical and Methodological Seminar of the Faculty of Mathematics and Computer Science of the Nicolaus Copernicus University, No. 2 1995. pp. 113–123 Bobiński Zbigniew, Burnicka Katarzyna, Jarek Paweł, Nodzyński Piotr, Świątek Adela, Uscki Mirosław: Mathematics with a cheerful Kangaroo: Maluch level, Benjamin. Toruń: Wydawnictwo Aksjomat, 1997. 127 pp. Bobiński Zbigniew, Jarek Paweł, Nodzyński Piotr, Świątek Adela, Uscki Mirosław: Mathematics with a cheerful Kangaroo: Cadet, Junior levels. Toruń: Wydawnictwo Aksjomat, 1997. 207 pp. Bobiński Zbigniew, Jarek Paweł, Nodzyński Piotr, Świątek Adela, Uscki Mirosław: Mathematics with a cheerful Kangaroo: Student level. Toruń: Wydawnictwo Aksjomat, 1997. 130 pp. Świątek Adela: On cyclic systems of equations / Jaroslav Švrček; translation Adela Świątek. In: Regular and semi-regular polygons; On irrational numbers; On cyclic systems of equations / Zbigniew Bobiński [et al.]. Toruń: Wydawnictwo Aksjomat, 2003. pp. 69–82, ill. (Mathematical Miniatures; 11) == References == |
Wikipedia:Aderemi Kuku#0 | Aderemi Oluyomi Kuku (20 March 1941 – 13 February 2022) was a Nigerian mathematician and academic, known for his contributions to the fields of algebraic K-theory and non-commutative geometry. Born in Ijebu-Ode, Ogun State, Nigeria, Kuku began his academic journey at Makerere University College and the University of Ibadan, where he earned his B.Sc. in Mathematics, followed by his M.Sc. and Ph.D. under Joshua Leslie and Hyman Bass. His doctoral research focused on the Whitehead group of p-adic integral group-rings of finite p-groups. Kuku held positions as a lecturer and professor at various Nigerian universities, including the University of Ife and the University of Ibadan, where he served as Head of the Department of Mathematics and Dean of the Postgraduate School. His research involved developing methods for computing higher K-theory of non-commutative rings and articulating higher algebraic K-theory in the language of Mackey functors. His work on equivariant higher algebraic K-theory and its generalisations impacted the field. During his career, Kuku was elected a Fellow of the African Academy of Sciences, the Nigerian Academy of Science, and the American Mathematical Society. He also received the Nigerian National Order of Merit and the Officer of the Order of the Niger. He served as president of the African Mathematical Union, where he worked to promote mathematics across Africa. Kuku's work extended beyond research, encompassing education and mentorship. He authored several books and articles, supervised graduate students, and fostered international collaborations. == Early life and education == Aderemi Oluyomi Kuku was born on 20 March 1941, in Ijebu-Ode, Ogun State, Nigeria. His father Busari Adeoye Kuku was a photographer, and mother Abusatu Oriaran Baruwa was a trader. Aderemi was the third of four brothers, all of whom pursued professional careers. Kuku began his education at Bishop Oluwole Memorial School in Agege, Lagos State, and continued at St James School Anglican primary school in Oke-Odan, Ogun State. He came first in the first school leaving certificate which led to his admission to Eko Boys' High School in Lagos, Nigeria, where he served as Head Boy in his final year, 1959. After completing his secondary education, Kuku moved to Abeokuta Grammar School to pursue his Higher School Certificate, focusing on Mathematics, Further Mathematics, and Physics. His performance earned him a scholarship from the African Scholarship Program of American Universities, administered by the United States Agency for International Development. However, he chose to attend Makerere University College in Kampala, Uganda, for his undergraduate studies. At Makerere University College, then part of the University of East Africa and an external college of the University of London, Kuku supplemented his coursework with self-study. He graduated in 1965 with a B.Sc. in Mathematics. Upon graduation, Kuku returned to Nigeria, where he was appointed as an Assistant Lecturer at the University of Ife in Ile-Ife, Osun State. Despite his lecturing duties, he registered at the University of Ibadan to pursue a Master's Degree. The University of Ibadan, established in 1948 as a College of the University of London and becoming an independent university in 1962, provided Kuku with the opportunity to delve deeper into his mathematical interests. Under the supervision of Joshua Leslie and with the external examination by Hyman Bass, Kuku submitted his M.Sc. thesis titled A survey of Algebraic K-theory, and bagged an M.Sc. in 1968. == Academic career == Aderemi Kuku pursued an academic career after earning his M.Sc. from the University of Ibadan. He was appointed as a Lecturer II at the University of Ife, where he began his teaching career. His dedication to education and research led to his promotion within the university in 1967. In 1968, Kuku transitioned to the University of Ibadan as a Lecturer II in Mathematics, succeeding Joshua Leslie. During this period, he married Felicia Osifunke Kalesanwo. Kuku's tenure at the University of Ibadan was marked by a focus on his research interests. He accepted an invitation from Hyman Bass to conduct research at Columbia University in New York City. This opportunity allowed him to spend the year 1970–71 at Columbia University, where he worked with Bass and submitted his thesis On the Whitehead group of p-adic integral group-rings of finite p-groups, earning his Ph.D. in 1971. Upon his return to Nigeria, Kuku was promoted to Senior Lecturer in Mathematics in 1976, to Reader in Mathematics in 1980, and to Professor of Mathematics at the University of Ibadan in 1982. Throughout the 1980s and 1990s, Kuku served as President of the African Mathematical Union, where he worked to establish commissions and networks to promote mathematics across the continent. He also organised the fourth Pan-African Congress of Mathematicians in Morocco in 1995, where he delivered a plenary lecture. Kuku's academic work extended beyond Nigeria. He held visiting professorships and research positions at various universities and research institutes. His international experience contributed to global mathematical discourse. == Research contributions == Kuku's work in mathematics, specifically algebraic K-theory and non-commutative geometry, has linked various mathematical disciplines, aiding in the understanding of algebraic structures and their applications. His research intersects algebra, number theory, and geometry, using K-theory and cyclic homology methodologies. He formulated higher algebraic K-theory using representation theory, specifically Mackey functors. This led to the development of equivariant higher algebraic K-theory and its relative generalisations in exact and Waldhausen categories. Kuku developed methods for computing higher K-theory of non-commutative rings, including non-commutative orders and group-rings, twisted polynomials, and Laurent series rings over orders. These methods are used in the calculations of higher K-theory of virtually infinite cyclic groups within the Farrell–Jones conjecture's context. In non-commutative geometry, Kuku's research includes entire/periodic cyclic homology and K-theory of involutive Banach algebras, C*-algebras, group C*-algebras, Hopf algebras, and quantum groups. He studied the connections between K-theory and cyclic homology of these structures. In collaboration with M. Mahdavi-Hezavehi, Kuku studied the algebraic structure of subgroups in the group of units of a non-commutative local ring. His work with N.Q. Tho and D.N. Diep on compact Lie group C*-algebras and compact quantum groups resulted in the construction and study of non-commutative Chern characters from K-theory to entire/periodic cyclic homology. Kuku's research includes the Baum–Connes conjecture in non-commutative geometry. He formulated this conjecture for the action of quantum groups and confirmed it in specific cases, such as for quantum SU2. In addition to theoretical work, Kuku has contributed to the computational aspects of algebraic K-theory, focusing on the computation of K-groups, periodic cyclic homology, and Chern characters of various non-commutative structures. His research has implications for other fields such as mathematical physics, dynamical systems, econometrics, and control theory, fostering collaborations across different research areas. Kuku has authored research articles, books, and monographs throughout his career. He has mentored M.Sc., M.Phil., and Ph.D. candidates, and guided postdoctoral researchers and mathematicians globally. His book Abstract Algebra, is used as a textbook for undergraduate and beginning graduate students. His advanced texts, such as "Representation Theory and Higher Algebraic K-theory," serve as resources for researchers and graduate students. == Awards and honours == Kuku has received various recognition for his works and contributions. He was the recipient of the Ogun State Special Merit Award in 1987, the Officer of the Order of the Niger in 2008, and the Nigerian National Order of Merit Award in 2009. Kuku has been a fellow of different academic institutions including the African Academy of Sciences, and a member of European Academy of Arts Science and Humanities in 1986, the Nigerian Academy of Science and The World Academy of Sciences in 1989, the Mongolian Academy of Sciences in 2005, and the American Mathematical Society in 2012. In 2011 during his 70th birthday, Kuku was honoured with an International Conference on Algebraic K-theory at Nanjing University in China. As the President of the African Mathematical Union, Kuku played a role in the establishment of networks to promote mathematics across the continent. == Death and legacy == Kuku died on 13 February 2022. His work in algebraic K-theory and non-commutative geometry has influenced mathematical sciences. His research has resulted in practical tools and methodologies for future research. Kuku has focused on education and has mentored mathematicians, particularly in Africa. He has helped establish networks that support students and researchers across the continent. Kuku has guided many students academically. The Ph.D. students and postdoctoral researchers he mentored have contributed to mathematics. Kuku's work has been recognised in conferences and special journal issues. Kuku has advocated for the application of mathematics to solve real-world problems. Kuku has written numerous publications and articles, which are often cited and used as references in mathematical research. Kuku's efforts to advance mathematics in Africa have impacted the continent's academic landscape. His work has raised the profile of African mathematics and encouraged international collaboration and recognition. Kuku's legacy includes his academic work, his role as a mentor and educator, and his influence on the development of mathematical sciences in Africa and beyond. == Selected publications == == References == === Citations === === Bibliography === Nigerian Society of Physical Sciences (15 November 2020). "Biography of the Distinguished Professor Aderemi Oluyomi Kuku". Journal of the Nigerian Society of Physical Sciences. Nigerian Society of Physical Sciences. doi:10.46481/jnsps.2020.4. ISSN 2714-4704. Houston, Johnny L. (26 November 2018). "Remembering Professor Aderemi O. Kuku (1941–2022), an Internationally Recognized Mathematician and Scholar" (PDF). American Mathematical Society. |
Wikipedia:Adhemar Bultheel#0 | Adhemar François Bultheel (born 1948) is a Belgian mathematician and computer scientist, the former president of the Belgian Mathematical Society. He is a prolific book reviewer for the Bulletin of the Belgian Mathematical Society and for the European Mathematical Society. His research concerns approximation theory. == Education and career == Bultheel was born in Zwijndrecht, Belgium on December 14, 1948. He earned a licenciate in mathematics in 1970 and another in industrial mathematics in 1971, both from KU Leuven. He remained at KU Leuven for a bachelor's degree in 1975 and a PhD in mathematics in 1979. His dissertation, Recursive Rational Approximation, was jointly supervised by Patrick M. Dewilde and Hugo Van de Vel. Except for a year of military service, he was employed at KU Leuven for his entire career, retiring as a professor emeritus of computer science in 2009. He was president of the Belgian Mathematical Society for 2002–2005. == Books == Bultheel is the author of: Laurent Series and their Padé Approximations (Operator Theory: Advances and Applications 27, Birkhäuser, 1987) Linear Algebra, Rational Approximation and Orthogonal Polynomials (with Marc van Barel, Studies in Computational Mathematics 6, North-Holland, 1997) Orthogonal Rational Functions (with Pablo González-Vera, Erik Hendriksen, and Olav Njåstad, Cambridge Monographs on Applied and Computational Mathematics 5, Cambridge University Press, 1999) Inleiding tot de numerieke wiskunde (Acco, 2006) == References == == External links == Home page Adhemar Bultheel publications indexed by Google Scholar |
Wikipedia:Adi Ben-Israel#0 | Adi Ben-Israel (born November 6, 1933) is a mathematician and an engineer, working in applied mathematics, optimization, statistics, operations research and other areas. He is a Professor of Operations Research at Rutgers University, New Jersey. == Research topics == Ben-Israel's research has included generalized inverses of matrices, in particular the Moore–Penrose pseudoinverse, and of operators, their extremal properties, computation and applications. as well as local inverses of nonlinear mappings. In the area of linear algebra, he studied the matrix volume and its applications, basic, approximate and least-norm solutions, and the geometry of subspaces. He wrote about ordered incidence geometry and the geometric foundations of convexity. In the topic of iterative methods, he published papers about the Newton method for systems of equations with rectangular or singular Jacobians, directional Newton methods, the quasi-Halley method, Newton and Halley methods for complex roots, and the inverse Newton transform. Ben-Israel's research into optimization included linear programming, a Newtonian bracketing method of convex minimization, input optimization, and risk modeling of dynamic programming, and the calculus of variations. He also studied various aspects of clustering and location theory, and investigated decisions under uncertainty. == Publications == Books Generalized Inverses: Theory and Applications, with T.N.E. Greville, J. Wiley, New York, 1974 Optimality in Nonlinear Programming: A Feasible Directions Approach, with A. Ben-Tal and S. Zlobec, J. Wiley, New York, 1981 Mathematik mit DERIVE (German), with W. Koepf and R.P. Gilbert, Vieweg-Verlag, Berlin, ISBN 3-528-06549-4, 1993 Computer Supported Calculus: With MACSYMA, with R.P. Gilbert, Springer-Verlag, Vienna, ISBN 3-211-82924-5, 2001 Generalized Inverses: Theory and Applications (2nd edition), with T.N.E. Greville, Springer-Verlag, New York, ISBN 0-387-00293-6, 2003 Selected articles Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math. 11(1963), 667–699, (with A. Charnes) A Newton–Raphson method for the solution of systems of equations, J. Math. Anal. Appl. 15(1966), 243–252 Linear equations and inequalities on finite-dimensional, real or complex, vector spaces: A unified theory, J. Math. Anal. Appl. 27(1969), 367–389 Ordered incidence geometry and the geometric foundations of convexity theory, J. Geometry 30(1987), 103–122, (with A. Ben-Tal) Input optimization for infinite horizon discounted programs, J. Optimiz. Th. Appl. 61(1989), 347–357, (with S.D. Flaam) Certainty equivalents and information measures: Duality and extremal principles, J. Math. Anal. Appl. 157(1991), 211–236 (with A. Ben-Tal and M. Teboulle). A volume associated with mxn matrices, Lin. Algeb. and Appl. 167(1992), 87–111. The Moore of the Moore–Penrose inverse, Electron. J. Lin. Algeb. 9(2002), 150–157. The Newton bracketing method for convex minimization, Comput. Optimiz. and Appl. 21(2002), 213–229 (with Y. Levin). An inverse Newton transform, Contemporary Math. 568(2012), 27–40. A concentrated Cauchy distribution with finite moments, Annals of Oper. Res. (to appear) == References == == External links == Adi Ben-Israel personal webpage Adi Ben-Israel at the Mathematics Genealogy Project |
Wikipedia:Adjacency algebra#0 | In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A. Some other similar mathematical objects are also called "adjacency algebra". == Properties == Properties of the adjacency algebra of G are associated with various spectral, adjacency and connectivity properties of G. Statement. The number of walks of length d between vertices i and j is equal to the (i, j)-th element of Ad. Statement. The dimension of the adjacency algebra of a connected graph of diameter d is at least d + 1. Corollary. A connected graph of diameter d has at least d + 1 distinct eigenvalues. == Spectral Properties == Adjacency algebra is closely linked with Spectral graph theory due to both them having the involvement of the Adjacency matrix of a graph and its eigenvalues. Spectral graph theory is about how eigenvalues, eigenvectors, and other linear-algebraic quantities give us useful information about a graph, for example about how well-connected it is, how well we can cluster or color the nodes, and how quickly random walks converge to a limiting distribution. In the context of Spectral graph theory the eigenvectors and the eigenvalues of graph's Adjacency matrix provide valid and essential insights into several structural properties such as connectivity, clustering, coloring, and the behavior of random walks, these insights are tightly tied to the adjacency algebra generated by the Adjacency matrix, including all its powers and linear combinations. Both concepts are concerned with the adjacency matrix but approach it differently and are looked with different perspectives, Spectral graph theory focuses more on the specific spectral properties (eigenvalues and eigenvectors) to extract information about the graph's connectivity. In contrast adjacency algebra works more with matrix powers and linear combinations to understand the graphs structure more algebraically. == Applications of Adjacency Algebra == Adjacency algebra has several applications both in mathematics and computer science. Adjacency algebra can be utilized in network analysis and graph theory, where the powers of the adjacency matrix can help to determine how connected a specific graph is by tracking paths of length k between vertices. It can also be used with graph partitioning and graph theory based AI and Large language models. == References == |
Wikipedia:Adjacency matrix#0 | In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix, which contains information about the degree of each vertex. == Definition == For a simple graph with vertex set U = {u1, ..., un}, the adjacency matrix is a square n × n matrix A such that its element Aij is 1 when there is an edge from vertex ui to vertex uj, and 0 when there is no edge. The diagonal elements of the matrix are all 0, since edges from a vertex to itself (loops) are not allowed in simple graphs. It is also sometimes useful in algebraic graph theory to replace the nonzero elements with algebraic variables. The same concept can be extended to multigraphs and graphs with loops by storing the number of edges between each two vertices in the corresponding matrix element, and by allowing nonzero diagonal elements. Loops may be counted either once (as a single edge) or twice (as two vertex-edge incidences), as long as a consistent convention is followed. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. === Of a bipartite graph === The adjacency matrix A of a bipartite graph whose two parts have r and s vertices can be written in the form A = ( 0 r , r B B T 0 s , s ) , {\displaystyle A={\begin{pmatrix}0_{r,r}&B\\B^{\mathsf {T}}&0_{s,s}\end{pmatrix}},} where B is an r × s matrix, and 0r,r and 0s,s represent the r × r and s × s zero matrices. In this case, the smaller matrix B uniquely represents the graph, and the remaining parts of A can be discarded as redundant. B is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U = {u1, ..., ur}, V = {v1, ..., vs} and edges E. The biadjacency matrix is the r × s 0–1 matrix B in which bi,j = 1 if and only if (ui, vj) ∈ E. If G is a bipartite multigraph or weighted graph, then the elements bi,j are taken to be the number of edges between the vertices or the weight of the edge (ui, vj), respectively. === Variations === An (a, b, c)-adjacency matrix A of a simple graph has Ai,j = a if (i, j) is an edge, b if it is not, and c on the diagonal. The Seidel adjacency matrix is a (−1, 1, 0)-adjacency matrix. This matrix is used in studying strongly regular graphs and two-graphs. The distance matrix has in position (i, j) the distance between vertices vi and vj. The distance is the length of a shortest path connecting the vertices. Unless lengths of edges are explicitly provided, the length of a path is the number of edges in it. The distance matrix resembles a high power of the adjacency matrix, but instead of telling only whether or not two vertices are connected (i.e., the connection matrix, which contains Boolean values), it gives the exact distance between them. == Examples == === Undirected graphs === The convention followed here (for undirected graphs) is that each edge adds 1 to the appropriate cell in the matrix, and each loop (an edge from a vertex to itself) adds 2 to the appropriate cell on the diagonal in the matrix. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix. === Directed graphs === The adjacency matrix of a directed graph can be asymmetric. One can define the adjacency matrix of a directed graph either such that a non-zero element Aij indicates an edge from i to j or it indicates an edge from j to i. The former definition is commonly used in graph theory and social network analysis (e.g., sociology, political science, economics, psychology). The latter is more common in other applied sciences (e.g., dynamical systems, physics, network science) where A is sometimes used to describe linear dynamics on graphs. Using the first definition, the in-degrees of a vertex can be computed by summing the entries of the corresponding column and the out-degree of vertex by summing the entries of the corresponding row. When using the second definition, the in-degree of a vertex is given by the corresponding row sum and the out-degree is given by the corresponding column sum. === Trivial graphs === The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros. The adjacency matrix of an empty graph is a zero matrix. == Properties == === Spectrum === The adjacency matrix of an undirected simple graph is symmetric, and therefore has a complete set of real eigenvalues and an orthogonal eigenvector basis. The set of eigenvalues of a graph is the spectrum of the graph. It is common to denote the eigenvalues by λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n . {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}.} The greatest eigenvalue λ 1 {\displaystyle \lambda _{1}} is bounded above by the maximum degree. This can be seen as result of the Perron–Frobenius theorem, but it can be proved easily. Let v be one eigenvector associated to λ 1 {\displaystyle \lambda _{1}} and x the entry in which v has maximum absolute value. Without loss of generality assume vx is positive since otherwise you simply take the eigenvector -v, also associated to λ 1 {\displaystyle \lambda _{1}} . Then λ 1 v x = ( A v ) x = ∑ y = 1 n A x , y v y ≤ ∑ y = 1 n A x , y v x = v x deg ( x ) . {\displaystyle \lambda _{1}v_{x}=(Av)_{x}=\sum _{y=1}^{n}A_{x,y}v_{y}\leq \sum _{y=1}^{n}A_{x,y}v_{x}=v_{x}\deg(x).} For d-regular graphs, d is the first eigenvalue of A for the vector v = (1, ..., 1) (it is easy to check that it is an eigenvalue and it is the maximum because of the above bound). The multiplicity of this eigenvalue is the number of connected components of G, in particular λ 1 > λ 2 {\displaystyle \lambda _{1}>\lambda _{2}} for connected graphs. It can be shown that for each eigenvalue λ i {\displaystyle \lambda _{i}} , its opposite − λ i = λ n + 1 − i {\displaystyle -\lambda _{i}=\lambda _{n+1-i}} is also an eigenvalue of A if G is a bipartite graph. In particular −d is an eigenvalue of any d-regular bipartite graph. The difference λ 1 − λ 2 {\displaystyle \lambda _{1}-\lambda _{2}} is called the spectral gap and it is related to the expansion of G. It is also useful to introduce the spectral radius of A {\displaystyle A} denoted by λ ( G ) = max | λ i | < d | λ i | {\displaystyle \lambda (G)=\max _{\left|\lambda _{i}\right|<d}|\lambda _{i}|} . This number is bounded by λ ( G ) ≥ 2 d − 1 − o ( 1 ) {\displaystyle \lambda (G)\geq 2{\sqrt {d-1}}-o(1)} . This bound is tight in the Ramanujan graphs. === Isomorphism and invariants === Suppose two directed or undirected graphs G1 and G2 with adjacency matrices A1 and A2 are given. G1 and G2 are isomorphic if and only if there exists a permutation matrix P such that P A 1 P − 1 = A 2 . {\displaystyle PA_{1}P^{-1}=A_{2}.} In particular, A1 and A2 are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore serve as isomorphism invariants of graphs. However, two graphs may possess the same set of eigenvalues but not be isomorphic. Such linear operators are said to be isospectral. === Matrix powers === If A is the adjacency matrix of the directed or undirected graph G, then the matrix An (i.e., the matrix product of n copies of A) has an interesting interpretation: the element (i, j) gives the number of (directed or undirected) walks of length n from vertex i to vertex j. If n is the smallest nonnegative integer, such that for some i, j, the element (i, j) of An is positive, then n is the distance between vertex i and vertex j. A great example of how this is useful is in counting the number of triangles in an undirected graph G, which is exactly the trace of A3 divided by 3 or 6 depending on whether the graph is directed or not. We divide by those values to compensate for the overcounting of each triangle. In an undirected graph, each triangle will be counted twice for all three nodes, because the path can be followed clockwise or counterclockwise : ijk or ikj. The adjacency matrix can be used to determine whether or not the graph is connected. If a directed graph has a nilpotent adjacency matrix (i.e., if there exists n such that An is the zero matrix), then it is a directed acyclic graph. == Data structures == The adjacency matrix may be used as a data structure for the representation of graphs in computer programs for manipulating graphs. The main alternative data structure, also in use for this application, is the adjacency list. The space needed to represent an adjacency matrix and the time needed to perform operations on them is dependent on the matrix representation chosen for the underlying matrix. Sparse matrix representations only store non-zero matrix entries and implicitly represent the zero entries. They can, for example, be used to represent sparse graphs without incurring the space overhead from storing the many zero entries in the adjacency matrix of the sparse graph. In the following section the adjacency matrix is assumed to be represented by an array data structure so that zero and non-zero entries are all directly represented in storage. Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only |V |2 / 8 bytes to represent a directed graph, or (by using a packed triangular format and only storing the lower triangular part of the matrix) approximately |V |2 / 16 bytes to represent an undirected graph. Although slightly more succinct representations are possible, this method gets close to the information-theoretic lower bound for the minimum number of bits needed to represent all n-vertex graphs. For storing graphs in text files, fewer bits per byte can be used to ensure that all bytes are text characters, for instance by using a Base64 representation. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a large sparse graph, adjacency lists require less storage space, because they do not waste any space representing edges that are not present. An alternative form of adjacency matrix (which, however, requires a larger amount of space) replaces the numbers in each element of the matrix with pointers to edge objects (when edges are present) or null pointers (when there is no edge). It is also possible to store edge weights directly in the elements of an adjacency matrix. Besides the space tradeoff, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list, and takes time proportional to the number of neighbors. With an adjacency matrix, an entire row must instead be scanned, which takes a larger amount of time, proportional to the number of vertices in the whole graph. On the other hand, testing whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list. == See also == Laplacian matrix Self-similarity matrix == References == == External links == Weisstein, Eric W. "Adjacency matrix". MathWorld. Fluffschack — an educational Java web start game demonstrating the relationship between adjacency matrices and graphs. Open Data Structures - Section 12.1 - AdjacencyMatrix: Representing a Graph by a Matrix, Pat Morin Café math : Adjacency Matrices of Graphs : Application of the adjacency matrices to the computation generating series of walks. |
Wikipedia:Adjugate matrix#0 | In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose. The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: A adj ( A ) = det ( A ) I , {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {I} ,} where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant. == Definition == The adjugate of A is the transpose of the cofactor matrix C of A, adj ( A ) = C T . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}.} In more detail, suppose R is a (unital) commutative ring and A is an n × n matrix with entries from R. The (i, j)-minor of A, denoted Mij, is the determinant of the (n − 1) × (n − 1) matrix that results from deleting row i and column j of A. The cofactor matrix of A is the n × n matrix C whose (i, j) entry is the (i, j) cofactor of A, which is the (i, j)-minor times a sign factor: C = ( ( − 1 ) i + j M i j ) 1 ≤ i , j ≤ n . {\displaystyle \mathbf {C} =\left((-1)^{i+j}\mathbf {M} _{ij}\right)_{1\leq i,j\leq n}.} The adjugate of A is the transpose of C, that is, the n × n matrix whose (i, j) entry is the (j,i) cofactor of A, adj ( A ) = C T = ( ( − 1 ) i + j M j i ) 1 ≤ i , j ≤ n . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}=\left((-1)^{i+j}\mathbf {M} _{ji}\right)_{1\leq i,j\leq n}.} === Important consequence === The adjugate is defined so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are the determinant det(A). That is, A adj ( A ) = adj ( A ) A = det ( A ) I , {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\operatorname {adj} (\mathbf {A} )\mathbf {A} =\det(\mathbf {A} )\mathbf {I} ,} where I is the n × n identity matrix. This is a consequence of the Laplace expansion of the determinant. The above formula implies one of the fundamental results in matrix algebra, that A is invertible if and only if det(A) is an invertible element of R. When this holds, the equation above yields adj ( A ) = det ( A ) A − 1 , A − 1 = det ( A ) − 1 adj ( A ) . {\displaystyle {\begin{aligned}\operatorname {adj} (\mathbf {A} )&=\det(\mathbf {A} )\mathbf {A} ^{-1},\\\mathbf {A} ^{-1}&=\det(\mathbf {A} )^{-1}\operatorname {adj} (\mathbf {A} ).\end{aligned}}} == Examples == === 1 × 1 generic matrix === Since the determinant of a 0 × 0 matrix is 1, the adjugate of any 1 × 1 matrix (complex scalar) is I = [ 1 ] {\displaystyle \mathbf {I} ={\begin{bmatrix}1\end{bmatrix}}} . Observe that A adj ( A ) = adj ( A ) A = ( det A ) I . {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\operatorname {adj} (\mathbf {A} )\mathbf {A} =(\det \mathbf {A} )\mathbf {I} .} === 2 × 2 generic matrix === The adjugate of the 2 × 2 matrix A = [ a b c d ] {\displaystyle \mathbf {A} ={\begin{bmatrix}a&b\\c&d\end{bmatrix}}} is adj ( A ) = [ d − b − c a ] . {\displaystyle \operatorname {adj} (\mathbf {A} )={\begin{bmatrix}d&-b\\-c&a\end{bmatrix}}.} By direct computation, A adj ( A ) = [ a d − b c 0 0 a d − b c ] = ( det A ) I . {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )={\begin{bmatrix}ad-bc&0\\0&ad-bc\end{bmatrix}}=(\det \mathbf {A} )\mathbf {I} .} In this case, it is also true that det(adj(A)) = det(A) and hence that adj(adj(A)) = A. === 3 × 3 generic matrix === Consider a 3 × 3 matrix A = [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] . {\displaystyle \mathbf {A} ={\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{bmatrix}}.} Its cofactor matrix is C = [ + | b 2 b 3 c 2 c 3 | − | b 1 b 3 c 1 c 3 | + | b 1 b 2 c 1 c 2 | − | a 2 a 3 c 2 c 3 | + | a 1 a 3 c 1 c 3 | − | a 1 a 2 c 1 c 2 | + | a 2 a 3 b 2 b 3 | − | a 1 a 3 b 1 b 3 | + | a 1 a 2 b 1 b 2 | ] , {\displaystyle \mathbf {C} ={\begin{bmatrix}+{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\\\\-{\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\end{vmatrix}}\\\\+{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\end{bmatrix}},} where | a b c d | = det [ a b c d ] . {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=\det \!{\begin{bmatrix}a&b\\c&d\end{bmatrix}}.} Its adjugate is the transpose of its cofactor matrix, adj ( A ) = C T = [ + | b 2 b 3 c 2 c 3 | − | a 2 a 3 c 2 c 3 | + | a 2 a 3 b 2 b 3 | − | b 1 b 3 c 1 c 3 | + | a 1 a 3 c 1 c 3 | − | a 1 a 3 b 1 b 3 | + | b 1 b 2 c 1 c 2 | − | a 1 a 2 c 1 c 2 | + | a 1 a 2 b 1 b 2 | ] . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}={\begin{bmatrix}+{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}\\&&\\-{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}\\&&\\+{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\end{bmatrix}}.} === 3 × 3 numeric matrix === As a specific example, we have adj [ − 3 2 − 5 − 1 0 − 2 3 − 4 1 ] = [ − 8 18 − 4 − 5 12 − 1 4 − 6 2 ] . {\displaystyle \operatorname {adj} \!{\begin{bmatrix}-3&2&-5\\-1&0&-2\\3&-4&1\end{bmatrix}}={\begin{bmatrix}-8&18&-4\\-5&12&-1\\4&-6&2\end{bmatrix}}.} It is easy to check the adjugate is the inverse times the determinant, −6. The −1 in the second row, third column of the adjugate was computed as follows. The (2,3) entry of the adjugate is the (3,2) cofactor of A. This cofactor is computed using the submatrix obtained by deleting the third row and second column of the original matrix A, [ − 3 − 5 − 1 − 2 ] . {\displaystyle {\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix}}.} The (3,2) cofactor is a sign times the determinant of this submatrix: ( − 1 ) 3 + 2 det [ − 3 − 5 − 1 − 2 ] = − ( − 3 ⋅ − 2 − − 5 ⋅ − 1 ) = − 1 , {\displaystyle (-1)^{3+2}\operatorname {det} \!{\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix}}=-(-3\cdot -2--5\cdot -1)=-1,} and this is the (2,3) entry of the adjugate. == Properties == For any n × n matrix A, elementary computations show that adjugates have the following properties: adj ( I ) = I {\displaystyle \operatorname {adj} (\mathbf {I} )=\mathbf {I} } , where I {\displaystyle \mathbf {I} } is the identity matrix. adj ( 0 ) = 0 {\displaystyle \operatorname {adj} (\mathbf {0} )=\mathbf {0} } , where 0 {\displaystyle \mathbf {0} } is the zero matrix, except that if n = 1 {\displaystyle n=1} then adj ( 0 ) = I {\displaystyle \operatorname {adj} (\mathbf {0} )=\mathbf {I} } . adj ( c A ) = c n − 1 adj ( A ) {\displaystyle \operatorname {adj} (c\mathbf {A} )=c^{n-1}\operatorname {adj} (\mathbf {A} )} for any scalar c. adj ( A T ) = adj ( A ) T {\displaystyle \operatorname {adj} (\mathbf {A} ^{\mathsf {T}})=\operatorname {adj} (\mathbf {A} )^{\mathsf {T}}} . det ( adj ( A ) ) = ( det A ) n − 1 {\displaystyle \det(\operatorname {adj} (\mathbf {A} ))=(\det \mathbf {A} )^{n-1}} . If A is invertible, then adj ( A ) = ( det A ) A − 1 {\displaystyle \operatorname {adj} (\mathbf {A} )=(\det \mathbf {A} )\mathbf {A} ^{-1}} . It follows that: adj(A) is invertible with inverse (det A)−1A. adj(A−1) = adj(A)−1. adj(A) is entrywise polynomial in A. In particular, over the real or complex numbers, the adjugate is a smooth function of the entries of A. Over the complex numbers, adj ( A ¯ ) = adj ( A ) ¯ {\displaystyle \operatorname {adj} ({\overline {\mathbf {A} }})={\overline {\operatorname {adj} (\mathbf {A} )}}} , where the bar denotes complex conjugation. adj ( A ∗ ) = adj ( A ) ∗ {\displaystyle \operatorname {adj} (\mathbf {A} ^{*})=\operatorname {adj} (\mathbf {A} )^{*}} , where the asterisk denotes conjugate transpose. Suppose that B is another n × n matrix. Then adj ( A B ) = adj ( B ) adj ( A ) . {\displaystyle \operatorname {adj} (\mathbf {AB} )=\operatorname {adj} (\mathbf {B} )\operatorname {adj} (\mathbf {A} ).} This can be proved in three ways. One way, valid for any commutative ring, is a direct computation using the Cauchy–Binet formula. The second way, valid for the real or complex numbers, is to first observe that for invertible matrices A and B, adj ( B ) adj ( A ) = ( det B ) B − 1 ( det A ) A − 1 = ( det A B ) ( A B ) − 1 = adj ( A B ) . {\displaystyle \operatorname {adj} (\mathbf {B} )\operatorname {adj} (\mathbf {A} )=(\det \mathbf {B} )\mathbf {B} ^{-1}(\det \mathbf {A} )\mathbf {A} ^{-1}=(\det \mathbf {AB} )(\mathbf {AB} )^{-1}=\operatorname {adj} (\mathbf {AB} ).} Because every non-invertible matrix is the limit of invertible matrices, continuity of the adjugate then implies that the formula remains true when one of A or B is not invertible. A corollary of the previous formula is that, for any non-negative integer k, adj ( A k ) = adj ( A ) k . {\displaystyle \operatorname {adj} (\mathbf {A} ^{k})=\operatorname {adj} (\mathbf {A} )^{k}.} If A is invertible, then the above formula also holds for negative k. From the identity ( A + B ) adj ( A + B ) B = det ( A + B ) B = B adj ( A + B ) ( A + B ) , {\displaystyle (\mathbf {A} +\mathbf {B} )\operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {B} =\det(\mathbf {A} +\mathbf {B} )\mathbf {B} =\mathbf {B} \operatorname {adj} (\mathbf {A} +\mathbf {B} )(\mathbf {A} +\mathbf {B} ),} we deduce A adj ( A + B ) B = B adj ( A + B ) A . {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {B} =\mathbf {B} \operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {A} .} Suppose that A commutes with B. Multiplying the identity AB = BA on the left and right by adj(A) proves that det ( A ) adj ( A ) B = det ( A ) B adj ( A ) . {\displaystyle \det(\mathbf {A} )\operatorname {adj} (\mathbf {A} )\mathbf {B} =\det(\mathbf {A} )\mathbf {B} \operatorname {adj} (\mathbf {A} ).} If A is invertible, this implies that adj(A) also commutes with B. Over the real or complex numbers, continuity implies that adj(A) commutes with B even when A is not invertible. Finally, there is a more general proof than the second proof, which only requires that an n × n matrix has entries over a field with at least 2n + 1 elements (e.g. a 5 × 5 matrix over the integers modulo 11). det(A+tI) is a polynomial in t with degree at most n, so it has at most n roots. Note that the ijth entry of adj((A+tI)(B)) is a polynomial of at most order n, and likewise for adj(A+tI)adj(B). These two polynomials at the ijth entry agree on at least n + 1 points, as we have at least n + 1 elements of the field where A+tI is invertible, and we have proven the identity for invertible matrices. Polynomials of degree n which agree on n + 1 points must be identical (subtract them from each other and you have n + 1 roots for a polynomial of degree at most n – a contradiction unless their difference is identically zero). As the two polynomials are identical, they take the same value for every value of t. Thus, they take the same value when t = 0. Using the above properties and other elementary computations, it is straightforward to show that if A has one of the following properties, then adjA does as well: upper triangular, lower triangular, diagonal, orthogonal, unitary, symmetric, Hermitian, normal. If A is skew-symmetric, then adj(A) is skew-symmetric for even n and symmetric for odd n. Similarly, if A is skew-Hermitian, then adj(A) is skew-Hermitian for even n and Hermitian for odd n. If A is invertible, then, as noted above, there is a formula for adj(A) in terms of the determinant and inverse of A. When A is not invertible, the adjugate satisfies different but closely related formulas. If rk(A) ≤ n − 2, then adj(A) = 0. If rk(A) = n − 1, then rk(adj(A)) = 1. (Some minor is non-zero, so adj(A) is non-zero and hence has rank at least one; the identity adj(A)A = 0 implies that the dimension of the nullspace of adj(A) is at least n − 1, so its rank is at most one.) It follows that adj(A) = αxyT, where α is a scalar and x and y are vectors such that Ax = 0 and AT y = 0. === Column substitution and Cramer's rule === Partition A into column vectors: A = [ a 1 ⋯ a n ] . {\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}.} Let b be a column vector of size n. Fix 1 ≤ i ≤ n and consider the matrix formed by replacing column i of A by b: ( A ← i b ) = def [ a 1 ⋯ a i − 1 b a i + 1 ⋯ a n ] . {\displaystyle (\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )\ {\stackrel {\text{def}}{=}}\ {\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{i-1}&\mathbf {b} &\mathbf {a} _{i+1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}.} Laplace expand the determinant of this matrix along column i. The result is entry i of the product adj(A)b. Collecting these determinants for the different possible i yields an equality of column vectors ( det ( A ← i b ) ) i = 1 n = adj ( A ) b . {\displaystyle \left(\det(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )\right)_{i=1}^{n}=\operatorname {adj} (\mathbf {A} )\mathbf {b} .} This formula has the following concrete consequence. Consider the linear system of equations A x = b . {\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} .} Assume that A is non-singular. Multiplying this system on the left by adj(A) and dividing by the determinant yields x = adj ( A ) b det A . {\displaystyle \mathbf {x} ={\frac {\operatorname {adj} (\mathbf {A} )\mathbf {b} }{\det \mathbf {A} }}.} Applying the previous formula to this situation yields Cramer's rule, x i = det ( A ← i b ) det A , {\displaystyle x_{i}={\frac {\det(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )}{\det \mathbf {A} }},} where xi is the ith entry of x. === Characteristic polynomial === Let the characteristic polynomial of A be p ( s ) = det ( s I − A ) = ∑ i = 0 n p i s i ∈ R [ s ] . {\displaystyle p(s)=\det(s\mathbf {I} -\mathbf {A} )=\sum _{i=0}^{n}p_{i}s^{i}\in R[s].} The first divided difference of p is a symmetric polynomial of degree n − 1, Δ p ( s , t ) = p ( s ) − p ( t ) s − t = ∑ 0 ≤ j + k < n p j + k + 1 s j t k ∈ R [ s , t ] . {\displaystyle \Delta p(s,t)={\frac {p(s)-p(t)}{s-t}}=\sum _{0\leq j+k<n}p_{j+k+1}s^{j}t^{k}\in R[s,t].} Multiply sI − A by its adjugate. Since p(A) = 0 by the Cayley–Hamilton theorem, some elementary manipulations reveal adj ( s I − A ) = Δ p ( s I , A ) . {\displaystyle \operatorname {adj} (s\mathbf {I} -\mathbf {A} )=\Delta p(s\mathbf {I} ,\mathbf {A} ).} In particular, the resolvent of A is defined to be R ( z ; A ) = ( z I − A ) − 1 , {\displaystyle R(z;\mathbf {A} )=(z\mathbf {I} -\mathbf {A} )^{-1},} and by the above formula, this is equal to R ( z ; A ) = Δ p ( z I , A ) p ( z ) . {\displaystyle R(z;\mathbf {A} )={\frac {\Delta p(z\mathbf {I} ,\mathbf {A} )}{p(z)}}.} === Jacobi's formula === The adjugate also appears in Jacobi's formula for the derivative of the determinant. If A(t) is continuously differentiable, then d ( det A ) d t ( t ) = tr ( adj ( A ( t ) ) A ′ ( t ) ) . {\displaystyle {\frac {d(\det \mathbf {A} )}{dt}}(t)=\operatorname {tr} \left(\operatorname {adj} (\mathbf {A} (t))\mathbf {A} '(t)\right).} It follows that the total derivative of the determinant is the transpose of the adjugate: d ( det A ) A 0 = adj ( A 0 ) T . {\displaystyle d(\det \mathbf {A} )_{\mathbf {A} _{0}}=\operatorname {adj} (\mathbf {A} _{0})^{\mathsf {T}}.} === Cayley–Hamilton formula === Let pA(t) be the characteristic polynomial of A. The Cayley–Hamilton theorem states that p A ( A ) = 0 . {\displaystyle p_{\mathbf {A} }(\mathbf {A} )=\mathbf {0} .} Separating the constant term and multiplying the equation by adj(A) gives an expression for the adjugate that depends only on A and the coefficients of pA(t). These coefficients can be explicitly represented in terms of traces of powers of A using complete exponential Bell polynomials. The resulting formula is adj ( A ) = ∑ s = 0 n − 1 A s ∑ k 1 , k 2 , … , k n − 1 ∏ ℓ = 1 n − 1 ( − 1 ) k ℓ + 1 ℓ k ℓ k ℓ ! tr ( A ℓ ) k ℓ , {\displaystyle \operatorname {adj} (\mathbf {A} )=\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{\ell =1}^{n-1}{\frac {(-1)^{k_{\ell }+1}}{\ell ^{k_{\ell }}k_{\ell }!}}\operatorname {tr} (\mathbf {A} ^{\ell })^{k_{\ell }},} where n is the dimension of A, and the sum is taken over s and all sequences of kl ≥ 0 satisfying the linear Diophantine equation s + ∑ ℓ = 1 n − 1 ℓ k ℓ = n − 1. {\displaystyle s+\sum _{\ell =1}^{n-1}\ell k_{\ell }=n-1.} For the 2 × 2 case, this gives adj ( A ) = I 2 ( tr A ) − A . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {I} _{2}(\operatorname {tr} \mathbf {A} )-\mathbf {A} .} For the 3 × 3 case, this gives adj ( A ) = 1 2 I 3 ( ( tr A ) 2 − tr A 2 ) − A ( tr A ) + A 2 . {\displaystyle \operatorname {adj} (\mathbf {A} )={\frac {1}{2}}\mathbf {I} _{3}\!\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)-\mathbf {A} (\operatorname {tr} \mathbf {A} )+\mathbf {A} ^{2}.} For the 4 × 4 case, this gives adj ( A ) = 1 6 I 4 ( ( tr A ) 3 − 3 tr A tr A 2 + 2 tr A 3 ) − 1 2 A ( ( tr A ) 2 − tr A 2 ) + A 2 ( tr A ) − A 3 . {\displaystyle \operatorname {adj} (\mathbf {A} )={\frac {1}{6}}\mathbf {I} _{4}\!\left((\operatorname {tr} \mathbf {A} )^{3}-3\operatorname {tr} \mathbf {A} \operatorname {tr} \mathbf {A} ^{2}+2\operatorname {tr} \mathbf {A} ^{3}\right)-{\frac {1}{2}}\mathbf {A} \!\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)+\mathbf {A} ^{2}(\operatorname {tr} \mathbf {A} )-\mathbf {A} ^{3}.} The same formula follows directly from the terminating step of the Faddeev–LeVerrier algorithm, which efficiently determines the characteristic polynomial of A. In general, adjugate matrix of arbitrary dimension N matrix can be computed by Einstein's convention. ( adj ( A ) ) i N j N = 1 ( N − 1 ) ! ϵ i 1 i 2 … i N ϵ j 1 j 2 … j N A j 1 i 1 A j 2 i 2 … A j N − 1 i N − 1 {\displaystyle (\operatorname {adj} (\mathbf {A} ))_{i_{N}}^{j_{N}}={\frac {1}{(N-1)!}}\epsilon _{i_{1}i_{2}\ldots i_{N}}\epsilon ^{j_{1}j_{2}\ldots j_{N}}A_{j_{1}}^{i_{1}}A_{j_{2}}^{i_{2}}\ldots A_{j_{N-1}}^{i_{N-1}}} == Relation to exterior algebras == The adjugate can be viewed in abstract terms using exterior algebras. Let V be an n-dimensional vector space. The exterior product defines a bilinear pairing V × ∧ n − 1 V → ∧ n V . {\displaystyle V\times \wedge ^{n-1}V\to \wedge ^{n}V.} Abstractly, ∧ n V {\displaystyle \wedge ^{n}V} is isomorphic to R, and under any such isomorphism the exterior product is a perfect pairing. That is, it yields an isomorphism ϕ : V → ≅ Hom ( ∧ n − 1 V , ∧ n V ) . {\displaystyle \phi \colon V\ {\xrightarrow {\cong }}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V).} This isomorphism sends each v ∈ V to the map ϕ v {\displaystyle \phi _{\mathbf {v} }} defined by ϕ v ( α ) = v ∧ α . {\displaystyle \phi _{\mathbf {v} }(\alpha )=\mathbf {v} \wedge \alpha .} Suppose that T : V → V is a linear transformation. Pullback by the (n − 1)th exterior power of T induces a morphism of Hom spaces. The adjugate of T is the composite V → ϕ Hom ( ∧ n − 1 V , ∧ n V ) → ( ∧ n − 1 T ) ∗ Hom ( ∧ n − 1 V , ∧ n V ) → ϕ − 1 V . {\displaystyle V\ {\xrightarrow {\phi }}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V)\ {\xrightarrow {(\wedge ^{n-1}T)^{*}}}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V)\ {\xrightarrow {\phi ^{-1}}}\ V.} If V = Rn is endowed with its canonical basis e1, ..., en, and if the matrix of T in this basis is A, then the adjugate of T is the adjugate of A. To see why, give ∧ n − 1 R n {\displaystyle \wedge ^{n-1}\mathbf {R} ^{n}} the basis { e 1 ∧ ⋯ ∧ e ^ k ∧ ⋯ ∧ e n } k = 1 n . {\displaystyle \{\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n}\}_{k=1}^{n}.} Fix a basis vector ei of Rn. The image of ei under ϕ {\displaystyle \phi } is determined by where it sends basis vectors: ϕ e i ( e 1 ∧ ⋯ ∧ e ^ k ∧ ⋯ ∧ e n ) = { ( − 1 ) i − 1 e 1 ∧ ⋯ ∧ e n , if k = i , 0 otherwise. {\displaystyle \phi _{\mathbf {e} _{i}}(\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n})={\begin{cases}(-1)^{i-1}\mathbf {e} _{1}\wedge \dots \wedge \mathbf {e} _{n},&{\text{if}}\ k=i,\\0&{\text{otherwise.}}\end{cases}}} On basis vectors, the (n − 1)st exterior power of T is e 1 ∧ ⋯ ∧ e ^ j ∧ ⋯ ∧ e n ↦ ∑ k = 1 n ( det A j k ) e 1 ∧ ⋯ ∧ e ^ k ∧ ⋯ ∧ e n . {\displaystyle \mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{j}\wedge \dots \wedge \mathbf {e} _{n}\mapsto \sum _{k=1}^{n}(\det A_{jk})\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n}.} Each of these terms maps to zero under ϕ e i {\displaystyle \phi _{\mathbf {e} _{i}}} except the k = i term. Therefore, the pullback of ϕ e i {\displaystyle \phi _{\mathbf {e} _{i}}} is the linear transformation for which e 1 ∧ ⋯ ∧ e ^ j ∧ ⋯ ∧ e n ↦ ( − 1 ) i − 1 ( det A j i ) e 1 ∧ ⋯ ∧ e n . {\displaystyle \mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{j}\wedge \dots \wedge \mathbf {e} _{n}\mapsto (-1)^{i-1}(\det A_{ji})\mathbf {e} _{1}\wedge \dots \wedge \mathbf {e} _{n}.} That is, it equals ∑ j = 1 n ( − 1 ) i + j ( det A j i ) ϕ e j . {\displaystyle \sum _{j=1}^{n}(-1)^{i+j}(\det A_{ji})\phi _{\mathbf {e} _{j}}.} Applying the inverse of ϕ {\displaystyle \phi } shows that the adjugate of T is the linear transformation for which e i ↦ ∑ j = 1 n ( − 1 ) i + j ( det A j i ) e j . {\displaystyle \mathbf {e} _{i}\mapsto \sum _{j=1}^{n}(-1)^{i+j}(\det A_{ji})\mathbf {e} _{j}.} Consequently, its matrix representation is the adjugate of A. If V is endowed with an inner product and a volume form, then the map φ can be decomposed further. In this case, φ can be understood as the composite of the Hodge star operator and dualization. Specifically, if ω is the volume form, then it, together with the inner product, determines an isomorphism ω ∨ : ∧ n V → R . {\displaystyle \omega ^{\vee }\colon \wedge ^{n}V\to \mathbf {R} .} This induces an isomorphism Hom ( ∧ n − 1 R n , ∧ n R n ) ≅ ∧ n − 1 ( R n ) ∨ . {\displaystyle \operatorname {Hom} (\wedge ^{n-1}\mathbf {R} ^{n},\wedge ^{n}\mathbf {R} ^{n})\cong \wedge ^{n-1}(\mathbf {R} ^{n})^{\vee }.} A vector v in Rn corresponds to the linear functional ( α ↦ ω ∨ ( v ∧ α ) ) ∈ ∧ n − 1 ( R n ) ∨ . {\displaystyle (\alpha \mapsto \omega ^{\vee }(\mathbf {v} \wedge \alpha ))\in \wedge ^{n-1}(\mathbf {R} ^{n})^{\vee }.} By the definition of the Hodge star operator, this linear functional is dual to *v. That is, ω∨∘ φ equals v ↦ *v∨. == Higher adjugates == Let A be an n × n matrix, and fix r ≥ 0. The rth higher adjugate of A is an ( n r ) × ( n r ) {\textstyle {\binom {n}{r}}\!\times \!{\binom {n}{r}}} matrix, denoted adjr A, whose entries are indexed by size r subsets I and J of {1, ..., m} . Let Ic and Jc denote the complements of I and J, respectively. Also let A I c , J c {\displaystyle \mathbf {A} _{I^{c},J^{c}}} denote the submatrix of A containing those rows and columns whose indices are in Ic and Jc, respectively. Then the (I, J) entry of adjr A is ( − 1 ) σ ( I ) + σ ( J ) det A J c , I c , {\displaystyle (-1)^{\sigma (I)+\sigma (J)}\det \mathbf {A} _{J^{c},I^{c}},} where σ(I) and σ(J) are the sum of the elements of I and J, respectively. Basic properties of higher adjugates include : adj0(A) = det A. adj1(A) = adj A. adjn(A) = 1. adjr(BA) = adjr(A) adjr(B). adj r ( A ) C r ( A ) = C r ( A ) adj r ( A ) = ( det A ) I ( n r ) {\displaystyle \operatorname {adj} _{r}(\mathbf {A} )C_{r}(\mathbf {A} )=C_{r}(\mathbf {A} )\operatorname {adj} _{r}(\mathbf {A} )=(\det \mathbf {A} )I_{\binom {n}{r}}} , where Cr(A) denotes the rth compound matrix. Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting ∧ r V {\displaystyle \wedge ^{r}V} and ∧ n − r V {\displaystyle \wedge ^{n-r}V} for V {\displaystyle V} and ∧ n − 1 V {\displaystyle \wedge ^{n-1}V} , respectively. == Iterated adjugates == Iteratively taking the adjugate of an invertible matrix A k times yields adj ⋯ adj ⏞ k ( A ) = det ( A ) ( n − 1 ) k − ( − 1 ) k n A ( − 1 ) k , {\displaystyle \overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} )=\det(\mathbf {A} )^{\frac {(n-1)^{k}-(-1)^{k}}{n}}\mathbf {A} ^{(-1)^{k}},} det ( adj ⋯ adj ⏞ k ( A ) ) = det ( A ) ( n − 1 ) k . {\displaystyle \det(\overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} ))=\det(\mathbf {A} )^{(n-1)^{k}}.} For example, adj ( adj ( A ) ) = det ( A ) n − 2 A . {\displaystyle \operatorname {adj} (\operatorname {adj} (\mathbf {A} ))=\det(\mathbf {A} )^{n-2}\mathbf {A} .} det ( adj ( adj ( A ) ) ) = det ( A ) ( n − 1 ) 2 . {\displaystyle \det(\operatorname {adj} (\operatorname {adj} (\mathbf {A} )))=\det(\mathbf {A} )^{(n-1)^{2}}.} == See also == Cayley–Hamilton theorem Cramer's rule Trace diagram Jacobi's formula Faddeev–LeVerrier algorithm Compound matrix == References == == Bibliography == Roger A. Horn and Charles R. Johnson (2013), Matrix Analysis, Second Edition. Cambridge University Press, ISBN 978-0-521-54823-6 Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1 == External links == Matrix Reference Manual Online matrix calculator (determinant, track, inverse, adjoint, transpose) Compute Adjugate matrix up to order 8 "Adjugate of { { a, b, c }, { d, e, f }, { g, h, i } }". Wolfram Alpha. |
Wikipedia:Adolf Kiefer#0 | Adolf Kiefer (22 June 1857 - 15 November 1929) was a Swiss mathematician, working mainly on geometry. == Life == Kiefer was born in 1857 in Selzach, Switzerland to Jakob, a farmer, village mayor and member of Solothurn parliament. In 1880 he graduated as a teacher of mathematics and physics. He taught, from 1881-2, at the Concordia Institute, in Zürich. Kiefer's 1881 doctorate was from the University of Zürich for the thesis Der Kontakt höherer Ordnung bei algebraischen Flächen. Between 1882 and 1894 he taught geometry and technical drawing at the canton school in Frauenfeld, becoming deputy head in 1886 and head in 1888. In 1894 he became director of the Concordia Institute. Concordia closed after the First World War, Kiefer taught elsewhere including Zurich teachers' college. Kiefer was a member of the committee of the first International Congress of Mathematicians. Kiefer retired in 1926 due to ill health. He became an honorary member of the Schweizerische Naturforschende Gesellschaft in 1928. He died 15 November 1929. == Work == === Books === Ueber die geraden Kegel und Cylinder, welche durch gegebene Punkte des Raumes gehen oder gegebene gerade Linien des Raumes berühren (1888) Kiefer published over thirty papers, mostly on geometry. === Papers === Über Kräftezerlegung (1904) Über die Kettenlinie (1915) Von der Cykloide (1917)* Zum Normalenproblem bei den Flächen zweiten Grades(1921) Zwei spezielle Tetraeder (1925). == References == == External links == S. U. Eminger, C. F. Geiser and R. Rudio (2014). The Men Behind the First International Congress of Mathematicians St Andrews (PDF) (Thesis). pp. 121–123. F. R. Scherrer (1930). "Adolf Kiefer". Verhandlungen der Schweizerischen Naturforschenden Gesellschaft. 111: 444–446. MacTutor biography |
Wikipedia:Adolfo Rumbos#0 | Adolfo J. Rumbos is an American mathematician whose research interests include nonlinear analysis and boundary value problems. He is the Joseph N. Fiske Professor of Mathematics at Pomona College in Claremont, California. Rumbos is a graduate of Humboldt State University. He completed his Ph.D. in 1989 from the University of California, Santa Cruz with the dissertation Applications of the Leray-Schauder topological Degree to Boundary Value Problems for Semilinear Differential Equations supervised by Edward M. Landesman. == References == == External links == Home page |
Wikipedia:Adriaan Cornelis Zaanen#0 | Adriaan Cornelis "Aad" Zaanen (14 June 1913 in Rotterdam – 1 April 2003 in Wassenaar) was a Dutch mathematician working in analysis. He is known for his books on Riesz spaces (together with Wim Luxemburg). == Biography == Zaanen was born in Rotterdam, where he attended the Hogere Burgerschool. He graduated in 1930 with excellent marks, and started his studies in mathematics at Leiden University. Having obtained his master's degree in 1935, he did research under the guidance of his doctoral advisor Johannes Droste, and was awarded a Ph.D. in 1938. His doctoral thesis dealt with the convergence of series of eigenvalues of boundary value problems of the Sturm–Liouville type. The same year he was appointed a mathematics teacher at the Hogere Burgerschool in Rotterdam, a profession that he continued until 1947. In the next years and also in the period of the German occupation of the Netherlands, Zaanen continued to do mathematical research in his spare time. He studied Stefan Banach's Théorie des Opérations Linéaires, the book that laid the foundations of functional analysis, and Marshall H. Stone's Linear Transformations in Hilbert Space. During this period he wrote nine scientific papers on integral equations with symmetrisable kernels that were published in the Proceedings of the Royal Netherlands Academy of Arts and Sciences in 1946-47. In parallel to his job as a secondary-school teacher, Zaanen was appointed in 1946 as a mathematics teacher for three hours per week at the Technische Hogeschool Delft, and as an unpaid privaatdocent at Leiden University where he taught a course on Lebesgue integration. In 1947 Zaanen accepted the position of Professor of Mathematics at the Technische Hogeschool Bandoeng. In 1950 he returned to the Netherlands where he was appointed Professor of Mathematics at the Technische Hogeschool Delft. In these years he continued his work on the book Linear Analysis, which was published in 1953 and for years was a prominent work on functional analysis and the theory of integral equations. In 1956 Zaanen was appointed Professor of Mathematics at Leiden University. There he started a large research programme into the theory of Riesz spaces, together with his first doctoral student Wim Luxemburg, Professor of Mathematics at the California Institute of Technology. Most of their results were published in a series of papers in the Proceedings of the Royal Netherlands Academy of Arts and Sciences. Unusually for mathematics research in the Netherlands at the time, Zaanen pursued a long-term research programme involving a number of collaborators and doctoral students. Eight doctoral theses on various topics in the theory of Riesz spaces were produced in this school. Zaanen took retirement in 1982. == Publications == Zaanen published almost 70 papers in scientific journals and reviewed conference proceedings. He is however best known for four large books that each took a prominent place in scientific literature: A.C. Zaanen, Linear Analysis, North-Holland Publishing Company, Amsterdam and P. Noordhoff, Groningen (1953, 1957, 1960), 600 pages Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Company, Amsterdam (1967) 604 pages. Revised and enlarged edition of An Introduction to the Theory of Integration (1958, 1961, 1965). W.A.J. Luxemburg and A.C. Zaanen, Riesz Spaces Volume I, North-Holland Publishing Company, Amsterdam London (1971), 514 pages A.C. Zaanen, Riesz Spaces II, North-Holland Publishing Company, Amsterdam New York Oxford (1983), 720 pages == Other functions and honours == Zaanen was elected a member of the Royal Netherlands Academy of Arts and Sciences in 1960. He served as President of the Dutch Mathematical Society from 1970 until 1972. He was an editor of the Society's journal Nieuw Archief voor Wiskunde from 1953 until 1982. In 1988 he was appointed an honorary member of the Society. He served as a member of the Curatorium of the Mathematisch Centrum from 1965 until 1979. On his retirement in 1982 Zaanen was appointed Knight of the Order of the Netherlands Lion. == References == |
Wikipedia:Adrian Constantin#0 | Adrian Constantin (born 22 April 1970) is a Romanian-Austrian mathematician who does research in the field of nonlinear partial differential equations. He is a professor at the University of Vienna and has made groundbreaking contributions to the mathematics of wave propagation. He is listed as an ISI Highly Cited Researcher with more than 160 publications and 11,000 citations. == Life and career == Adrian Constantin was born in Timișoara, Romania, where he studied at the Nikolaus Lenau High School. He was later educated at the University of Nice Sophia Antipolis (BSc 1991, MSc 1992) and at New York University (NYU), where he got his PhD in 1996 under Henry McKean with the thesis "The Periodic Problem for the Camassa–Holm equation". He did post-doctoral work at the University of Basel and at the University of Zurich. After a short period as a lecturer at the University of Newcastle upon Tyne, he became a professor at the University of Lund in 2000, and then was Erasmus Smith's Professor of Mathematics at Trinity College Dublin (TCD) from 2004 to 2008, and was made a fellow in 2005. Since then he has been university professor for partial differential equations at the University of Vienna, and also had a chair at King's College London during the period 2011-2014. Constantin specializes in the role of mathematics in geophysics using nonlinear partial differential equations to mathematically model currents and waves in the oceans and in the atmosphere. These flows and waves play an important role in the El Niño climate phenomenon and in natural disasters such as tsunamis. His approach takes into account the fact that the surface of the earth is curved and the importance of the Coriolis force. == Awards and honours == 2000: Highly Cited Researcher with more than 160 publications and 11,000 citations 2005: Göran Gustafsson Prize from the Royal Swedish Academy of Sciences 2007: Friedrich Wilhelm Bessel Research Prize from the Alexander von Humboldt Foundation 2009: Fluid Dynamics Research Prize from the Japan Society of Fluid Mechanics 2010: Advanced Grant from the European Research Council (ERC) 2012: Plenary lecture at the European Congress of Mathematicians (ECM) in Krakow 2020: Wittgenstein Award from The Austrian Ministry for Science 2022: Elected corresponding member of the Austrian Academy of Sciences, 22 April 2022 2022: Elected member of the German National Academy of Sciences Leopoldina, 16 March 2022 2022: Made an honorary citizen of the city of Timișoara 2024: Elected full member of the Austrian Academy of Sciences, 15 April 2024 == Selected publications == papers 1998: Wave breaking for nonlinear nonlocal shallow water equations (with J. Escher), Acta Mathematica 181 229–243. 1999: A shallow water equation on the circle (with H. P. McKean), Comm. Pure Appl. Math. 52 949–982. 2000: Stability of peakons (with W. Strauss), Comm. Pure Appl. Math. 53 603–610. 2004: Exact steady periodic water waves with vorticity (with W. Strauss), Comm. Pure Appl. Math. 57 481–527. 2006: The trajectories of particles in Stokes waves, Invent. Math. 166 523–535. 2007: Global conservative solutions of the Camassa-Holm equation (with A. Bressan), Arch. Ration. Mech. Anal. 183 215–239. 2011: Analyticity of periodic traveling free surface water waves with vorticity (with J. Escher), Ann. of Math. 173 559–568. 2016: Global bifurcation of steady gravity water waves with critical layers (with W. Strauss and E. Varvaruca), Acta Mathematica 217 195–262. 2019: Equatorial wave-current interactions (with R. I. Ivanov), Comm. Math. Phys. 370 1–48. 2022: On the propagation of nonlinear waves in the atmosphere (with R. S. Johnson), Proceedings of the Royal Society A 478 (2260), 20210895 2022: Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere (with P. Germain), Arch. Ration. Mech. Anal., (245 587–644) Books 2011: "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis", Society for Industrial and Applied Mathematics, Philadelphia, ISBN 978-1611971866 2016: "Fourier Analysis. Part 1. Theory", London Mathematical Society, Cambridge University Press, ISBN 978-1107044104 2024: "Analysis I", Springer Spektrum, Berlin, Heidelberg, ISBN 978-3-662-68219-7 == References == == External links == Adrian Constantin at the Mathematics Genealogy Project Adrian Constantin's homepage Literature by and about Adrian Constantin in the catalog of the German National Library |
Wikipedia:Adrian Krzyżanowski#0 | Adrian Krzyżanowski (born 8 September 1788 in Dębowo - died 21 August 1852 in Warsaw) was a Polish mathematician and translator of German literature. == Life == From 1805 to 1810 he taught in a school in Warsaw, then was a professor of mathematics in Radzyń and Płock before studying from 1817 to 1820 in Paris. He was also a professor at the Warsaw Lyceum, which had been founded by Prussia, and at the University of Warsaw. Krzyżanowski was also involved in the November Uprising after which the university was closed by the Russians. He made a living by translating literature into Polish, for example the German novels of Alexander von Oppeln-Bronikowski. He also published several articles on Nicolaus Copernicus. Krzyżanowski died of cholera in 1852 and was buried at the Powązki Cemetery (163, VI). == Works == De construendis camaris ellipsoidicis ope projectionis graphicae. Dissertatio inauguralis matematico philosophica ąuam pro summis in philosophia honoribus rite conseąuendis, 1821 (O budowie sklepień eliptycznych za pomocą rzutów ukośnych. Rozprawa doktorska.) Teorya równań wszech-stopni podług binomu Newtona, 1816 Geometry a analityczna linii i powierzchni drugiego rzędu, 1822 O trudnościach zachodzących w wykładzie zasad geometryi elementarnej, 1825 O rodzinach spółczesnych i zażyłych w Krakowie z Kopernikani, Biblioteka Warszawska (1841) No. 3, pp. 27–40 Kopernik w Walhalli, in: Rozmaitości, Pismo dodatkowe do Gazety Lwowskiej (1843), No. 16 Kopernik gehört nicht in die Walhalla. In: Jahrbücher für slavische Literatur, Kunst u. Wissenschaft (Leipzig) 1 (1843), pp. 247–252. Copernicus in Walhalla (Kopernik w Walhalli) and Rozprawa o Koperniku (The Life and the Writings of Copernicus) by Jan Sniadecki (1756–1830), in New Quarterly Review; or, home, foreign and colonial journal, Volume 3, 1844 pp. 361–393 == References == == External links == (in Polish) Biografia Adriana Krzyżanowskiego |
Wikipedia:Adriana Garroni#0 | Adriana Garroni (born 1966) is an Italian mathematician specializing in mathematical analysis, including the calculus of variations, geometric measure theory, potential theory, and applications to the mathematical modeling of materials including plasticity and fracture. She is a professor in mathematics at Sapienza University of Rome. == Education and career == Garroni was born on 22 March 1966 in Rome, and despite having mathematics professor Maria Giovanna Garroni as her mother, she grew up torn between mathematics and the arts. After earning a laurea in mathematics in 1991 from Sapienza University of Rome, she went to the International School for Advanced Studies (SISSA) in Trieste for graduate study in functional analysis with Gianni Dal Maso, earning an M.Phil. in 1993 and completing her Ph.D. in 1994. She returned to Sapienza University as a researcher in 1995, became an associate professor in 1998, and was named full professor in 2017. == References == == External links == Home page Adriana Garroni publications indexed by Google Scholar Interview with Garroni at the 20th Congress of the Italian Mathematical Union, 2015 (in Italian) |
Wikipedia:Adriana Pesci#0 | Adriana Irma Pesci is an Argentine applied mathematician and mathematical physicist at the University of Cambridge, specialising in fluid dynamics. Her research topics have included lattice models of polymer solutions,[A] Hele-Shaw flow,[B] flagellar motion of organisms in fluids,[C] soap films on Möbius strips,[D] and the Leidenfrost effect.[E] == Education and career == Pesci is originally from Argentina, and earned her Ph.D. in 1986 at the National University of La Plata in Argentina. She was a postdoctoral researcher at the University of Chicago, under the mentorship of Leo Kadanoff and Norman Lebovitz. She joined the University of Arizona as a lecturer in physics in 1999, becoming a senior lecturer in 2003. In 2007 she moved to the University of Cambridge, where she is a senior research associate in the Department of Applied Mathematics and Theoretical Physics, a fellow of King's College, and a former Darley Fellow in Mathematics of Downing College. == Personal life == Pesci married Raymond E. Goldstein, a frequent coauthor who was also a postdoctoral researcher in Chicago and moved with her to Arizona and Cambridge. == Selected publications == == References == |
Wikipedia:Adriana Salerno#0 | Adriana Julia Salerno Domínguez (born 1979) is a Venezuelan-American mathematician, a professor of mathematics at Bates College, and a program director at the National Science Foundation. Her research interests include arithmetic geometry and arithmetic dynamics in number theory. She is also a mathematics blogger, the co-founder of the American Mathematical Society blogs "Ph.D. plus epsilon" and "inclusion/exclusion". == Education and career == Salerno was born in Caracas in 1979, and earned a licenciatura in mathematics from Simón Bolívar University (Venezuela) in 2001, advised by Pedro Berrizbeitia. She completed a Ph.D. in mathematics in 2009 at the University of Texas at Austin, with the dissertation Hypergeometric Functions in Arithmetic Geometry supervised by Fernando Rodríguez-Villegas. She joined Bates College as an assistant professor in 2009. In 2016, she visited the Mathematical Association of America (MAA) headquarters in Washington, D.C., as Dolciani Visiting Mathematician. After serving as department chair, she took a leave from Bates College beginning in 2021 to become a program officer for the National Science Foundation, where she is a program director for algebra and number theory. In 2021, she was also elected vice president of the MAA. == Recognition == Salerno is a 2023 recipient of one of the Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics. == References == == External links == Home page Adriana Salerno, Calendar 2017, Latinxs and Hispanics in the Mathematical Sciences |
Wikipedia:Adrien Pouliot#0 | Adrien Pouliot, (January 4, 1896 – March 10, 1980) was a Canadian mathematician and educator. Born in Île d'Orléans, Quebec. He married Laure Clark and was cousin of André Hudon. He obtained a B.A. in applied sciences from the École Polytechnique de Montréal in 1919. He helped to create the department of mathematics at Université Laval where he began teaching in 1922. He was president of the Canadian Mathematical Society from 1949 to 1953. He was made a Companion of the Order of Canada in 1972. He was head of the Faculty of Science at Laval from 1940 to 1956. A building on the Laval campus has been named in his honour. The Canadian Mathematical Society's Adrien Pouliot Award is named in his honour. == References == "Adrien Pouliot Award". Canadian Mathematical Society. Retrieved 31 March 2005. The Archives of Université Laval has important funds for him. == External links == Adrien Pouliot at The Canadian Encyclopedia |
Wikipedia:Advanced Extension Award#0 | The Advanced Extension Awards are a type of school-leaving qualification in England, Wales and Northern Ireland, usually taken in the final year of schooling (age 17/18), and designed to allow students to "demonstrate their knowledge, understanding and skills to the full". Currently, it is only available for Mathematics and offered by the exam board Edexcel. They were introduced in 2002, in response to the UK Government's Excellence in Cities report, as a successor to the S-level examination, and aimed at the top 10% of students in A level tests. They are assessed entirely by external examinations. Due to introduction of the A* grade for A level courses starting September 2008 (first certification 2010), they have since been phased out, with the exception of the Advanced Extension Award in Mathematics which continues to be available to students. == Results == According to EducationGuardian.co.uk, in 2004, 50.4% of the 7246 entrants failed to achieve a grade at all (fail), indicating that the awards are fulfilling their role in separating the elite. Only 18.3% of students attained the top of the two grades available, the Distinction, with the next 31.3% of students receiving the grade of Merit. Given that only the top students in the country sat these examinations, these results indicate that the AEAs were successful in rewarding only the 50-100 students that were most able in a particular subject. It was possible to obtain an AEA distinction in more than one subject; however, given the rarity of AEA distinctions, this was uncommon. == Available subjects == Due to the small numbers of candidates for each subject, the exam boards divided the subjects offered amongst themselves, so that – unlike A-levels – each AEA was only offered by one board. Biology (including Human Biology) (AQA) Business (OCR) Chemistry (AQA) Critical Thinking (OCR) Economics (AQA) English (OCR) French (OCR) Geography (WJEC) German (CCEA) History (Edexcel) Irish (CCEA) Latin (OCR) Mathematics (Edexcel) Physics (CCEA) Psychology (AQA) Religious Studies (Edexcel) Spanish (Edexcel) Welsh (WJEC) Welsh as a second language (WJEC) == Partial withdrawal == The last AEA examinations across the full range of subjects took place in June 2009, with results issued in August 2009. The Advanced Extension Award was then withdrawn for all subjects except mathematics. This came after the Joint Council for Qualifications (JCQ) decided that the new A* grade being offered at A level would overlap with the purpose of the AEA, rendering them unnecessary. However, the AEA in mathematics was extended until June 2012, as confirmed by Edexcel and the QCA. This was because it met a "definite need", since the A* grade was still not viewed as being challenging enough. In June 2011 Edexcel announced that the AEA was being extended further for mathematics, until June 2015, which was later extended until 2018. In 2018, Edexcel introduced a new specification, meaning the Advanced Extension Award in mathematics would continue to be available to students in 2019 and beyond, as a qualification aimed at the top 10% of students at A level. All other subjects remain withdrawn, though opportunity exists for examination boards to offer AEAs in other subjects should they choose to in the future, subject to certain expectations. == Effect of 2020 pandemic == In response to the COVID-19 pandemic, in summer 2020 AEA (mathematics) grades were awarded according to assessments made by teachers. == See also == Sixth Term Examination Paper == References == == External links == Directgov: Advanced Extension Awards Official website, archived December 2007 |
Wikipedia:Advanced level mathematics#0 | Advanced Level (A-Level) Mathematics is a qualification of further education taken in the United Kingdom (and occasionally other countries as well). In the UK, A-Level exams are traditionally taken by 17-18 year-olds after a two-year course at a sixth form or college. Advanced Level Further Mathematics is often taken by students who wish to study a mathematics-based degree at university, or related degree courses such as physics or computer science. Like other A-level subjects, mathematics has been assessed in a modular system since the introduction of Curriculum 2000, whereby each candidate must take six modules, with the best achieved score in each of these modules (after any retake) contributing to the final grade. Most students will complete three modules in one year, which will create an AS-level qualification in their own right and will complete the A-level course the following year—with three more modules. The system in which mathematics is assessed is changing for students starting courses in 2017 (as part of the A-level reforms first introduced in 2015), where the reformed specifications have reverted to a linear structure with exams taken only at the end of the course in a single sitting. In addition, while schools could choose freely between taking Statistics, Mechanics or Discrete Mathematics (also known as Decision Mathematics) modules with the ability to specialise in one branch of applied Mathematics in the older modular specification, in the new specifications, both Mechanics and Statistics were made compulsory, with Discrete Mathematics being made exclusive as an option to students pursuing a Further Mathematics course. The first assessment opportunity for the new specification is 2018 and 2019 for A-levels in Mathematics and Further Mathematics, respectively. == 2000s specification == Prior to the 2017 reform, the basic A-Level course consisted of six modules, four pure modules (C1, C2, C3, and C4) and two applied modules in Statistics, Mechanics and/or Decision Mathematics. The C1 through C4 modules are referred to by A-level textbooks as "Core" modules, encompassing the major topics of mathematics such as logarithms, differentiation/integration and geometric/arithmetic progressions. The two chosen modules for the final two parts of the A-Level are determined either by a student's personal choices, or the course choice of their school/college, though it commonly took the form of S1 (Statistics) and M1 (Mechanics). === Further mathematics === Students that were studying for (or had completed) an A-level in Mathematics had the opportunity to study an A-level in Further Mathematics, which required taking a further 6 modules to give a second qualification. The grades of the two A-levels will be independent of each other, with Further Mathematics requiring students to take a minimum of two Further Pure modules, one of which must be FP1, and the other either FP2 or FP3, which are simply extensions of the four Core modules from the normal Maths A-Level. Four more modules need to be taken; those available vary with different specifications. Not all schools are able to offer Further Mathematics, due to a low student number (meaning that the course is not financially viable) or a lack of suitably experienced teachers. To fulfil the demand, extra tutoring is available, with providers such as the Further Mathematics Support Programme. Some students had the opportunity to take a third maths qualification, "Additional Further Mathematics", which added more modules from those not used for Mathematics or Further Mathematics. Schools that offer this qualification usually only took this to AS-level, taking three modules, although some students went further, taking the extra six modules to gain another full A-Level qualification. Additional Further Mathematics is offered by Edexcel only, and a Pure Mathematics A-level is available for students who—on the Edexcel exam board—take the modules C1, C2, C3, C4, FP1 and either FP2 or FP3. No comparable qualification has been available since the 2017 reforms. === Results and statistics === Each module carried a maximum of 100 UMS points towards the total grade, and each module is also given a separate grade depending on its score. The number of points required for different grades were defined as follows: The proportion of candidates acquiring these grades in 2007 are below: === Mathematics === === Further mathematics === == 2017 specification == A new specification was introduced in 2017 for first examination in summer 2019. Under this specification, there are three papers which must all be taken in the same year. There are three overarching themes - “Argument, language and proof”, “Problem solving” and “Modelling” throughout the assessment. Each board structures the three papers as follows: === AQA === Paper 1: Pure Mathematics Paper 2: Content on Paper 1 plus Mechanics Paper 3: Content on Paper 1 plus Statistics === Edexcel === Paper 1: Pure Mathematics 1 Paper 2: Pure Mathematics 2 Paper 3: Statistics and Mechanics === OCR A === Paper 1: Pure Mathematics Paper 2: Pure Mathematics and Statistics Paper 3: Pure Mathematics and Mechanics === OCR B (MEI) === Paper 1: Pure Mathematics and Mechanics Paper 2: Pure Mathematics and Statistics Paper 3: Pure Mathematics and Comprehension == Grading == It was suggested by the Department for Education that the high proportion of candidates who obtain grade A makes it difficult for universities to distinguish between the most able candidates. As a result, the 2010 exam session introduced the grade A*—which serves to distinguish between the better candidates. Prior to the 2017 reforms, the A* grade in maths was awarded to candidates who achieve an A (480/600) in their overall A Level, as well as achieving a combined score of 180/200 in modules Core 3 and Core 4. For the reformed specification, the A* is given by a more traditional grade boundary based on the raw mark achieved by the candidate over their papers. The A* grade in further maths was awarded slightly differently. The same minimum score of 480/600 was required across all six modules. However, a 90% average (or a score of 270/300) had to be obtained across the candidate's best 'A2' modules. A2 modules included any modules other than those with a '1' (FP1, S1, M1 and D1 are not A2 modules, whereas FP2, FP3, FP4 (from AQA only), S2, S3, S4, M2, M3 and D2 are). == List of subjects == 1. Core Mathematics: Covers foundational topics like algebra, calculus, trigonometry, and coordinate geometry. 2. Further Mathematics: Expands upon Core Mathematics with additional areas such as complex numbers, matrices, differential equations, and numerical methods. 3. Pure Mathematics: Explores advanced topics in algebra, calculus, and mathematical proofs. 4. Applied Mathematics: Focuses on practical applications of mathematical concepts to solve real-world problems in various fields. 5. Mechanics: Focuses on the study of motion, forces, and vectors, particularly relevant for physics or engineering interests. 6. Statistics: Involves collecting, analysing, and interpreting data, including topics like probability, hypothesis testing, regression analysis, and sampling. 7. Discrete Mathematics: Deals with separate and distinct mathematical structures, including topics such as combinatorics, graph theory, and algorithms. 8. Decision Mathematics: Applies mathematical techniques to solve real-world problems related to optimisation, networks, and decision-making. 9. Financial Mathematics: Applies mathematical concepts to analyse financial markets, investments, and risk management. == References == == External links == Underground Mathematics (Resources on A-level mathematics) A Level Maths Revision (Free resources for A Level Maths) |
Wikipedia:Advances in Applied Clifford Algebras#0 | Advances in Applied Clifford Algebras is a peer-reviewed scientific journal that publishes original research papers and also notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops in the area of Clifford algebras and their applications to other branches of mathematics and physics, and in certain cognate areas. There is a vibrant and interdisciplinary community around Clifford and Geometric Algebras with a wide range of applications. The main conferences in this subject include The International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA) and Applications of Geometric Algebra in Computer Science and Engineering (AGACSE) series. The journal was established in 1991 by Jaime Keller who was its editor-in-chief until his death in 2011. The second editor-in-chief of the journal was Waldyr Alves Rodrigues Jr. (Universidade Estadual de Campinas), and the current editor-in-chief is Uwe Kähler from University of Aveiro. The journal is published by Springer Science+Business Media under its Birkhäuser Verlag imprint. == References == == External links == Official website |
Wikipedia:Advances in Difference Equations#0 | Advances in Difference Equations is a peer-reviewed mathematics journal covering research on difference equations, published by Springer Open. The journal was established in 2004 and publishes articles on theory, methodology, and application of difference and differential equations. Originally published by Hindawi Publishing Corporation, the journal was acquired by Springer Science+Business Media in early 2011. The editors-in-chief are Ravi Agarwal, Martin Bohner, and Elena Braverman. == Abstracting and indexing == The journal is abstracted and indexed by the Science Citation Index Expanded, Current Contents/Physical, Chemical & Earth Sciences, and Zentralblatt MATH. According to the Journal Citation Reports, the journal has a 2021 impact factor of 2.803. from July 1, the journal has been transitioning to a new title that opens the scope of the journal to broader developments in theory and applications of models. Under the new title, Advances in Continuous and Discrete Models: Theory and Modern Applications, the journal will cover developments in machine learning, data driven modeling, differential equations, numerical analysis, scientific computing, control, optimization, and computing. == References == == External links == Official website |
Wikipedia:Advances in Group Theory and Applications#0 | Advances in Group Theory and Applications (AGTA) is a peer reviewed, open access research journal in mathematics, specifically group theory. It was founded in 2015 by the council of the no-profit association AGTA - Advances in Group Theory and Applications, and is published by Aracne. The journal is composed of three sections. The main one contains mathematical research papers, while the two other sections are respectively devoted to historical papers and open problems. The journal is published as a diamond open access journal, meaning that the content is immediately freely available to the readers, and the authors do not have to pay any author publication fees. The journal is abstracted and indexed by Mathematical Reviews and Zentralblatt MATH. == Sections == === ADV – The History behind Group Theory === The purpose of this section is to present historical documents, biographical notes and discussions on relevant aspects of group theory and its applications. === ADV – Perspectives in Group Theory === The open problems submitted to the journal are published in this section. == Reinhold Baer Prize sponsorship == The journal started co-sponsoring the Reinhold Baer Prize in 2017. == Proceedings == Volume 15 publishes the proceedings of the conference Groups & Algebras in Bicocca for Young algebraists 2022. == References == == External links == Advances in Group Theory and Applications at Aracne Journal website |
Wikipedia:Advisory Committee on Mathematics Education#0 | The Advisory Committee on Mathematics Education (ACME) is a British policy council for the Royal Society based in London, England. Founded in 2002 by the Royal Society and the Joint Mathematical Council, ACME analyzes mathematics education practices and provides advice on education policy. ACME is funded by the Gatsby Charitable Foundation (2002-2015) and the Department for Education. == Members == The committee chair is appointed for a three-year term. As of 2018, the membership is composed of: Frank Kelly (Chair) Martin Bridson Paul Glaister Paul Golby Jeremy Hodgen Mary McAlinden Lynne McClure Emma McCoy Jil Matheson David Spiegelhalter Sally Bridgeland == References == |
Wikipedia:Aequationes Mathematicae#0 | Aequationes Mathematicae is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics, it also regularly reports on international symposia on functional equations and produces bibliographies on the subject. János Aczél founded the journal in 1968 at the University of Waterloo, in part because of the long publication delays of up to four years in other journals at the time of its founding. It is currently published by Springer Science+Business Media, with Zsolt Páles of the University of Debrecen as its editor in chief. János Aczél remains its honorary editor in chief. It is frequently listed as a second-quartile mathematics journal by SCImago Journal Rank. == References == |
Wikipedia:Affine action#0 | In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through k + 1 points in general position, a k-dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines within the same plane intersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction. Unlike for vectors in a vector space, in an affine space there is no distinguished point that serves as an origin. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are called free vectors, displacement vectors, translation vectors or simply translations. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. While points cannot be arbitrarily added together, it is meaningful to take affine combinations of points: weighted sums with numerical coefficients summing to 1, resulting in another point. These coefficients define a barycentric coordinate system for the flat through the points. Any vector space may be viewed as an affine space; this amounts to "forgetting" the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane. == Informal description == The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it p—is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed p + (a − p) + (b − p). Similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer. If Alice travels to λa + (1 − λ)b then Bob can similarly travel to p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b. Under this condition, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, despite using different origins. While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space. == Definition == While affine space can be defined axiomatically (see § Axioms below), analogously to the definition of Euclidean space implied by Euclid's Elements, for convenience most modern sources define affine spaces in terms of the well developed vector space theory. An affine space is a set A together with a vector space A → {\displaystyle {\overrightarrow {A}}} , and a transitive and free action of the additive group of A → {\displaystyle {\overrightarrow {A}}} on the set A. The elements of the affine space A are called points. The vector space A → {\displaystyle {\overrightarrow {A}}} is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes free vectors. Explicitly, the definition above means that the action is a mapping, generally denoted as an addition, A × A → → A ( a , v ) ↦ a + v , {\displaystyle {\begin{aligned}A\times {\overrightarrow {A}}&\to A\\(a,v)\;&\mapsto a+v,\end{aligned}}} that has the following properties. Right identity: ∀ a ∈ A , a + 0 = a {\displaystyle \forall a\in A,\;a+0=a} , where 0 is the zero vector in A → {\displaystyle {\overrightarrow {A}}} Associativity: ∀ v , w ∈ A → , ∀ a ∈ A , ( a + v ) + w = a + ( v + w ) {\displaystyle \forall v,w\in {\overrightarrow {A}},\forall a\in A,\;(a+v)+w=a+(v+w)} (here the last + is the addition in A → {\displaystyle {\overrightarrow {A}}} ) Free and transitive action: For every a ∈ A {\displaystyle a\in A} , the mapping A → → A : v ↦ a + v {\displaystyle {\overrightarrow {A}}\to A\colon v\mapsto a+v} is a bijection. The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above: Existence of one-to-one translations For all v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , the mapping A → A : a ↦ a + v {\displaystyle A\to A\colon a\mapsto a+v} is a bijection. Property 3 is often used in the following equivalent form (the 5th property). Subtraction: For every a, b in A, there exists a unique v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , denoted b – a, such that b = a + v {\displaystyle b=a+v} . Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are, by definition, endowed with a transitive group action, and for a principal homogeneous space, such a transitive action is, by definition, free. === Subtraction and Weyl's axioms === The properties of the group action allows for the definition of subtraction for any given ordered pair (b, a) of points in A, producing a vector of A → {\displaystyle {\overrightarrow {A}}} . This vector, denoted b − a {\displaystyle b-a} or a b → {\displaystyle {\overrightarrow {ab}}} , is defined to be the unique vector in A → {\displaystyle {\overrightarrow {A}}} such that a + ( b − a ) = b . {\displaystyle a+(b-a)=b.} Existence follows from the transitivity of the action, and uniqueness follows because the action is free. This subtraction has the two following properties, called Weyl's axioms: ∀ a ∈ A , ∀ v ∈ A → {\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}} , there is a unique point b ∈ A {\displaystyle b\in A} such that b − a = v . {\displaystyle b-a=v.} ∀ a , b , c ∈ A , ( c − b ) + ( b − a ) = c − a . {\displaystyle \forall a,b,c\in A,\;(c-b)+(b-a)=c-a.} The parallelogram property is satisfied in affine spaces, where it is expressed as: given four points a , b , c , d , {\displaystyle a,b,c,d,} the equalities b − a = d − c {\displaystyle b-a=d-c} and c − a = d − b {\displaystyle c-a=d-b} are equivalent. This results from the second Weyl's axiom, since d − a = ( d − b ) + ( b − a ) = ( d − c ) + ( c − a ) . {\displaystyle d-a=(d-b)+(b-a)=(d-c)+(c-a).} Affine spaces can be equivalently defined as a point set A, together with a vector space A → {\displaystyle {\overrightarrow {A}}} , and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms. == Affine subspaces and parallelism == An affine subspace (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) B of an affine space A is a subset of A such that, given a point a ∈ B {\displaystyle a\in B} , the set of vectors B → = { b − a ∣ b ∈ B } {\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}} is a linear subspace of A → {\displaystyle {\overrightarrow {A}}} . This property, which does not depend on the choice of a, implies that B is an affine space, which has B → {\displaystyle {\overrightarrow {B}}} as its associated vector space. The affine subspaces of A are the subsets of A of the form a + V = { a + w : w ∈ V } , {\displaystyle a+V=\{a+w:w\in V\},} where a is a point of A, and V a linear subspace of A → {\displaystyle {\overrightarrow {A}}} . The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said to be parallel. This implies the following generalization of Playfair's axiom: Given a direction V, for any point a of A there is one and only one affine subspace of direction V, which passes through a, namely the subspace a + V. Every translation A → A : a ↦ a + v {\displaystyle A\to A:a\mapsto a+v} maps any affine subspace to a parallel subspace. The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other. == Affine map == Given two affine spaces A and B whose associated vector spaces are A → {\displaystyle {\overrightarrow {A}}} and B → {\displaystyle {\overrightarrow {B}}} , an affine map or affine homomorphism from A to B is a map f : A → B {\displaystyle f:A\to B} such that f → : A → → B → b − a ↦ f ( b ) − f ( a ) {\displaystyle {\begin{aligned}{\overrightarrow {f}}:{\overrightarrow {A}}&\to {\overrightarrow {B}}\\b-a&\mapsto f(b)-f(a)\end{aligned}}} is a well defined linear map. By f {\displaystyle f} being well defined is meant that b – a = d – c implies f(b) – f(a) = f(d) – f(c). This implies that, for a point a ∈ A {\displaystyle a\in A} and a vector v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , one has f ( a + v ) = f ( a ) + f → ( v ) . {\displaystyle f(a+v)=f(a)+{\overrightarrow {f}}(v).} Therefore, since for any given b in A, b = a + v for a unique v, f is completely defined by its value on a single point and the associated linear map f → {\displaystyle {\overrightarrow {f}}} . === Endomorphisms === An affine transformation or endomorphism of an affine space A {\displaystyle A} is an affine map from that space to itself. One important family of examples is the translations: given a vector v → {\displaystyle {\overrightarrow {v}}} , the translation map T v → : A → A {\displaystyle T_{\overrightarrow {v}}:A\rightarrow A} that sends a ↦ a + v → {\displaystyle a\mapsto a+{\overrightarrow {v}}} for every a {\displaystyle a} in A {\displaystyle A} is an affine map. Another important family of examples are the linear maps centred at an origin: given a point b {\displaystyle b} and a linear map M {\displaystyle M} , one may define an affine map L M , b : A → A {\displaystyle L_{M,b}:A\rightarrow A} by L M , b ( a ) = b + M ( a − b ) {\displaystyle L_{M,b}(a)=b+M(a-b)} for every a {\displaystyle a} in A {\displaystyle A} . After making a choice of origin b {\displaystyle b} , any affine map may be written uniquely as a combination of a translation and a linear map centred at b {\displaystyle b} . == Vector spaces as affine spaces == Every vector space V may be considered as an affine space over itself. This means that every element of V may be considered either as a point or as a vector. This affine space is sometimes denoted (V, V) for emphasizing the double role of the elements of V. When considered as a point, the zero vector is commonly denoted o (or O, when upper-case letters are used for points) and called the origin. If A is another affine space over the same vector space (that is V = A → {\displaystyle V={\overrightarrow {A}}} ) the choice of any point a in A defines a unique affine isomorphism, which is the identity of V and maps a to o. In other words, the choice of an origin a in A allows us to identify A and (V, V) up to a canonical isomorphism. The counterpart of this property is that the affine space A may be identified with the vector space V in which "the place of the origin has been forgotten". == Relation to Euclidean spaces == === Definition of Euclidean spaces === Euclidean spaces (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces. Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form q(x). The inner product of two vectors x and y is the value of the symmetric bilinear form x ⋅ y = 1 2 ( q ( x + y ) − q ( x ) − q ( y ) ) . {\displaystyle x\cdot y={\frac {1}{2}}(q(x+y)-q(x)-q(y)).} The usual Euclidean distance between two points A and B is d ( A , B ) = q ( B − A ) . {\displaystyle d(A,B)={\sqrt {q(B-A)}}.} In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of ordered pairs of points under equipollence (the pairs (A, B) and (C, D) are equipollent if the points A, B, D, C (in this order) form a parallelogram). It is straightforward to verify that the vectors form a vector space, the square of the Euclidean distance is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent. === Affine properties === In Euclidean geometry, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples are parallelism, and the definition of a tangent. A non-example is the definition of a normal. Equivalently, an affine property is a property that is invariant under affine transformations of the Euclidean space. == Affine combinations and barycenter == Let a1, ..., an be a collection of n points in an affine space, and λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} be n elements of the ground field. Suppose that λ 1 + ⋯ + λ n = 0 {\displaystyle \lambda _{1}+\dots +\lambda _{n}=0} . For any two points o and o' one has λ 1 o a 1 → + ⋯ + λ n o a n → = λ 1 o ′ a 1 → + ⋯ + λ n o ′ a n → . {\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}=\lambda _{1}{\overrightarrow {o'a_{1}}}+\dots +\lambda _{n}{\overrightarrow {o'a_{n}}}.} Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted λ 1 a 1 + ⋯ + λ n a n . {\displaystyle \lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.} When n = 2 , λ 1 = 1 , λ 2 = − 1 {\displaystyle n=2,\lambda _{1}=1,\lambda _{2}=-1} , one retrieves the definition of the subtraction of points. Now suppose instead that the field elements satisfy λ 1 + ⋯ + λ n = 1 {\displaystyle \lambda _{1}+\dots +\lambda _{n}=1} . For some choice of an origin o, denote by g {\displaystyle g} the unique point such that λ 1 o a 1 → + ⋯ + λ n o a n → = o g → . {\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}={\overrightarrow {og}}.} One can show that g {\displaystyle g} is independent from the choice of o. Therefore, if λ 1 + ⋯ + λ n = 1 , {\displaystyle \lambda _{1}+\dots +\lambda _{n}=1,} one may write g = λ 1 a 1 + ⋯ + λ n a n . {\displaystyle g=\lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.} The point g {\displaystyle g} is called the barycenter of the a i {\displaystyle a_{i}} for the weights λ i {\displaystyle \lambda _{i}} . One says also that g {\displaystyle g} is an affine combination of the a i {\displaystyle a_{i}} with coefficients λ i {\displaystyle \lambda _{i}} . == Examples == When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space. Time can be modelled as a one-dimensional affine space. Specific points in time (such as a date on the calendar) are points in the affine space, while durations (such as a number of days) are displacements. The space of energies is an affine space for R {\displaystyle \mathbb {R} } , since it is often not meaningful to talk about absolute energy, but it is meaningful to talk about energy differences. The vacuum energy when it is defined picks out a canonical origin. Physical space is often modelled as an affine space for R 3 {\displaystyle \mathbb {R} ^{3}} in non-relativistic settings and R 1 , 3 {\displaystyle \mathbb {R} ^{1,3}} in the relativistic setting. To distinguish them from the vector space these are sometimes called Euclidean spaces E ( 3 ) {\displaystyle {\text{E}}(3)} and E ( 1 , 3 ) {\displaystyle {\text{E}}(1,3)} . Any coset of a subspace V of a vector space is an affine space over that subspace. In particular, a line in the plane that doesn't pass through the origin is an affine space that is not a vector space relative to the operations it inherits from R 2 {\displaystyle \mathbb {R} ^{2}} , although it can be given a canonical vector space structure by picking the point closest to the origin as the zero vector; likewise in higher dimensions and for any normed vector space If T is a matrix and b lies in its column space, the set of solutions of the equation Tx = b is an affine space over the subspace of solutions of Tx = 0. The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. Generalizing all of the above, if T : V → W is a linear map and y lies in its image, the set of solutions x ∈ V to the equation Tx = y is a coset of the kernel of T , and is therefore an affine space over Ker T . The space of (linear) complementary subspaces of a vector subspace V in a vector space W is an affine space, over Hom(W/V, V). That is, if 0 → V → W → X → 0 is a short exact sequence of vector spaces, then the space of all splittings of the exact sequence naturally carries the structure of an affine space over Hom(X, V). The space of connections (viewed from the vector bundle E → π M {\displaystyle E\xrightarrow {\pi } M} , where M {\displaystyle M} is a smooth manifold) is an affine space for the vector space of End ( E ) {\displaystyle {\text{End}}(E)} valued 1-forms. The space of connections (viewed from the principal bundle P → π M {\displaystyle P\xrightarrow {\pi } M} ) is an affine space for the vector space of ad ( P ) {\displaystyle {\text{ad}}(P)} -valued 1-forms, where ad ( P ) {\displaystyle {\text{ad}}(P)} is the associated adjoint bundle. == Affine span and bases == For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X. It is the intersection of all affine subspaces containing X, and its direction is the intersection of the directions of the affine subspaces that contain X. The affine span of X is the set of all (finite) affine combinations of points of X, and its direction is the linear span of the x − y for x and y in X. If one chooses a particular point x0, the direction of the affine span of X is also the linear span of the x – x0 for x in X. One says also that the affine span of X is generated by X and that X is a generating set of its affine span. A set X of points of an affine space is said to be affinely independent or, simply, independent, if the affine span of any strict subset of X is a strict subset of the affine span of X. An affine basis or barycentric frame (see § Barycentric coordinates, below) of an affine space is a generating set that is also independent (that is a minimal generating set). Recall that the dimension of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimension n are the independent subsets of n + 1 elements, or, equivalently, the generating subsets of n + 1 elements. Equivalently, {x0, ..., xn} is an affine basis of an affine space if and only if {x1 − x0, ..., xn − x0} is a linear basis of the associated vector space. == Coordinates == There are two strongly related kinds of coordinate systems that may be defined on affine spaces. === Barycentric coordinates === Let A be an affine space of dimension n over a field k, and { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} be an affine basis of A. The properties of an affine basis imply that for every x in A there is a unique (n + 1)-tuple ( λ 0 , … , λ n ) {\displaystyle (\lambda _{0},\dots ,\lambda _{n})} of elements of k such that λ 0 + ⋯ + λ n = 1 {\displaystyle \lambda _{0}+\dots +\lambda _{n}=1} and x = λ 0 x 0 + ⋯ + λ n x n . {\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.} The λ i {\displaystyle \lambda _{i}} are called the barycentric coordinates of x over the affine basis { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} . If the xi are viewed as bodies that have weights (or masses) λ i {\displaystyle \lambda _{i}} , the point x is thus the barycenter of the xi, and this explains the origin of the term barycentric coordinates. The barycentric coordinates define an affine isomorphism between the affine space A and the affine subspace of kn + 1 defined by the equation λ 0 + ⋯ + λ n = 1 {\displaystyle \lambda _{0}+\dots +\lambda _{n}=1} . For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero. === Affine coordinates === An affine frame is a coordinate frame of an affine space, consisting of a point, called the origin, and a linear basis of the associated vector space. More precisely, for an affine space A with associated vector space A → {\displaystyle {\overrightarrow {A}}} , the origin o belongs to A, and the linear basis is a basis (v1, ..., vn) of A → {\displaystyle {\overrightarrow {A}}} (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar). For each point p of A, there is a unique sequence λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} of elements of the ground field such that p = o + λ 1 v 1 + ⋯ + λ n v n , {\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},} or equivalently o p → = λ 1 v 1 + ⋯ + λ n v n . {\displaystyle {\overrightarrow {op}}=\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n}.} The λ i {\displaystyle \lambda _{i}} are called the affine coordinates of p over the affine frame (o, v1, ..., vn). Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame (o, v1, ..., vn) such that (v1, ..., vn) is an orthonormal basis. === Relationship between barycentric and affine coordinates === Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent. In fact, given a barycentric frame ( x 0 , … , x n ) , {\displaystyle (x_{0},\dots ,x_{n}),} one deduces immediately the affine frame ( x 0 , x 0 x 1 → , … , x 0 x n → ) = ( x 0 , x 1 − x 0 , … , x n − x 0 ) , {\displaystyle (x_{0},{\overrightarrow {x_{0}x_{1}}},\dots ,{\overrightarrow {x_{0}x_{n}}})=\left(x_{0},x_{1}-x_{0},\dots ,x_{n}-x_{0}\right),} and, if ( λ 0 , λ 1 , … , λ n ) {\displaystyle \left(\lambda _{0},\lambda _{1},\dots ,\lambda _{n}\right)} are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are ( λ 1 , … , λ n ) . {\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right).} Conversely, if ( o , v 1 , … , v n ) {\displaystyle \left(o,v_{1},\dots ,v_{n}\right)} is an affine frame, then ( o , o + v 1 , … , o + v n ) {\displaystyle \left(o,o+v_{1},\dots ,o+v_{n}\right)} is a barycentric frame. If ( λ 1 , … , λ n ) {\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right)} are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are ( 1 − λ 1 − ⋯ − λ n , λ 1 , … , λ n ) . {\displaystyle \left(1-\lambda _{1}-\dots -\lambda _{n},\lambda _{1},\dots ,\lambda _{n}\right).} Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example. ==== Example of the triangle ==== The vertices of a non-flat triangle form an affine basis of the Euclidean plane. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distances: The vertices are the points of barycentric coordinates (1, 0, 0), (0, 1, 0) and (0, 0, 1). The lines supporting the edges are the points that have a zero coordinate. The edges themselves are the points that have one zero coordinate and two nonnegative coordinates. The interior of the triangle are the points whose coordinates are all positive. The medians are the points that have two equal coordinates, and the centroid is the point of coordinates (1/3, 1/3, 1/3). === Change of coordinates === ==== Case of barycentric coordinates ==== Barycentric coordinates are readily changed from one basis to another. Let { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} and { x 0 ′ , … , x n ′ } {\displaystyle \{x'_{0},\dots ,x'_{n}\}} be affine bases of A. For every x in A there is some tuple { λ 0 , … , λ n } {\displaystyle \{\lambda _{0},\dots ,\lambda _{n}\}} for which x = λ 0 x 0 + ⋯ + λ n x n . {\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.} Similarly, for every x i ∈ { x 0 , … , x n } {\displaystyle x_{i}\in \{x_{0},\dots ,x_{n}\}} from the first basis, we now have in the second basis x i = λ i , 0 x 0 ′ + ⋯ + λ i , j x j ′ + ⋯ + λ i , n x n ′ {\displaystyle x_{i}=\lambda _{i,0}x'_{0}+\dots +\lambda _{i,j}x'_{j}+\dots +\lambda _{i,n}x'_{n}} for some tuple { λ i , 0 , … , λ i , n } {\displaystyle \{\lambda _{i,0},\dots ,\lambda _{i,n}\}} . Now we can rewrite our expression in the first basis as one in the second with x = ∑ i = 0 n λ i x i = ∑ i = 0 n λ i ∑ j = 0 n λ i , j x j ′ = ∑ j = 0 n ( ∑ i = 0 n λ i λ i , j ) x j ′ , {\displaystyle \,x=\sum _{i=0}^{n}\lambda _{i}x_{i}=\sum _{i=0}^{n}\lambda _{i}\sum _{j=0}^{n}\lambda _{i,j}x'_{j}=\sum _{j=0}^{n}{\biggl (}\sum _{i=0}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}x'_{j}\,,} giving us coordinates in the second basis as the tuple { ∑ i λ i λ i , 0 , … , {\textstyle {\bigl \{}\sum _{i}\lambda _{i}\lambda _{i,0},\,\dots ,\,{}} ∑ i λ i λ i , n } {\textstyle \sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}} . ==== Case of affine coordinates ==== Affine coordinates are also readily changed from one basis to another. Let o {\displaystyle o} , { v 1 , … , v n } {\displaystyle \{v_{1},\dots ,v_{n}\}} and o ′ {\displaystyle o'} , { v 1 ′ , … , v n ′ } {\displaystyle \{v'_{1},\dots ,v'_{n}\}} be affine frames of A. For each point p of A, there is a unique sequence λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} of elements of the ground field such that p = o + λ 1 v 1 + ⋯ + λ n v n , {\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},} and similarly, for every v i ∈ { v 1 , … , v n } {\displaystyle v_{i}\in \{v_{1},\dots ,v_{n}\}} from the first basis, we now have in the second basis o = o ′ + λ o , 1 v 1 ′ + ⋯ + λ o , j v j ′ + ⋯ + λ o , n v n ′ {\displaystyle o=o'+\lambda _{o,1}v'_{1}+\dots +\lambda _{o,j}v'_{j}+\dots +\lambda _{o,n}v'_{n}\,} v i = λ i , 1 v 1 ′ + ⋯ + λ i , j v j ′ + ⋯ + λ i , n v n ′ {\displaystyle v_{i}=\lambda _{i,1}v'_{1}+\dots +\lambda _{i,j}v'_{j}+\dots +\lambda _{i,n}v'_{n}} for tuple { λ o , 1 , … , λ o , n } {\displaystyle \{\lambda _{o,1},\dots ,\lambda _{o,n}\}} and tuples { λ i , 1 , … , λ i , n } {\displaystyle \{\lambda _{i,1},\dots ,\lambda _{i,n}\}} . Now we can rewrite our expression in the first basis as one in the second with p = o + ∑ i = 1 n λ i v i = ( o ′ + ∑ j = 1 n λ o , j v j ′ ) + ∑ i = 1 n λ i ∑ j = 1 n λ i , j v j ′ = o ′ + ∑ j = 1 n ( λ o , j + ∑ i = 1 n λ i λ i , j ) v j ′ , {\displaystyle {\begin{aligned}\,p&=o+\sum _{i=1}^{n}\lambda _{i}v_{i}={\biggl (}o'+\sum _{j=1}^{n}\lambda _{o,j}v'_{j}{\biggr )}+\sum _{i=1}^{n}\lambda _{i}\sum _{j=1}^{n}\lambda _{i,j}v'_{j}\\&=o'+\sum _{j=1}^{n}{\biggl (}\lambda _{o,j}+\sum _{i=1}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}v'_{j}\,,\end{aligned}}} giving us coordinates in the second basis as the tuple { λ o , 1 + ∑ i λ i λ i , 1 , … , {\textstyle {\bigl \{}\lambda _{o,1}+\sum _{i}\lambda _{i}\lambda _{i,1},\,\dots ,\,{}} λ o , n + ∑ i λ i λ i , n } {\textstyle \lambda _{o,n}+\sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}} . == Properties of affine homomorphisms == === Matrix representation === An affine transformation T {\displaystyle T} is executed on a projective space P 3 {\displaystyle \mathbb {P} ^{3}} of R 3 {\displaystyle \mathbb {R} ^{3}} , by a 4 by 4 matrix with a special fourth column: A = [ a 11 a 12 a 13 0 a 21 a 22 a 23 0 a 31 a 32 a 33 0 a 41 a 42 a 43 1 ] = [ T ( 1 , 0 , 0 ) 0 T ( 0 , 1 , 0 ) 0 T ( 0 , 0 , 1 ) 0 T ( 0 , 0 , 0 ) 1 ] {\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&0\\a_{21}&a_{22}&a_{23}&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&1\end{bmatrix}}={\begin{bmatrix}T(1,0,0)&0\\T(0,1,0)&0\\T(0,0,1)&0\\T(0,0,0)&1\end{bmatrix}}} The transformation is affine instead of linear due to the inclusion of point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} , the transformed output of which reveals the affine shift. === Image and fibers === Let f : E → F {\displaystyle f\colon E\to F} be an affine homomorphism, with f → : E → → F → {\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}} its associated linear map. The image of f is the affine subspace f ( E ) = { f ( a ) ∣ a ∈ E } {\displaystyle f(E)=\{f(a)\mid a\in E\}} of F, which has f → ( E → ) {\displaystyle {\overrightarrow {f}}({\overrightarrow {E}})} as associated vector space. As an affine space does not have a zero element, an affine homomorphism does not have a kernel. However, the linear map f → {\displaystyle {\overrightarrow {f}}} does, and if we denote by K = { v ∈ E → ∣ f → ( v ) = 0 } {\displaystyle K=\{v\in {\overrightarrow {E}}\mid {\overrightarrow {f}}(v)=0\}} its kernel, then for any point x of f ( E ) {\displaystyle f(E)} , the inverse image f − 1 ( x ) {\displaystyle f^{-1}(x)} of x is an affine subspace of E whose direction is K {\displaystyle K} . This affine subspace is called the fiber of x. === Projection === An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact that Euclidean spaces are affine spaces, and that these kinds of projections are fundamental in Euclidean geometry. More precisely, given an affine space E with associated vector space E → {\displaystyle {\overrightarrow {E}}} , let F be an affine subspace of direction F → {\displaystyle {\overrightarrow {F}}} , and D be a complementary subspace of F → {\displaystyle {\overrightarrow {F}}} in E → {\displaystyle {\overrightarrow {E}}} (this means that every vector of E → {\displaystyle {\overrightarrow {E}}} may be decomposed in a unique way as the sum of an element of F → {\displaystyle {\overrightarrow {F}}} and an element of D). For every point x of E, its projection to F parallel to D is the unique point p(x) in F such that p ( x ) − x ∈ D . {\displaystyle p(x)-x\in D.} This is an affine homomorphism whose associated linear map p → {\displaystyle {\overrightarrow {p}}} is defined by p → ( x − y ) = p ( x ) − p ( y ) , {\displaystyle {\overrightarrow {p}}(x-y)=p(x)-p(y),} for x and y in E. The image of this projection is F, and its fibers are the subspaces of direction D. === Quotient space === Although kernels are not defined for affine spaces, quotient spaces are defined. This results from the fact that "belonging to the same fiber of an affine homomorphism" is an equivalence relation. Let E be an affine space, and D be a linear subspace of the associated vector space E → {\displaystyle {\overrightarrow {E}}} . The quotient E/D of E by D is the quotient of E by the equivalence relation such that x and y are equivalent if x − y ∈ D . {\displaystyle x-y\in D.} This quotient is an affine space, which has E → / D {\displaystyle {\overrightarrow {E}}/D} as associated vector space. For every affine homomorphism E → F {\displaystyle E\to F} , the image is isomorphic to the quotient of E by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces. == Axioms == Affine spaces are usually studied by analytic geometry using coordinates, or equivalently vector spaces. They can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space. Coxeter (1969, p. 192) axiomatizes the special case of affine geometry over the reals as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line. Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint): Any two distinct points lie on a unique line. Given a point and line there is a unique line that contains the point and is parallel to the line There exist three non-collinear points. As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional affine spaces. Purely axiomatic affine geometry is more general than affine spaces and is treated in a separate article. == Relation to projective spaces == Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines. Similar constructions hold in higher dimensions. Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity. == Affine algebraic geometry == In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariate polynomial function of the coordinates of the point. As a change of affine coordinates may be expressed by linear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates. The choice of a system of affine coordinates for an affine space A k n {\displaystyle \mathbb {A} _{k}^{n}} of dimension n over a field k induces an affine isomorphism between A k n {\displaystyle \mathbb {A} _{k}^{n}} and the affine coordinate space kn. This explains why, for simplification, many textbooks write A k n = k n {\displaystyle \mathbb {A} _{k}^{n}=k^{n}} , and introduce affine algebraic varieties as the common zeros of polynomial functions over kn. As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties. === Ring of polynomial functions === By the definition above, the choice of an affine frame of an affine space A k n {\displaystyle \mathbb {A} _{k}^{n}} allows one to identify the polynomial functions on A k n {\displaystyle \mathbb {A} _{k}^{n}} with polynomials in n variables, the ith variable representing the function that maps a point to its ith coordinate. It follows that the set of polynomial functions over A k n {\displaystyle \mathbb {A} _{k}^{n}} is a k-algebra, denoted k [ A k n ] {\displaystyle k\left[\mathbb {A} _{k}^{n}\right]} , which is isomorphic to the polynomial ring k [ X 1 , … , X n ] {\displaystyle k\left[X_{1},\dots ,X_{n}\right]} . When one changes coordinates, the isomorphism between k [ A k n ] {\displaystyle k\left[\mathbb {A} _{k}^{n}\right]} and k [ X 1 , … , X n ] {\displaystyle k[X_{1},\dots ,X_{n}]} changes accordingly, and this induces an automorphism of k [ X 1 , … , X n ] {\displaystyle k\left[X_{1},\dots ,X_{n}\right]} , which maps each indeterminate to a polynomial of degree one. It follows that the total degree defines a filtration of k [ A k n ] {\displaystyle k\left[\mathbb {A} _{k}^{n}\right]} , which is independent from the choice of coordinates. The total degree defines also a graduation, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-homogeneous polynomials. === Zariski topology === Affine spaces over topological fields, such as the real or the complex numbers, have a natural topology. The Zariski topology, which is defined for affine spaces over any field, allows use of topological methods in any case. Zariski topology is the unique topology on an affine space whose closed sets are affine algebraic sets (that is sets of the common zeros of polynomial functions over the affine set). As, over a topological field, polynomial functions are continuous, every Zariski closed set is closed for the usual topology, if any. In other words, over a topological field, Zariski topology is coarser than the natural topology. There is a natural injective function from an affine space into the set of prime ideals (that is the spectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates ( a 1 , … , a n ) {\displaystyle \left(a_{1},\dots ,a_{n}\right)} to the maximal ideal ⟨ X 1 − a 1 , … , X n − a n ⟩ {\displaystyle \left\langle X_{1}-a_{1},\dots ,X_{n}-a_{n}\right\rangle } . This function is a homeomorphism (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function. The case of an algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz). This is the starting idea of scheme theory of Grothendieck, which consists, for studying algebraic varieties, of considering as "points", not only the points of the affine space, but also all the prime ideals of the spectrum. This allows gluing together algebraic varieties in a similar way as, for manifolds, charts are glued together for building a manifold. === Cohomology === Like all affine varieties, local data on an affine space can always be patched together globally: the cohomology of affine space is trivial. More precisely, H i ( A k n , F ) = 0 {\displaystyle H^{i}\left(\mathbb {A} _{k}^{n},\mathbf {F} \right)=0} for all coherent sheaves F, and integers i > 0 {\displaystyle i>0} . This property is also enjoyed by all other affine varieties (see Serre's theorem on affineness). But also all of the étale cohomology groups on affine space are trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that every algebraic vector bundle over an affine space is trivial. == See also == Affine hull – Smallest affine subspace that contains a subset Barycentric coordinate system – Coordinate system that is defined by points instead of vectors Complex affine space – Affine space over the complex numbers Dimensional analysis § Geometry: position vs. displacement Exotic affine space – Real affine space of even dimension that is not isomorphic to a complex affine space Space (mathematics) – Mathematical set with some added structure Skew coordinates == Notes == == References == Berger, Marcel (1984), "Affine spaces", Problems in Geometry, Springer-Verlag, ISBN 978-0-387-90971-4 Berger, Marcel (1987), Geometry I, Berlin: Springer, ISBN 3-540-11658-3 Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019 Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 Dolgachev, I.V.; Shirokov, A.P. (2001) [1994], "Affine space", Encyclopedia of Mathematics, EMS Press Hartshorne, Robin (1977). Algebraic Geometry. Springer-Verlag. ISBN 978-0-387-90244-9. Zbl 0367.14001. Nomizu, K.; Sasaki, S. (1994), Affine Differential Geometry (New ed.), Cambridge University Press, ISBN 978-0-521-44177-3 Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry (Dover edition, first published in 1989 ed.), Dover Publications, ISBN 0-486-66108-3 Reventós Tarrida, Agustí (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-0-85729-709-9 |
Wikipedia:Affine representation#0 | In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) on the Euclidean space En. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space; in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general. == See also == Group action Projective representation == References == Remm, Elisabeth; Goze, Michel (2003), "Affine Structures on abelian Lie Groups", Linear Algebra and Its Applications, 360: 215–230, arXiv:math/0105023, doi:10.1016/S0024-3795(02)00452-4. |
Wikipedia:Affine space#0 | In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through k + 1 points in general position, a k-dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines within the same plane intersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction. Unlike for vectors in a vector space, in an affine space there is no distinguished point that serves as an origin. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are called free vectors, displacement vectors, translation vectors or simply translations. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. While points cannot be arbitrarily added together, it is meaningful to take affine combinations of points: weighted sums with numerical coefficients summing to 1, resulting in another point. These coefficients define a barycentric coordinate system for the flat through the points. Any vector space may be viewed as an affine space; this amounts to "forgetting" the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane. == Informal description == The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it p—is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed p + (a − p) + (b − p). Similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer. If Alice travels to λa + (1 − λ)b then Bob can similarly travel to p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b. Under this condition, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, despite using different origins. While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space. == Definition == While affine space can be defined axiomatically (see § Axioms below), analogously to the definition of Euclidean space implied by Euclid's Elements, for convenience most modern sources define affine spaces in terms of the well developed vector space theory. An affine space is a set A together with a vector space A → {\displaystyle {\overrightarrow {A}}} , and a transitive and free action of the additive group of A → {\displaystyle {\overrightarrow {A}}} on the set A. The elements of the affine space A are called points. The vector space A → {\displaystyle {\overrightarrow {A}}} is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes free vectors. Explicitly, the definition above means that the action is a mapping, generally denoted as an addition, A × A → → A ( a , v ) ↦ a + v , {\displaystyle {\begin{aligned}A\times {\overrightarrow {A}}&\to A\\(a,v)\;&\mapsto a+v,\end{aligned}}} that has the following properties. Right identity: ∀ a ∈ A , a + 0 = a {\displaystyle \forall a\in A,\;a+0=a} , where 0 is the zero vector in A → {\displaystyle {\overrightarrow {A}}} Associativity: ∀ v , w ∈ A → , ∀ a ∈ A , ( a + v ) + w = a + ( v + w ) {\displaystyle \forall v,w\in {\overrightarrow {A}},\forall a\in A,\;(a+v)+w=a+(v+w)} (here the last + is the addition in A → {\displaystyle {\overrightarrow {A}}} ) Free and transitive action: For every a ∈ A {\displaystyle a\in A} , the mapping A → → A : v ↦ a + v {\displaystyle {\overrightarrow {A}}\to A\colon v\mapsto a+v} is a bijection. The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above: Existence of one-to-one translations For all v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , the mapping A → A : a ↦ a + v {\displaystyle A\to A\colon a\mapsto a+v} is a bijection. Property 3 is often used in the following equivalent form (the 5th property). Subtraction: For every a, b in A, there exists a unique v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , denoted b – a, such that b = a + v {\displaystyle b=a+v} . Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are, by definition, endowed with a transitive group action, and for a principal homogeneous space, such a transitive action is, by definition, free. === Subtraction and Weyl's axioms === The properties of the group action allows for the definition of subtraction for any given ordered pair (b, a) of points in A, producing a vector of A → {\displaystyle {\overrightarrow {A}}} . This vector, denoted b − a {\displaystyle b-a} or a b → {\displaystyle {\overrightarrow {ab}}} , is defined to be the unique vector in A → {\displaystyle {\overrightarrow {A}}} such that a + ( b − a ) = b . {\displaystyle a+(b-a)=b.} Existence follows from the transitivity of the action, and uniqueness follows because the action is free. This subtraction has the two following properties, called Weyl's axioms: ∀ a ∈ A , ∀ v ∈ A → {\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}} , there is a unique point b ∈ A {\displaystyle b\in A} such that b − a = v . {\displaystyle b-a=v.} ∀ a , b , c ∈ A , ( c − b ) + ( b − a ) = c − a . {\displaystyle \forall a,b,c\in A,\;(c-b)+(b-a)=c-a.} The parallelogram property is satisfied in affine spaces, where it is expressed as: given four points a , b , c , d , {\displaystyle a,b,c,d,} the equalities b − a = d − c {\displaystyle b-a=d-c} and c − a = d − b {\displaystyle c-a=d-b} are equivalent. This results from the second Weyl's axiom, since d − a = ( d − b ) + ( b − a ) = ( d − c ) + ( c − a ) . {\displaystyle d-a=(d-b)+(b-a)=(d-c)+(c-a).} Affine spaces can be equivalently defined as a point set A, together with a vector space A → {\displaystyle {\overrightarrow {A}}} , and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms. == Affine subspaces and parallelism == An affine subspace (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) B of an affine space A is a subset of A such that, given a point a ∈ B {\displaystyle a\in B} , the set of vectors B → = { b − a ∣ b ∈ B } {\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}} is a linear subspace of A → {\displaystyle {\overrightarrow {A}}} . This property, which does not depend on the choice of a, implies that B is an affine space, which has B → {\displaystyle {\overrightarrow {B}}} as its associated vector space. The affine subspaces of A are the subsets of A of the form a + V = { a + w : w ∈ V } , {\displaystyle a+V=\{a+w:w\in V\},} where a is a point of A, and V a linear subspace of A → {\displaystyle {\overrightarrow {A}}} . The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said to be parallel. This implies the following generalization of Playfair's axiom: Given a direction V, for any point a of A there is one and only one affine subspace of direction V, which passes through a, namely the subspace a + V. Every translation A → A : a ↦ a + v {\displaystyle A\to A:a\mapsto a+v} maps any affine subspace to a parallel subspace. The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other. == Affine map == Given two affine spaces A and B whose associated vector spaces are A → {\displaystyle {\overrightarrow {A}}} and B → {\displaystyle {\overrightarrow {B}}} , an affine map or affine homomorphism from A to B is a map f : A → B {\displaystyle f:A\to B} such that f → : A → → B → b − a ↦ f ( b ) − f ( a ) {\displaystyle {\begin{aligned}{\overrightarrow {f}}:{\overrightarrow {A}}&\to {\overrightarrow {B}}\\b-a&\mapsto f(b)-f(a)\end{aligned}}} is a well defined linear map. By f {\displaystyle f} being well defined is meant that b – a = d – c implies f(b) – f(a) = f(d) – f(c). This implies that, for a point a ∈ A {\displaystyle a\in A} and a vector v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , one has f ( a + v ) = f ( a ) + f → ( v ) . {\displaystyle f(a+v)=f(a)+{\overrightarrow {f}}(v).} Therefore, since for any given b in A, b = a + v for a unique v, f is completely defined by its value on a single point and the associated linear map f → {\displaystyle {\overrightarrow {f}}} . === Endomorphisms === An affine transformation or endomorphism of an affine space A {\displaystyle A} is an affine map from that space to itself. One important family of examples is the translations: given a vector v → {\displaystyle {\overrightarrow {v}}} , the translation map T v → : A → A {\displaystyle T_{\overrightarrow {v}}:A\rightarrow A} that sends a ↦ a + v → {\displaystyle a\mapsto a+{\overrightarrow {v}}} for every a {\displaystyle a} in A {\displaystyle A} is an affine map. Another important family of examples are the linear maps centred at an origin: given a point b {\displaystyle b} and a linear map M {\displaystyle M} , one may define an affine map L M , b : A → A {\displaystyle L_{M,b}:A\rightarrow A} by L M , b ( a ) = b + M ( a − b ) {\displaystyle L_{M,b}(a)=b+M(a-b)} for every a {\displaystyle a} in A {\displaystyle A} . After making a choice of origin b {\displaystyle b} , any affine map may be written uniquely as a combination of a translation and a linear map centred at b {\displaystyle b} . == Vector spaces as affine spaces == Every vector space V may be considered as an affine space over itself. This means that every element of V may be considered either as a point or as a vector. This affine space is sometimes denoted (V, V) for emphasizing the double role of the elements of V. When considered as a point, the zero vector is commonly denoted o (or O, when upper-case letters are used for points) and called the origin. If A is another affine space over the same vector space (that is V = A → {\displaystyle V={\overrightarrow {A}}} ) the choice of any point a in A defines a unique affine isomorphism, which is the identity of V and maps a to o. In other words, the choice of an origin a in A allows us to identify A and (V, V) up to a canonical isomorphism. The counterpart of this property is that the affine space A may be identified with the vector space V in which "the place of the origin has been forgotten". == Relation to Euclidean spaces == === Definition of Euclidean spaces === Euclidean spaces (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces. Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form q(x). The inner product of two vectors x and y is the value of the symmetric bilinear form x ⋅ y = 1 2 ( q ( x + y ) − q ( x ) − q ( y ) ) . {\displaystyle x\cdot y={\frac {1}{2}}(q(x+y)-q(x)-q(y)).} The usual Euclidean distance between two points A and B is d ( A , B ) = q ( B − A ) . {\displaystyle d(A,B)={\sqrt {q(B-A)}}.} In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of ordered pairs of points under equipollence (the pairs (A, B) and (C, D) are equipollent if the points A, B, D, C (in this order) form a parallelogram). It is straightforward to verify that the vectors form a vector space, the square of the Euclidean distance is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent. === Affine properties === In Euclidean geometry, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples are parallelism, and the definition of a tangent. A non-example is the definition of a normal. Equivalently, an affine property is a property that is invariant under affine transformations of the Euclidean space. == Affine combinations and barycenter == Let a1, ..., an be a collection of n points in an affine space, and λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} be n elements of the ground field. Suppose that λ 1 + ⋯ + λ n = 0 {\displaystyle \lambda _{1}+\dots +\lambda _{n}=0} . For any two points o and o' one has λ 1 o a 1 → + ⋯ + λ n o a n → = λ 1 o ′ a 1 → + ⋯ + λ n o ′ a n → . {\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}=\lambda _{1}{\overrightarrow {o'a_{1}}}+\dots +\lambda _{n}{\overrightarrow {o'a_{n}}}.} Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted λ 1 a 1 + ⋯ + λ n a n . {\displaystyle \lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.} When n = 2 , λ 1 = 1 , λ 2 = − 1 {\displaystyle n=2,\lambda _{1}=1,\lambda _{2}=-1} , one retrieves the definition of the subtraction of points. Now suppose instead that the field elements satisfy λ 1 + ⋯ + λ n = 1 {\displaystyle \lambda _{1}+\dots +\lambda _{n}=1} . For some choice of an origin o, denote by g {\displaystyle g} the unique point such that λ 1 o a 1 → + ⋯ + λ n o a n → = o g → . {\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}={\overrightarrow {og}}.} One can show that g {\displaystyle g} is independent from the choice of o. Therefore, if λ 1 + ⋯ + λ n = 1 , {\displaystyle \lambda _{1}+\dots +\lambda _{n}=1,} one may write g = λ 1 a 1 + ⋯ + λ n a n . {\displaystyle g=\lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.} The point g {\displaystyle g} is called the barycenter of the a i {\displaystyle a_{i}} for the weights λ i {\displaystyle \lambda _{i}} . One says also that g {\displaystyle g} is an affine combination of the a i {\displaystyle a_{i}} with coefficients λ i {\displaystyle \lambda _{i}} . == Examples == When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space. Time can be modelled as a one-dimensional affine space. Specific points in time (such as a date on the calendar) are points in the affine space, while durations (such as a number of days) are displacements. The space of energies is an affine space for R {\displaystyle \mathbb {R} } , since it is often not meaningful to talk about absolute energy, but it is meaningful to talk about energy differences. The vacuum energy when it is defined picks out a canonical origin. Physical space is often modelled as an affine space for R 3 {\displaystyle \mathbb {R} ^{3}} in non-relativistic settings and R 1 , 3 {\displaystyle \mathbb {R} ^{1,3}} in the relativistic setting. To distinguish them from the vector space these are sometimes called Euclidean spaces E ( 3 ) {\displaystyle {\text{E}}(3)} and E ( 1 , 3 ) {\displaystyle {\text{E}}(1,3)} . Any coset of a subspace V of a vector space is an affine space over that subspace. In particular, a line in the plane that doesn't pass through the origin is an affine space that is not a vector space relative to the operations it inherits from R 2 {\displaystyle \mathbb {R} ^{2}} , although it can be given a canonical vector space structure by picking the point closest to the origin as the zero vector; likewise in higher dimensions and for any normed vector space If T is a matrix and b lies in its column space, the set of solutions of the equation Tx = b is an affine space over the subspace of solutions of Tx = 0. The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. Generalizing all of the above, if T : V → W is a linear map and y lies in its image, the set of solutions x ∈ V to the equation Tx = y is a coset of the kernel of T , and is therefore an affine space over Ker T . The space of (linear) complementary subspaces of a vector subspace V in a vector space W is an affine space, over Hom(W/V, V). That is, if 0 → V → W → X → 0 is a short exact sequence of vector spaces, then the space of all splittings of the exact sequence naturally carries the structure of an affine space over Hom(X, V). The space of connections (viewed from the vector bundle E → π M {\displaystyle E\xrightarrow {\pi } M} , where M {\displaystyle M} is a smooth manifold) is an affine space for the vector space of End ( E ) {\displaystyle {\text{End}}(E)} valued 1-forms. The space of connections (viewed from the principal bundle P → π M {\displaystyle P\xrightarrow {\pi } M} ) is an affine space for the vector space of ad ( P ) {\displaystyle {\text{ad}}(P)} -valued 1-forms, where ad ( P ) {\displaystyle {\text{ad}}(P)} is the associated adjoint bundle. == Affine span and bases == For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X. It is the intersection of all affine subspaces containing X, and its direction is the intersection of the directions of the affine subspaces that contain X. The affine span of X is the set of all (finite) affine combinations of points of X, and its direction is the linear span of the x − y for x and y in X. If one chooses a particular point x0, the direction of the affine span of X is also the linear span of the x – x0 for x in X. One says also that the affine span of X is generated by X and that X is a generating set of its affine span. A set X of points of an affine space is said to be affinely independent or, simply, independent, if the affine span of any strict subset of X is a strict subset of the affine span of X. An affine basis or barycentric frame (see § Barycentric coordinates, below) of an affine space is a generating set that is also independent (that is a minimal generating set). Recall that the dimension of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimension n are the independent subsets of n + 1 elements, or, equivalently, the generating subsets of n + 1 elements. Equivalently, {x0, ..., xn} is an affine basis of an affine space if and only if {x1 − x0, ..., xn − x0} is a linear basis of the associated vector space. == Coordinates == There are two strongly related kinds of coordinate systems that may be defined on affine spaces. === Barycentric coordinates === Let A be an affine space of dimension n over a field k, and { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} be an affine basis of A. The properties of an affine basis imply that for every x in A there is a unique (n + 1)-tuple ( λ 0 , … , λ n ) {\displaystyle (\lambda _{0},\dots ,\lambda _{n})} of elements of k such that λ 0 + ⋯ + λ n = 1 {\displaystyle \lambda _{0}+\dots +\lambda _{n}=1} and x = λ 0 x 0 + ⋯ + λ n x n . {\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.} The λ i {\displaystyle \lambda _{i}} are called the barycentric coordinates of x over the affine basis { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} . If the xi are viewed as bodies that have weights (or masses) λ i {\displaystyle \lambda _{i}} , the point x is thus the barycenter of the xi, and this explains the origin of the term barycentric coordinates. The barycentric coordinates define an affine isomorphism between the affine space A and the affine subspace of kn + 1 defined by the equation λ 0 + ⋯ + λ n = 1 {\displaystyle \lambda _{0}+\dots +\lambda _{n}=1} . For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero. === Affine coordinates === An affine frame is a coordinate frame of an affine space, consisting of a point, called the origin, and a linear basis of the associated vector space. More precisely, for an affine space A with associated vector space A → {\displaystyle {\overrightarrow {A}}} , the origin o belongs to A, and the linear basis is a basis (v1, ..., vn) of A → {\displaystyle {\overrightarrow {A}}} (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar). For each point p of A, there is a unique sequence λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} of elements of the ground field such that p = o + λ 1 v 1 + ⋯ + λ n v n , {\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},} or equivalently o p → = λ 1 v 1 + ⋯ + λ n v n . {\displaystyle {\overrightarrow {op}}=\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n}.} The λ i {\displaystyle \lambda _{i}} are called the affine coordinates of p over the affine frame (o, v1, ..., vn). Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame (o, v1, ..., vn) such that (v1, ..., vn) is an orthonormal basis. === Relationship between barycentric and affine coordinates === Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent. In fact, given a barycentric frame ( x 0 , … , x n ) , {\displaystyle (x_{0},\dots ,x_{n}),} one deduces immediately the affine frame ( x 0 , x 0 x 1 → , … , x 0 x n → ) = ( x 0 , x 1 − x 0 , … , x n − x 0 ) , {\displaystyle (x_{0},{\overrightarrow {x_{0}x_{1}}},\dots ,{\overrightarrow {x_{0}x_{n}}})=\left(x_{0},x_{1}-x_{0},\dots ,x_{n}-x_{0}\right),} and, if ( λ 0 , λ 1 , … , λ n ) {\displaystyle \left(\lambda _{0},\lambda _{1},\dots ,\lambda _{n}\right)} are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are ( λ 1 , … , λ n ) . {\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right).} Conversely, if ( o , v 1 , … , v n ) {\displaystyle \left(o,v_{1},\dots ,v_{n}\right)} is an affine frame, then ( o , o + v 1 , … , o + v n ) {\displaystyle \left(o,o+v_{1},\dots ,o+v_{n}\right)} is a barycentric frame. If ( λ 1 , … , λ n ) {\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right)} are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are ( 1 − λ 1 − ⋯ − λ n , λ 1 , … , λ n ) . {\displaystyle \left(1-\lambda _{1}-\dots -\lambda _{n},\lambda _{1},\dots ,\lambda _{n}\right).} Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example. ==== Example of the triangle ==== The vertices of a non-flat triangle form an affine basis of the Euclidean plane. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distances: The vertices are the points of barycentric coordinates (1, 0, 0), (0, 1, 0) and (0, 0, 1). The lines supporting the edges are the points that have a zero coordinate. The edges themselves are the points that have one zero coordinate and two nonnegative coordinates. The interior of the triangle are the points whose coordinates are all positive. The medians are the points that have two equal coordinates, and the centroid is the point of coordinates (1/3, 1/3, 1/3). === Change of coordinates === ==== Case of barycentric coordinates ==== Barycentric coordinates are readily changed from one basis to another. Let { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} and { x 0 ′ , … , x n ′ } {\displaystyle \{x'_{0},\dots ,x'_{n}\}} be affine bases of A. For every x in A there is some tuple { λ 0 , … , λ n } {\displaystyle \{\lambda _{0},\dots ,\lambda _{n}\}} for which x = λ 0 x 0 + ⋯ + λ n x n . {\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.} Similarly, for every x i ∈ { x 0 , … , x n } {\displaystyle x_{i}\in \{x_{0},\dots ,x_{n}\}} from the first basis, we now have in the second basis x i = λ i , 0 x 0 ′ + ⋯ + λ i , j x j ′ + ⋯ + λ i , n x n ′ {\displaystyle x_{i}=\lambda _{i,0}x'_{0}+\dots +\lambda _{i,j}x'_{j}+\dots +\lambda _{i,n}x'_{n}} for some tuple { λ i , 0 , … , λ i , n } {\displaystyle \{\lambda _{i,0},\dots ,\lambda _{i,n}\}} . Now we can rewrite our expression in the first basis as one in the second with x = ∑ i = 0 n λ i x i = ∑ i = 0 n λ i ∑ j = 0 n λ i , j x j ′ = ∑ j = 0 n ( ∑ i = 0 n λ i λ i , j ) x j ′ , {\displaystyle \,x=\sum _{i=0}^{n}\lambda _{i}x_{i}=\sum _{i=0}^{n}\lambda _{i}\sum _{j=0}^{n}\lambda _{i,j}x'_{j}=\sum _{j=0}^{n}{\biggl (}\sum _{i=0}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}x'_{j}\,,} giving us coordinates in the second basis as the tuple { ∑ i λ i λ i , 0 , … , {\textstyle {\bigl \{}\sum _{i}\lambda _{i}\lambda _{i,0},\,\dots ,\,{}} ∑ i λ i λ i , n } {\textstyle \sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}} . ==== Case of affine coordinates ==== Affine coordinates are also readily changed from one basis to another. Let o {\displaystyle o} , { v 1 , … , v n } {\displaystyle \{v_{1},\dots ,v_{n}\}} and o ′ {\displaystyle o'} , { v 1 ′ , … , v n ′ } {\displaystyle \{v'_{1},\dots ,v'_{n}\}} be affine frames of A. For each point p of A, there is a unique sequence λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} of elements of the ground field such that p = o + λ 1 v 1 + ⋯ + λ n v n , {\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},} and similarly, for every v i ∈ { v 1 , … , v n } {\displaystyle v_{i}\in \{v_{1},\dots ,v_{n}\}} from the first basis, we now have in the second basis o = o ′ + λ o , 1 v 1 ′ + ⋯ + λ o , j v j ′ + ⋯ + λ o , n v n ′ {\displaystyle o=o'+\lambda _{o,1}v'_{1}+\dots +\lambda _{o,j}v'_{j}+\dots +\lambda _{o,n}v'_{n}\,} v i = λ i , 1 v 1 ′ + ⋯ + λ i , j v j ′ + ⋯ + λ i , n v n ′ {\displaystyle v_{i}=\lambda _{i,1}v'_{1}+\dots +\lambda _{i,j}v'_{j}+\dots +\lambda _{i,n}v'_{n}} for tuple { λ o , 1 , … , λ o , n } {\displaystyle \{\lambda _{o,1},\dots ,\lambda _{o,n}\}} and tuples { λ i , 1 , … , λ i , n } {\displaystyle \{\lambda _{i,1},\dots ,\lambda _{i,n}\}} . Now we can rewrite our expression in the first basis as one in the second with p = o + ∑ i = 1 n λ i v i = ( o ′ + ∑ j = 1 n λ o , j v j ′ ) + ∑ i = 1 n λ i ∑ j = 1 n λ i , j v j ′ = o ′ + ∑ j = 1 n ( λ o , j + ∑ i = 1 n λ i λ i , j ) v j ′ , {\displaystyle {\begin{aligned}\,p&=o+\sum _{i=1}^{n}\lambda _{i}v_{i}={\biggl (}o'+\sum _{j=1}^{n}\lambda _{o,j}v'_{j}{\biggr )}+\sum _{i=1}^{n}\lambda _{i}\sum _{j=1}^{n}\lambda _{i,j}v'_{j}\\&=o'+\sum _{j=1}^{n}{\biggl (}\lambda _{o,j}+\sum _{i=1}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}v'_{j}\,,\end{aligned}}} giving us coordinates in the second basis as the tuple { λ o , 1 + ∑ i λ i λ i , 1 , … , {\textstyle {\bigl \{}\lambda _{o,1}+\sum _{i}\lambda _{i}\lambda _{i,1},\,\dots ,\,{}} λ o , n + ∑ i λ i λ i , n } {\textstyle \lambda _{o,n}+\sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}} . == Properties of affine homomorphisms == === Matrix representation === An affine transformation T {\displaystyle T} is executed on a projective space P 3 {\displaystyle \mathbb {P} ^{3}} of R 3 {\displaystyle \mathbb {R} ^{3}} , by a 4 by 4 matrix with a special fourth column: A = [ a 11 a 12 a 13 0 a 21 a 22 a 23 0 a 31 a 32 a 33 0 a 41 a 42 a 43 1 ] = [ T ( 1 , 0 , 0 ) 0 T ( 0 , 1 , 0 ) 0 T ( 0 , 0 , 1 ) 0 T ( 0 , 0 , 0 ) 1 ] {\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&0\\a_{21}&a_{22}&a_{23}&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&1\end{bmatrix}}={\begin{bmatrix}T(1,0,0)&0\\T(0,1,0)&0\\T(0,0,1)&0\\T(0,0,0)&1\end{bmatrix}}} The transformation is affine instead of linear due to the inclusion of point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} , the transformed output of which reveals the affine shift. === Image and fibers === Let f : E → F {\displaystyle f\colon E\to F} be an affine homomorphism, with f → : E → → F → {\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}} its associated linear map. The image of f is the affine subspace f ( E ) = { f ( a ) ∣ a ∈ E } {\displaystyle f(E)=\{f(a)\mid a\in E\}} of F, which has f → ( E → ) {\displaystyle {\overrightarrow {f}}({\overrightarrow {E}})} as associated vector space. As an affine space does not have a zero element, an affine homomorphism does not have a kernel. However, the linear map f → {\displaystyle {\overrightarrow {f}}} does, and if we denote by K = { v ∈ E → ∣ f → ( v ) = 0 } {\displaystyle K=\{v\in {\overrightarrow {E}}\mid {\overrightarrow {f}}(v)=0\}} its kernel, then for any point x of f ( E ) {\displaystyle f(E)} , the inverse image f − 1 ( x ) {\displaystyle f^{-1}(x)} of x is an affine subspace of E whose direction is K {\displaystyle K} . This affine subspace is called the fiber of x. === Projection === An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact that Euclidean spaces are affine spaces, and that these kinds of projections are fundamental in Euclidean geometry. More precisely, given an affine space E with associated vector space E → {\displaystyle {\overrightarrow {E}}} , let F be an affine subspace of direction F → {\displaystyle {\overrightarrow {F}}} , and D be a complementary subspace of F → {\displaystyle {\overrightarrow {F}}} in E → {\displaystyle {\overrightarrow {E}}} (this means that every vector of E → {\displaystyle {\overrightarrow {E}}} may be decomposed in a unique way as the sum of an element of F → {\displaystyle {\overrightarrow {F}}} and an element of D). For every point x of E, its projection to F parallel to D is the unique point p(x) in F such that p ( x ) − x ∈ D . {\displaystyle p(x)-x\in D.} This is an affine homomorphism whose associated linear map p → {\displaystyle {\overrightarrow {p}}} is defined by p → ( x − y ) = p ( x ) − p ( y ) , {\displaystyle {\overrightarrow {p}}(x-y)=p(x)-p(y),} for x and y in E. The image of this projection is F, and its fibers are the subspaces of direction D. === Quotient space === Although kernels are not defined for affine spaces, quotient spaces are defined. This results from the fact that "belonging to the same fiber of an affine homomorphism" is an equivalence relation. Let E be an affine space, and D be a linear subspace of the associated vector space E → {\displaystyle {\overrightarrow {E}}} . The quotient E/D of E by D is the quotient of E by the equivalence relation such that x and y are equivalent if x − y ∈ D . {\displaystyle x-y\in D.} This quotient is an affine space, which has E → / D {\displaystyle {\overrightarrow {E}}/D} as associated vector space. For every affine homomorphism E → F {\displaystyle E\to F} , the image is isomorphic to the quotient of E by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces. == Axioms == Affine spaces are usually studied by analytic geometry using coordinates, or equivalently vector spaces. They can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space. Coxeter (1969, p. 192) axiomatizes the special case of affine geometry over the reals as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line. Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint): Any two distinct points lie on a unique line. Given a point and line there is a unique line that contains the point and is parallel to the line There exist three non-collinear points. As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional affine spaces. Purely axiomatic affine geometry is more general than affine spaces and is treated in a separate article. == Relation to projective spaces == Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines. Similar constructions hold in higher dimensions. Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity. == Affine algebraic geometry == In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariate polynomial function of the coordinates of the point. As a change of affine coordinates may be expressed by linear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates. The choice of a system of affine coordinates for an affine space A k n {\displaystyle \mathbb {A} _{k}^{n}} of dimension n over a field k induces an affine isomorphism between A k n {\displaystyle \mathbb {A} _{k}^{n}} and the affine coordinate space kn. This explains why, for simplification, many textbooks write A k n = k n {\displaystyle \mathbb {A} _{k}^{n}=k^{n}} , and introduce affine algebraic varieties as the common zeros of polynomial functions over kn. As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties. === Ring of polynomial functions === By the definition above, the choice of an affine frame of an affine space A k n {\displaystyle \mathbb {A} _{k}^{n}} allows one to identify the polynomial functions on A k n {\displaystyle \mathbb {A} _{k}^{n}} with polynomials in n variables, the ith variable representing the function that maps a point to its ith coordinate. It follows that the set of polynomial functions over A k n {\displaystyle \mathbb {A} _{k}^{n}} is a k-algebra, denoted k [ A k n ] {\displaystyle k\left[\mathbb {A} _{k}^{n}\right]} , which is isomorphic to the polynomial ring k [ X 1 , … , X n ] {\displaystyle k\left[X_{1},\dots ,X_{n}\right]} . When one changes coordinates, the isomorphism between k [ A k n ] {\displaystyle k\left[\mathbb {A} _{k}^{n}\right]} and k [ X 1 , … , X n ] {\displaystyle k[X_{1},\dots ,X_{n}]} changes accordingly, and this induces an automorphism of k [ X 1 , … , X n ] {\displaystyle k\left[X_{1},\dots ,X_{n}\right]} , which maps each indeterminate to a polynomial of degree one. It follows that the total degree defines a filtration of k [ A k n ] {\displaystyle k\left[\mathbb {A} _{k}^{n}\right]} , which is independent from the choice of coordinates. The total degree defines also a graduation, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-homogeneous polynomials. === Zariski topology === Affine spaces over topological fields, such as the real or the complex numbers, have a natural topology. The Zariski topology, which is defined for affine spaces over any field, allows use of topological methods in any case. Zariski topology is the unique topology on an affine space whose closed sets are affine algebraic sets (that is sets of the common zeros of polynomial functions over the affine set). As, over a topological field, polynomial functions are continuous, every Zariski closed set is closed for the usual topology, if any. In other words, over a topological field, Zariski topology is coarser than the natural topology. There is a natural injective function from an affine space into the set of prime ideals (that is the spectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates ( a 1 , … , a n ) {\displaystyle \left(a_{1},\dots ,a_{n}\right)} to the maximal ideal ⟨ X 1 − a 1 , … , X n − a n ⟩ {\displaystyle \left\langle X_{1}-a_{1},\dots ,X_{n}-a_{n}\right\rangle } . This function is a homeomorphism (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function. The case of an algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz). This is the starting idea of scheme theory of Grothendieck, which consists, for studying algebraic varieties, of considering as "points", not only the points of the affine space, but also all the prime ideals of the spectrum. This allows gluing together algebraic varieties in a similar way as, for manifolds, charts are glued together for building a manifold. === Cohomology === Like all affine varieties, local data on an affine space can always be patched together globally: the cohomology of affine space is trivial. More precisely, H i ( A k n , F ) = 0 {\displaystyle H^{i}\left(\mathbb {A} _{k}^{n},\mathbf {F} \right)=0} for all coherent sheaves F, and integers i > 0 {\displaystyle i>0} . This property is also enjoyed by all other affine varieties (see Serre's theorem on affineness). But also all of the étale cohomology groups on affine space are trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that every algebraic vector bundle over an affine space is trivial. == See also == Affine hull – Smallest affine subspace that contains a subset Barycentric coordinate system – Coordinate system that is defined by points instead of vectors Complex affine space – Affine space over the complex numbers Dimensional analysis § Geometry: position vs. displacement Exotic affine space – Real affine space of even dimension that is not isomorphic to a complex affine space Space (mathematics) – Mathematical set with some added structure Skew coordinates == Notes == == References == Berger, Marcel (1984), "Affine spaces", Problems in Geometry, Springer-Verlag, ISBN 978-0-387-90971-4 Berger, Marcel (1987), Geometry I, Berlin: Springer, ISBN 3-540-11658-3 Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019 Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 Dolgachev, I.V.; Shirokov, A.P. (2001) [1994], "Affine space", Encyclopedia of Mathematics, EMS Press Hartshorne, Robin (1977). Algebraic Geometry. Springer-Verlag. ISBN 978-0-387-90244-9. Zbl 0367.14001. Nomizu, K.; Sasaki, S. (1994), Affine Differential Geometry (New ed.), Cambridge University Press, ISBN 978-0-521-44177-3 Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry (Dover edition, first published in 1989 ed.), Dover Publications, ISBN 0-486-66108-3 Reventós Tarrida, Agustí (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-0-85729-709-9 |
Wikipedia:Agata Ciabattoni#0 | Agata Ciabattoni is an Italian mathematical logician specializing in non-classical logic. She is a full professor at the Institute of Logic and Computation of the Faculty of Informatics at the Vienna University of Technology (TU Wien), and a co-chair of the Vienna Center for Logic and Algorithms of TU Wien (VCLA). == Education and career == Ciabattoni is originally from Ripatransone. She studied computer science at the University of Bologna, and completed her Ph.D. in 2000 at the University of Milan. Her dissertation, Proof-theory in many-valued logics, was supervised by Daniele Mundici. She moved to Vienna in 2000 with the support of an EU Marie Curie Fellowship, and In 2007, she earned her habilitation at TU Wien. She remains affiliated with TU Wien, as a professor in the faculty of informatics. She also serves as the Collegium Logicum lecture series chair for the Kurt Gödel Society. == Contributions == One of Ciabattoni's projects at TU Wien involves using mathematical logic to formalize the ethical reasoning in the Vedas, a body of Indian sacred texts. == Recognition == In 2011, Ciabattoni won the Start-Preis of the Austrian Science Fund, the only woman to win the prize that year. == References == == External links == Home page Agata Ciabattoni publications indexed by Google Scholar |
Wikipedia:Agata Smoktunowicz#0 | Agata Smoktunowicz FRSE (born 12 October 1973) is a Polish mathematician who works as a professor at the University of Edinburgh. Her research is in abstract algebra. == Contributions == Smoktunowicz's contributions to mathematics include constructing noncommutative nil rings, solving a "famous problem" formulated in 1970 by Irving Kaplansky. She proved the Artin–Stafford gap conjecture according to which the Gelfand–Kirillov dimension of a graded domain cannot fall within the open interval (2,3). She also found an example of a nil ideal of a ring R that does not lift to a nil ideal of the polynomial ring R[X], disproving a conjecture of Amitsur and hinting that the Köthe conjecture might be false. == Awards and honours == Smoktunowicz was an invited speaker at the International Congress of Mathematicians in 2006. She won the Whitehead Prize of the London Mathematical Society in 2006, the European Mathematical Society Prize in 2008, and the Sir Edmund Whittaker Memorial Prize of the Edinburgh Mathematical Society in 2009. In 2009, she was elected as a fellow of the Royal Society of Edinburgh, and in 2012, she became one of the inaugural fellows of the American Mathematical Society. She also won the Polish Academy of Sciences annual research prize in 2018. She was awarded the Senior Whitehead Prize of the London Mathematical Society in 2023. == Education and career == Smoktunowicz earned a master's degree from the University of Warsaw in 1997, a PhD in 1999 from the Institute of Mathematics of the Polish Academy of Sciences, and a habilitation in 2007, again from the Polish Academy of Sciences. After temporary positions at Yale University and the University of California, San Diego, she joined the University of Edinburgh in 2005, and was promoted to professor there in 2007. == Selected publications == Smoktunowicz, Agata (2000), "Polynomial rings over nil rings need not be nil", Journal of Algebra, 233 (2): 427–436, doi:10.1006/jabr.2000.8451, MR 1793911. Huh, Chan; Lee, Yang; Smoktunowicz, Agata (2002), "Armendariz rings and semicommutative rings", Communications in Algebra, 30 (2): 751–761, doi:10.1081/AGB-120013179, MR 1883022, S2CID 121438679. Smoktunowicz, Agata (2002), "A simple nil ring exists", Communications in Algebra, 30 (1): 27–59, doi:10.1081/AGB-120006478, MR 1880660, S2CID 121093658. Smoktunowicz, Agata (2006), "There are no graded domains with GK dimension strictly between 2 and 3", Inventiones Mathematicae, 164 (3): 635–640, Bibcode:2006InMat.164..635S, doi:10.1007/s00222-005-0489-1, MR 2221134, S2CID 119680902. == References == |
Wikipedia:Agmon's inequality#0 | In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space L ∞ {\displaystyle L^{\infty }} and the Sobolev spaces H s {\displaystyle H^{s}} . It is useful in the study of partial differential equations. Let u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) {\displaystyle u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} where Ω ⊂ R 3 {\displaystyle \Omega \subset \mathbb {R} ^{3}} . Then Agmon's inequalities in 3D state that there exists a constant C {\displaystyle C} such that ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) 1 / 2 ‖ u ‖ H 2 ( Ω ) 1 / 2 , {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2},} and ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ L 2 ( Ω ) 1 / 4 ‖ u ‖ H 2 ( Ω ) 3 / 4 . {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/4}\|u\|_{H^{2}(\Omega )}^{3/4}.} In 2D, the first inequality still holds, but not the second: let u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) {\displaystyle u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} where Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} . Then Agmon's inequality in 2D states that there exists a constant C {\displaystyle C} such that ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ L 2 ( Ω ) 1 / 2 ‖ u ‖ H 2 ( Ω ) 1 / 2 . {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2}.} For the n {\displaystyle n} -dimensional case, choose s 1 {\displaystyle s_{1}} and s 2 {\displaystyle s_{2}} such that s 1 < n 2 < s 2 {\displaystyle s_{1}<{\tfrac {n}{2}}<s_{2}} . Then, if 0 < θ < 1 {\displaystyle 0<\theta <1} and n 2 = θ s 1 + ( 1 − θ ) s 2 {\displaystyle {\tfrac {n}{2}}=\theta s_{1}+(1-\theta )s_{2}} , the following inequality holds for any u ∈ H s 2 ( Ω ) {\displaystyle u\in H^{s_{2}}(\Omega )} ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H s 1 ( Ω ) θ ‖ u ‖ H s 2 ( Ω ) 1 − θ {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{s_{1}}(\Omega )}^{\theta }\|u\|_{H^{s_{2}}(\Omega )}^{1-\theta }} == See also == Ladyzhenskaya inequality Brezis–Gallouet inequality == Notes == == References == Agmon, Shmuel (2010). Lectures on elliptic boundary value problems. Providence, RI: AMS Chelsea Publishing. ISBN 978-0-8218-4910-1. Foias, Ciprian; Manley, O.; Rosa, R.; Temam, R. (2001). Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press. ISBN 0-521-36032-3. |
Wikipedia:Agnes Bell Collier#0 | Agnes Bell Collier (31 January 1860 – 2 January 1930) was a British mathematician who was a pioneer female mathematician, associated with Newnham College, Cambridge. Collier was born in Hyde, Cheshire. She was educated at Ellerslie Ladies' College, Manchester and Newnham College, Cambridge from 1880 to 1883, passing the Mathematical Tripos in 1883. She was College Lecturer in Mathematics from 1883 to 1925 and Director of Studies 1883–1920. She was College Vice-Principal 1920–25, and a College Associate from 1893 to 1917. == References == |
Wikipedia:Agnew's theorem#0 | Agnew's theorem, proposed by American mathematician Ralph Palmer Agnew, characterizes reorderings of terms of infinite series that preserve convergence for all series. == Statement == We call a permutation p : N → N {\displaystyle p:\mathbb {N} \to \mathbb {N} } an Agnew permutation if there exists K ∈ N {\displaystyle K\in \mathbb {N} } such that any interval that starts with 1 is mapped by p to a union of at most K intervals, i.e., ∃ K ∈ N : ∀ n ∈ N # [ ] ( p ( [ 1 , n ] ) ) ≤ K {\textstyle \exists K\in \mathbb {N} \,:\;\forall n\in \mathbb {N} \;\;\#_{[\,]}(p([1,\,n]))\leq K\,} , where # [ ] {\displaystyle \#_{[\,]}} counts the number of intervals. Agnew's theorem. p {\displaystyle p} is an Agnew permutation ⟺ {\displaystyle \iff } for all converging series of real or complex terms ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}\,} , the series ∑ i = 1 ∞ a p ( i ) {\textstyle \sum _{i=1}^{\infty }a_{p(i)}} converges to the same sum. Corollary 1. p − 1 {\displaystyle p^{-1}} (the inverse of p {\displaystyle p} ) is an Agnew permutation ⟹ {\displaystyle \implies } for all diverging series of real or complex terms ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}\,} , the series ∑ i = 1 ∞ a p ( i ) {\textstyle \sum _{i=1}^{\infty }a_{p(i)}} diverges. Corollary 2. p {\displaystyle p} and p − 1 {\displaystyle p^{-1}} are Agnew permutations ⟹ {\displaystyle \implies } for all series of real or complex terms ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}\,} , the convergence type of the series ∑ i = 1 ∞ a p ( i ) {\textstyle \sum _{i=1}^{\infty }a_{p(i)}} is the same. == Usage == Agnew's theorem is useful when the convergence of ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}} has already been established: any Agnew permutation can be used to rearrange its terms while preserving convergence to the same sum. The Corollary 2 is useful when the convergence type of ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}} is unknown: the convergence type of ∑ i = 1 ∞ a p ( i ) {\textstyle \sum _{i=1}^{\infty }a_{p(i)}} is the same as that of the original series. == Examples == An important class of permutations is infinite compositions of permutations p = ⋯ ∘ p k ∘ ⋯ ∘ p 1 {\displaystyle p=\cdots \circ p_{k}\circ \cdots \circ p_{1}} in which each constituent permutation p k {\displaystyle p_{k}} acts only on its corresponding interval [ g k + 1 , g k + 1 ] {\displaystyle [g_{k}+1,\,g_{k+1}]} (with g 1 = 0 {\displaystyle g_{1}=0} ). Since p ( [ 1 , n ] ) = [ 1 , g k ] ∪ p k ( [ g k + 1 , n ] ) {\displaystyle p([1,\,n])=[1,\,g_{k}]\cup p_{k}([g_{k}+1,\,n])} for g k + 1 ≤ n < g k + 1 {\displaystyle g_{k}+1\leq n<g_{k+1}} , we only need to consider the behavior of p k {\displaystyle p_{k}} as n {\displaystyle n} increases. === Bounded groups of consecutive terms === When the sizes of all groups of consecutive terms are bounded by a constant, i.e., g k + 1 − g k ≤ L {\displaystyle g_{k+1}-g_{k}\leq L\,} , p {\displaystyle p} and its inverse are Agnew permutations (with K = ⌊ L 2 ⌋ {\textstyle K=\left\lfloor {\frac {L}{2}}\right\rfloor } ), i.e., arbitrary reorderings can be applied within the groups with the convergence type preserved. === Unbounded groups of consecutive terms === When the sizes of groups of consecutive terms grow without bounds, it is necessary to look at the behavior of p k {\displaystyle p_{k}} . Mirroring permutations and circular shift permutations, as well as their inverses, add at most 1 interval to the main interval [ 1 , g k ] {\displaystyle [1,\,g_{k}]} , hence p {\displaystyle p} and its inverse are Agnew permutations (with K = 2 {\displaystyle K=2} ), i.e., mirroring and circular shifting can be applied within the groups with the convergence type preserved. A block reordering permutation with B > 1 blocks and its inverse add at most ⌈ B 2 ⌉ {\textstyle \left\lceil {\frac {B}{2}}\right\rceil } intervals (when g k + 1 − g k {\textstyle g_{k+1}-g_{k}} is large) to the main interval [ 1 , g k ] {\displaystyle [1,\,g_{k}]} , hence p {\displaystyle p} and its inverse are Agnew permutations, i.e., block reordering can be applied within the groups with the convergence type preserved. == Notes == == References == |
Wikipedia:Agnès Beaudry#0 | Agnès France Marie Beaudry is a Canadian mathematician specializing in algebraic topology, including stable homotopy theory, chromatic homotopy theory, equivariant homotopy theory, and applications of these theories to condensed matter physics. She is an associate professor of mathematics at the University of Colorado Boulder. == Education and career == Beaudry majored in mathematics at McGill University in Canada, with a minor in philosophy. After graduating in 2008, she went to Northwestern University for doctoral study in mathematics. She completed her Ph.D. in 2013, with the dissertation The Duality Resolution Spectral Sequence for the Moore Spectrum at the Prime 2 supervised by Paul Goerss. After working at the University of Chicago from 2013 to 2016 as an L. E. Dickson Instructor in mathematics, she joined the Department of Mathematics at the University of Colorado as an assistant professor in 2016. She was promoted to associate professor in 2022. == Recognition == Beaudry was elected as a Fellow of the American Mathematical Society in the 2024 class of fellows. == References == == External links == Home page Agnès Beaudry publications indexed by Google Scholar |
Wikipedia:Agnès Sulem#0 | Agnès Sulem (born 1959) is a French applied mathematician whose research topics include stochastic control, jump diffusion, and mathematical finance. == Education == Sulem earned a Ph.D. in 1983 at Paris Dauphine University, with the dissertation Résolution explicite d'Inéquations Quasi-Variationnelles associées à des problèmes de gestion de stock supervised by Alain Bensoussan. == Career == She is a director of research at the French Institute for Research in Computer Science and Automation (INRIA) in Paris, where she heads the MATHRISK project on mathematical risk handling. She is currently a professor at the University of Luxembourg in the Mathematics department. She is a coauthor of the book Applied Stochastic Control of Jump Diffusions (with Bernt Øksendal, Springer, 2005; 2nd ed., 2007; 3rd ed., 2019). Sulem is also an associate editor at the Journal of Mathematical Analysis and Applications and at the SIAM Journal on Financial Mathematics. == References == == External links == Agnès Sulem publications indexed by INRIA |
Wikipedia:Agranovich–Dynin formula#0 | In mathematical analysis, the Agranovich–Dynin formula is a formula for the index of an elliptic system of differential operators, introduced by Agranovich and Dynin (1962). == References == Dynin, A. S.; Agranovich, M. S. (1962), "General boundary-value problems for elliptic systems in higher-dimensional regions", Doklady Akademii Nauk SSSR, 146: 511–514, ISSN 0002-3264, MR 0140820 |
Wikipedia:Ahlfors finiteness theorem#0 | In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by Lars Ahlfors (1964, 1965), apart from a gap that was filled by Greenberg (1967). The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed. == Bers area inequality == The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by Lipman Bers (1967a). It states that if Γ is a non-elementary finitely-generated Kleinian group with N generators and with region of discontinuity Ω, then Area(Ω/Γ) ≤ 4π(N − 1) with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for Fuchsian groups of the first kind (so in particular there can be at most two invariant components). == References == Ahlfors, Lars V. (1964), "Finitely generated Kleinian groups", American Journal of Mathematics, 86 (2): 413–429, doi:10.2307/2373173, ISSN 0002-9327, JSTOR 2373173, MR 0167618 Ahlfors, Lars (1965), "Correction to "Finitely generated Kleinian groups"", American Journal of Mathematics, 87 (3): 759, doi:10.2307/2373073, ISSN 0002-9327, JSTOR 2373073, MR 0180675 Bers, Lipman (1967a), "Inequalities for finitely generated Kleinian groups", Journal d'Analyse Mathématique, 18: 23–41, doi:10.1007/BF02798032, ISSN 0021-7670, MR 0229817 Bers, Lipman (1967b), "On Ahlfors' finiteness theorem", American Journal of Mathematics, 89 (4): 1078–1082, doi:10.2307/2373419, ISSN 0002-9327, JSTOR 2373419, MR 0222282 Greenberg, L. (1967), "On a theorem of Ahlfors and conjugate subgroups of Kleinian groups", American Journal of Mathematics, 89 (1): 56–68, doi:10.2307/2373096, ISSN 0002-9327, JSTOR 2373096, MR 0209471 |
Wikipedia:Ahlfors measure conjecture#0 | In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by Ahlfors (1966), who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. Canary (1993) proved the Ahlfors conjecture for topologically tame groups, by showing that a topologically tame Kleinian group is geometrically tame, so the Ahlfors conjecture follows from Marden's tameness conjecture that hyperbolic 3-manifolds with finitely generated fundamental groups are topologically tame (homeomorphic to the interior of compact 3-manifolds). This latter conjecture was proved, independently, by Agol (2004) and by Calegari & Gabai (2006). Canary (1993) also showed that in the case when the limit set is the whole sphere, the action of the Kleinian group on the limit set is ergodic. == References == Agol, Ian (2004), Tameness of hyperbolic 3-manifolds, arXiv:math/0405568, Bibcode:2004math......5568A Ahlfors, Lars V. (1966), "Fundamental polyhedrons and limit point sets of Kleinian groups", Proceedings of the National Academy of Sciences of the United States of America, 55 (2): 251–254, Bibcode:1966PNAS...55..251A, doi:10.1073/pnas.55.2.251, ISSN 0027-8424, JSTOR 57511, MR 0194970, PMC 224131, PMID 16591331 Calegari, Danny; Gabai, David (2006), "Shrinkwrapping and the taming of hyperbolic 3-manifolds", Journal of the American Mathematical Society, 19 (2): 385–446, arXiv:math/0407161, doi:10.1090/S0894-0347-05-00513-8, ISSN 0894-0347, MR 2188131, S2CID 1053364 Canary, Richard D. (1993), "Ends of hyperbolic 3-manifolds", Journal of the American Mathematical Society, 6 (1): 1–35, doi:10.2307/2152793, ISSN 0894-0347, JSTOR 2152793, MR 1166330 |
Wikipedia:Ahmed Chalabi#0 | Ahmed Abdel Hadi Chalabi (Arabic: أحمد عبد الهادي الجلبي; 30 October 1945 – 3 November 2015) was an Iraqi dissident politician, convicted fraudster and founder of the Iraqi National Congress (INC) who served as the President of the Governing Council of Iraq (37th Prime Minister of Iraq) and a Deputy Prime Minister of Iraq under Ibrahim al-Jaafari. He is believed to have been an Iranian agent and had ties with Iran. Chalabi was interim Minister of Oil in Iraq in April–May 2005 and December 2005 – January 2006 and Deputy Prime Minister from May 2005 to May 2006. Chalabi failed to win a seat in parliament in the December 2005 elections, and when the new Iraqi cabinet was announced in May 2006, he was not given a post. Once dubbed the "George Washington of Iraq" by American supporters, he was initially a CIA-backed operative, who later fell out of favor, with U.S. Special Forces raiding his private residence in Baghdad only one year after the invasion of Iraq. He later came under investigation by several U.S. government agencies after switching his allegiances to become an instrument of pro-Iranian influence in Iraqi politics. In the lead-up to the 2003 invasion of Iraq, the Iraqi National Congress (INC), with the assistance of lobbying powerhouse BKSH & Associates, provided a major portion of the information on which the Office of Special Plans based its condemnation of the Iraqi President Saddam Hussein, including reports of weapons of mass destruction and alleged ties to al-Qaeda. Most, if not all, of this information has turned out to be false and Chalabi has been called a fabricator. Along with this, Chalabi also subsequently boasted, in an interview with the British Sunday Telegraph, about the impact that their faulty intelligence had on American policy. These factors led to a falling-out between him and the U.S. government. Furthermore, Chalabi was found guilty of bank fraud in the Petra Bank scandal in Jordan and he was sentenced to 22 years imprisonment in absentia. In 2008, Office for Reconstruction and Humanitarian Assistance Director Jay Garner stated that he believed Chalabi was an Iranian agent. In January 2012, a French intelligence official stated that he believed Chalabi to be "acting on behalf of Iran". == Early life == Chalabi was the son of a prominent Shia family, one of the wealthy power elite of Baghdad. He was born in Kadhimiya in 1945. His family, who dated back 300 years to the Sultanate, ran Iraq's oldest commercial bank under the British-backed Kingdom of Iraq. His father, Abdel Hadi Chalabi, a wealthy grain merchant and member of the Iraqi parliament, became head of the senate when King Abdullah I of Jordan was assassinated. His family retired from public life to a farmhouse near Baghdad when the military seized power. Chalabi left Iraq with his family in 1958, following the 14 July Revolution, and spent most of his life in the United States and the United Kingdom. He was educated at Baghdad College and Seaford College in Sussex, England before leaving for America. === Western education === In exile, following the Ba'ath party takeover, his family acted as the Iraqi Shia clergy's bankers. In the mid-1960s, he studied with cryptographer Whitfield Diffie at the Massachusetts Institute of Technology from which he received a Bachelor of Science degree in mathematics. In 1969, he earned a PhD in mathematics from the University of Chicago under the direction of George Glauberman on the Theory of Knots. After which he took a position in the mathematics department at the American University of Beirut. He published three mathematics papers between 1973 and 1980, in the field of abstract algebra. In 1971, Chalabi married Leila Osseiran, daughter of Lebanese politician Adel Osseiran. They had four children, Tamara, Mariam, Hashem and Hadi. Whilst still at Beirut the civil war broke out in 1975, so he moved to Jordan and found work as an interpreter. == Business career == Chalabi was a bold and shrewd investor, amassing a fortune of $100 million . During his life he was accused of corruption many times. In 1977, he founded the Petra Bank in Jordan with Crown Prince Hassan, the King's brother. In May 1989, the Governor of the Central Bank of Jordan, Mohammed Said Nabulsi, issued a decree ordering all banks in the country to deposit 35% of their reserves with the Central Bank. Petra Bank was the only bank that was unable to meet this requirement. An investigation was launched which led to accusations of embezzlement and false accounting. The bank failed, causing a $350 million bail-out by the Central Bank, after which Chalabi fled the country. It was widely reported that Chalabi escaped Jordan locked in the trunk of a car owned by Prince Hassan of Jordan. Chalabi was convicted and sentenced in absentia for bank fraud by a Jordanian military tribunal to 22 years in prison. Chalabi maintained that his prosecution was a politically motivated effort to discredit him sponsored by Saddam Hussein. === Exile in the UK === Living abroad by 1992 in London, and unable to return home for fear of his life, he set up the Iraqi National Congress (INC) with an agenda of regime change for his homeland. The organization was open to all ethnic and religious groups of Iraq, including Arabs, Kurds, Turkmen, Sunni and Shia Muslims. Already a fluent English speaker, he turned his attention to Washington, D.C. In 1995, after preparation and lobbying, he persuaded President Bill Clinton to fund an expedition into northern Iraq to use subterfuge to start an insurgency. Chalabi was convinced that the Iraqi military would rise up to overthrow the dictator. The commanders to whom he had spoken, were the same who openly supported Saddam and crushed his opponents in the Kurdish and Shia revolts. The insurgency failed, lacking the promised ground troops, and 100 insurgents were killed by the military. The command structure of INC fell apart with factional infighting. Chalabi was banned from those frequent visits to CIA headquarters at Langley, Virginia. Nonetheless, Chalabi was doggedly determined: in 1998 Congress passed the Iraq Liberation Act passing into American law the objective of "regime change" in Iraq. It was reported by BBC News in May 2005 that the Jordanian government was considering whether to pardon Chalabi, in part to ease the relations between Jordan and the new Iraqi government of which Chalabi was a member. According to one report, Chalabi proposed a $32 million compensation fund for depositors affected by Petra Bank's failure. The website for Petra Bank contains a press release stating that Chalabi would refuse the pardon. Although Chalabi always maintained the case was a plot to frame him by Baghdad, the issue was revisited when the U.S. State Department raised questions about the accounting practices of the Iraqi National Congress (INC). According to The New York Times, "Chalabi insisted on a public apology, which the Jordanians refused to give." Conservative pundit Arnaud de Borchgrave stated that following his escape from Jordan as a wanted fugitive, Chalabi established the Iraqi National Congress by sending out "scores of all-expenses-paid invitations to Iraqi exiles to a conference in Vienna a month later. The conference created INC - and made Mr Chalabi its president". Around the same time, the Jordan's Central Bank Governor, Mohammed Said Nabulsi, had referred to Chalabi as "one of the most notorious crooks in the history of the Middle East". Chalabi headed the executive council of the INC, an umbrella Iraqi opposition group created in 1992 for the purpose of fomenting the overthrow of Saddam Hussein. The INC received major funding and assistance from the United States. Chalabi was involved in organizing a resistance movement among Kurds in northern Iraq in the early mid-1990s. When that effort was crushed and hundreds of his supporters were killed, Chalabi fled the country. Chalabi lobbied in Washington for the passage of the Iraq Liberation Act (passed October 1998). This earmarked US$97 million to support Iraqi opposition groups. Chalabi and the INC were widely reported to be active CIA assets in the pre-2003 period, with one report in the British Guardian newspaper estimating that CIA operatives had funnelled an estimated $100m to the INC, which had helped to finance the epic failure of the 1996 coup attempt by Kurdish forces. In 2001 it was revealed that INC was accused of false accounting and irregularities. During the period from March 2000 to September 2003, the U.S. State Department paid nearly $33 million to the INC, according to a General Accounting Office report released in 2004, some of which was used to purchase office artefacts. == Invasion of Iraq == Before the Iraq War (2003), Chalabi enjoyed close political and business relationships with some members of the U.S. government, including some prominent neoconservatives within the Pentagon. Chalabi was said to have had political contacts within the Project for the New American Century, most notably with Paul Wolfowitz, a student of nuclear strategist Albert Wohlstetter, and Richard Perle. He also enjoyed considerable support among politicians and political pundits in the United States, most notably Jim Hoagland of The Washington Post, who held him up as a notable force for democracy in Iraq. He was a special guest of First Lady Laura Bush at the 2004 State of the Union Address. Chalabi was flown back to Iraq after the invasion with a force of 700 US-trained militia and was, according to the Washington Times, being paid over $300,000 a month by the US government, with the direct support of Vice President Dick Cheney, Deputy Defence Secretary Paul Wolfowitz, Richard Perle and Douglas Feith. The CIA was largely skeptical of Chalabi and the INC, but information allegedly from his group (most famously from a defector codenamed "Curveball") made its way into intelligence dossiers used by President George W. Bush and British Prime Minister Tony Blair to justify an invasion of Iraq. "Curveball", Rafid Ahmed Alwan al-Janabi, fed officials hundreds of pages of bogus "firsthand" descriptions of mobile biological weapons factories on wheels and rails. Secretary of State Colin Powell later used this information in a U.N. presentation trying to garner support for the war, despite warnings from German intelligence that "Curveball" was fabricating claims. Since then, the CIA has admitted that the defector made up the story, and Powell said in 2011 the information should not have been used in his presentation. A later congressionally appointed investigation (Robb-Silberman) concluded that Curveball had no relation whatsoever to the INC, and that press reports linking Curveball to the INC were erroneous. The INC often worked with the media, most notably with Judith Miller, concerning her WMD stories for The New York Times starting on 26 February 1998. After the war, given the lack of discovery of WMDs, most of the WMD claims of the INC were shown to have been either misleading, exaggerated, or completely made up while INC information about the whereabouts of Saddam Hussein's loyalists and Chalabi's personal enemies were accurate. Another of Chalabi's advocates was American Enterprise Institute's Iraq specialist Danielle Pletka. Chalabi received advice on media and television presentation techniques from the Irish scriptwriter and commentator Eoghan Harris prior to the invasion of Iraq. During the initial ten days of the U.S. invasion of Iraq, American intelligence discovered signs that Ahmed Chalabi was communicating with the Iranians and sharing details about US troop movements with their government. While this act of betrayal angered the Americans, it was largely downplayed at the time. The U.S. aimed to prevent the conflict from spreading beyond Iraq's borders, so it was seen as beneficial that the Iranians were aware that U.S. troops had no intention of entering Iran. When questioned later about his dealings with the Iranian government during that period, Chalabi provided an ambiguous response: "I did not pass any information to Iran that compromised any national security information of the United States." In April as U.S. forces took control during the 2003 Invasion of Iraq, Chalabi entered with allied troops the southern town of Shatrah. 300 US-trained FIF (Freedom for Iraq Fighters) expected opposition, but none emerged. Thousands of Iraqis cheered the troops. Chalabi returned under their aegis and was given a position on the Iraq interim governing council by the Coalition Provisional Authority. He served as president of the council in September 2003. He denounced a plan to let the UN choose an interim government for Iraq. "We are grateful to President Bush for liberating Iraq, but it is time for the Iraqi people to run their affairs," he was quoted as saying in The New York Times. In August 2003, Chalabi was the only candidate whose unfavorable ratings exceeded his favorable ones with Iraqis in a State Department poll. In a survey of nearly 3,000 Iraqis in February 2004 (by Oxford Research International, sponsored by the BBC in the United Kingdom, ABC in the U.S., ARD of Germany, and the NHK in Japan), only 0.2 percent of respondents said he was the most trustworthy leader in Iraq (see survey link below, question #13). A secret document written in 2002 by the British Overseas and Defence Secretariat reportedly described Chalabi as "a convicted fraudster popular on Capitol Hill." In response to the WMD controversy, Chalabi told London's The Daily Telegraph in February 2004, We are heroes in error. As far as we're concerned, we've been entirely successful. That tyrant Saddam is gone and the Americans are in Baghdad. What was said before is not important. The Bush administration is looking for a scapegoat. == Falling out with the U.S., 2004–05 == As Chalabi's position of trust with the Pentagon crumbled, he found a new political position as a champion of Iraq's Shia (Chalabi himself was Shia). Beginning 25 January 2004, Chalabi and his close associates promoted the claim that leaders around the world were illegally profiting from the Oil for Food program. These charges were around the same time that UN envoy Lakhdar Brahimi indicated that Chalabi would likely not be welcome in a future Iraqi government. Up until this time, Chalabi had been mentioned formally several times in connection with possible future leadership positions. Chalabi contended that documents in his possession detailed the misconduct, but he did not provide any documents or other evidence. The U.S. sharply criticized Chalabi's Oil for Food investigation as undermining the credibility of its own. Additionally, Chalabi and other members of the INC were investigated for fraud involving the exchange of Iraqi currency, grand theft of both national and private assets, and many other criminal charges in Iraq. On 19 May 2004 the U.S. government discontinued their regular payments to Chalabi for information he provided. Iraqi police, supported by U.S. soldiers, raided his offices and residence on 20 May, taking documents and computers, presumably to be used as evidence. A major target of the raid was Aras Habib, Chalabi's long-term director of intelligence, who controlled the vast network of agents bankrolled by U.S. funding. The U.S. announced that they had stopped funding the INC, having previously paid the organization $330,000 per month. In June 2004, it was reported that Chalabi gave U.S. state secrets to Iran in April, including the fact that one of the United States' most valuable sources of Iranian intelligence was a broken Iranian code used by their spy services. Chalabi allegedly learned of the code through a drunk American involved in the code-breaking operation. Chalabi denied all of the charges, which nothing ever came of. An arrest warrant for alleged counterfeiting was issued for Chalabi on 8 August 2004, while at the same time a warrant was issued on murder charges against his nephew Salem Chalabi (at the time, head of the Iraqi Special Tribunal), while they both were out of the country. Chalabi returned to Iraq on 10 August planning to make himself available to Iraqi government officials, but he was never arrested. Charges were later dropped against Chalabi, with Judge Zuhair al-Maliki citing lack of evidence. On 1 September 2004, Chalabi told reporters of an assassination attempt made on him near Latifiya, a town south of Baghdad. Chalabi reported he was returning from a meeting with Ayatollah Ali al-Sistani (whose trust Chalabi enjoyed) in Najaf, where a few days earlier a cease-fire had taken effect, ending three weeks of confrontations between followers of Muqtada al-Sadr and the U.S. military, at the time. He regained enough credibility to be made deputy prime minister on 28 April 2005. At the same time he was made acting oil minister, before the appointment of Ibrahim Bahr al-Uloum in May 2005. On protesting IMF austerity measures, Al-Uloum was instructed to extend his vacation by a month in December 2005 by Prime Minister Ibrahim al-Jaafari, and Chalabi was reappointed as acting oil minister. Al-Uloum returned to the post in January 2006. In November 2005, Chalabi traveled to the U.S. and met with top U.S. government officials, including Vice President Dick Cheney, Secretary of State Condoleezza Rice, and National Security Advisor Stephen Hadley. At this time Chalabi also traveled to Iran to meet with Iranian president Mahmoud Ahmadinejad. == Political activity in Iraq, 2005–15 == The Iraqi National Congress, headed by Chalabi, was a part of the United Iraqi Alliance in the 2005 legislative election. After the election, Chalabi claimed that he had the support of the majority of elected members of United Iraqi Alliance and staked claim to be the first democratically elected Prime Minister of Iraq; however, Ibrahim al-Jaafari later emerged as the consensus candidate for prime minister. Prior to the December 2005 elections, the Iraqi National Congress had left the United Iraqi Alliance and formed the National Congress Coalition, which ran in the elections but failed to win a single seat in Parliament, gaining less than 0.5% of the vote. Other groups joining the INC in this list included: Democratic Iraqi Grouping, Democratic Joint Action Front, First Democratic National Party, Independent List, Iraqi Constitutional Movement, Iraqi Constitutional Party, Tariq Abd al-Karim Al Shahd al-Budairi, and the Turkoman Decision Party. He was refused a seat in the cabinet. Dogged by allegations, still unproven, of corruption he retorted that he had never "participated in any scheme of intelligence against the United States." Chalabi attended the 2006 Bilderberg Conference meeting outside of Ottawa, Ontario, Canada. In October 2007, Chalabi was appointed by Prime Minister Nouri al Maliki to head the Iraqi services committee, a consortium of eight service ministries and two Baghdad municipal posts tasked with the "surge" plan's next phase, restoring electricity, health, education and local security services to Baghdad neighborhoods. "The key is going to be getting the concerned local citizens—and all the citizens—feeling that this government is reconnected with them.... [Chalabi] agrees with that," said Gen. David Petraeus. Chalabi "is an important part of the process," said Col. Steven Boylan, Petraeus' spokesman. "He has a lot of energy." In April 2008, journalist Melik Kaylan wrote about Chalabi, "Arguably, he has, more than anyone in the country, evolved a detailed sense of what ails Baghdadis and how to fix things." After the invasion Chalabi was placed in charge of "de-Ba'athification"—the removal of senior office holders judged to have been close supporters of the deposed Saddam Hussein. The role fell into disuse, but in early 2010 Chalabi was accused of reviving this dormant post to eliminate his political enemies, especially Sunnis. The banning of some 500 candidates prior to the general election of 7 March 2010 at the initiative of Chalabi and his Iraqi National Congress was reported to have badly damaged previously improving relations between the Shia and Sunni. On 26 January 2012, The New York Times reported Western intelligence officials expressing concern that Chalabi was working with the leading opposition group in Bahrain, Al Wefaq National Islamic Society. A French intelligence official said, "When we hear that some members of the opposition are in touch with Hezbollah or with shady figures like the Iraqi Ahmed Chalabi, of whom we think he is acting on behalf of Iran, then this worries us". The connection between Chalabi and Al Wefaq was acknowledged by Jawad Fairooz, secretary general of Wefaq and a former member of Parliament in Bahrain. Fairooz said, "Mr Chalabi has helped us with contacts in Washington like other people have done and we thank them." During an interview in 2014, he was shown to be frail and depressed about his country's future: Iraq is a mess. Daesh is organised, with one command, united and well run, and we are so fragmented. We have no discipline, no command structure, no effective plans. == Death == The period leading up to Chalabi's death was marked by regular pronouncements released by Chalabi in various formats in which he would expose alleged corruption of highest officials in the Paul Bremer-led Provisional Coalition Authority and in the Iraqi Government led by Nouri Al Maliki. Chalabi died on 3 November 2015, four days after his 70th birthday, having apparently suffered a heart attack at his home in Kadhimiya, Baghdad. Iraqi Press speculated at the time of his death that it came about as a result of being poisoned due to his ongoing efforts to expose regime corruption. He died while serving as member of the Iraqi Parliament and chaired its Finance Committee. Ahmed Chalabi was laid to rest at the Kadhimiya Holy Shrine, a high honour bestowed by Iraq's influential Shia theocratic establishment. == See also == Curveball (informant) == References == == External links == Appearances on C-SPAN |
Wikipedia:Ahto Buldas#0 | Ahto Buldas (born 17 January 1967) is an Estonian computer scientist. He is the inventor of Keyless Signature Infrastructure, Co-Founder and Chief Scientist at Guardtime and Chair of the OpenKSI foundation. == Life and education == Buldas was born in Tallinn. After graduating from high school, he was conscripted in to the Soviet Army where he spent 2 years as an artillery officer in Siberia. After being discharged, he started studies in Tallinn University of Technology, where he defended his MSc degree in 1993 and his PhD in 1999. He currently lives in Tallinn with his wife and four children. == Career == Buldas was a leading contributor to the Estonian Digital Signature Act and ID-card from 1996 to 2002, currently the only national-level public-key infrastructure (PKI) which has achieved widespread adoption by a country's population for legally binding digital signatures. He published his first timestamping related research in 1998 and has published over 30 academic papers on the subject. His experience of implementing a national level PKI led him to invent Keyless Signature Infrastructure, a digital signature/timestamping system for electronic data that uses only hash-function based cryptography. By using hash-functions as the only cryptographic primitive the complexities of key management are eliminated and the system remains secure from quantum cryptographic attacks. His invention led to the founding of keyless signature technology company Guardtime in 2006. He is the Chair of Information Security at Tallinn University of Technology. Buldas has been a supervisor for 15 MSc dissertations and 4 PhD theses. == Awards == 2002: Young Scientist Award by the Cultural Foundation of the President of Estonia. 2015: Order of the White Star, IV class. == References == == Academic work == doi:10.1007/978-3-642-34266-0_6 doi:10.1007/978-3-642-14081-5_20 doi:10.1007/978-3-642-04642-1_19 doi:10.1007/978-3-642-02620-1_19 doi:10.1007/978-3-540-88733-1_18 doi:10.1007/978-3-540-75670-5_9 doi:10.1007/978-3-540-71677-8_11 == External links == Ahto Buldas' personal website Ahto Buldas' series of mini-lectures about cryptographic hash functions Archived 2012-12-06 at archive.today Ahto Buldas' TTU lecture on keyless signatures |
Wikipedia:Ailana Fraser#0 | Ailana Margaret Fraser is a Canadian mathematician and professor of mathematics at the University of British Columbia. She is known for her work in geometric analysis and the theory of minimal surfaces. Her research is particularly focused on extremal eigenvalue problems and sharp eigenvalue estimates for surfaces, min-max minimal surface theory, free boundary minimal surfaces, and positive isotropic curvature. == Early life and education == Fraser was born in Toronto, Ontario. She received her Ph.D. from Stanford University in 1998 under the supervision of Richard Schoen. After postdoctoral studies at the Courant Institute of New York University, she taught at Brown University before moving to UBC. == Major work == Fraser is well-known for her 2011 work with Schoen on the first "Steklov eigenvalue" of a compact Riemannian manifold-with-boundary. This is defined as the minimal nonzero eigenvalue of the "Dirichlet to Neumann" operator which sends a function on the boundary to the normal derivative of its harmonic extension into the interior. In the two-dimensional case, Fraser and Schoen were able to adapt Paul Yang and Shing-Tung Yau's use of the Hersch trick in order to approximate the product of the first Steklov eigenvalue with the length of the boundary from above, by topological data. Under an ansatz of rotational symmetry, Fraser and Schoen carefully analyzed the case of an annulus, showing that the metric optimizing the above eigenvalue-length product is obtained as the intrinsic geometry of a geometrically meaningful part of the catenoid. By use of the uniformization theorem for surfaces with boundary, they were able to remove the condition of rotational symmetry, replacing it by certain weaker conditions; however, they conjectured that their result should be unconditional. In general dimensions, Fraser and Schoen developed a "boundary" version of Peter Li and Yau's "conformal volume." By building upon some of Li and Yau's arguments, they gave lower bounds for the first Steklov eigenvalue in terms of conformal volumes, in addition to isoperimetric inequalities for certain minimal surfaces of the unit ball. == Awards and honors == Fraser won the Krieger–Nelson Prize of the Canadian Mathematical Society in 2012 and became a fellow of the American Mathematical Society in 2013. In 2018 the Canadian Mathematical Society listed her in their inaugural class of fellows and in 2021 awarded her, along with Marco Gualtieri, the Cathleen Synge Morawetz Prize. In 2022 she was awarded a Simons Fellowship. == Major publications == Fraser, Ailana; Schoen, Richard (2011). "The first Steklov eigenvalue, conformal geometry, and minimal surfaces". Advances in Mathematics. 226 (5): 4011–4030. arXiv:0912.5392. doi:10.1016/j.aim.2010.11.007. MR 2770439. Zbl 1215.53052. Fraser, Ailana; Schoen, Richard (2016). "Sharp eigenvalue bounds and minimal surfaces in the ball". Inventiones Mathematicae. 203 (3): 823–890. arXiv:1209.3789. Bibcode:2016InMat.203..823F. doi:10.1007/s00222-015-0604-x. MR 3461367. S2CID 119615775. Zbl 1337.35099. == References == |
Wikipedia:Ailsa Land#0 | Ailsa Horton Land (née Dicken; 14 June 1927 – 16 May 2021) was a professor of Operational Research in the Department of Management at the London School of Economics and was the first woman professor of Operational Research in Britain. She is most well known for co-defining the branch and bound algorithm along with Alison Doig whilst carrying out research at the London School of Economics in 1960. She was married to Frank Land, who is an emeritus Professor at the LSE. == Early life == Ailsa Horton Dicken was born on 14 June 1927 in West Bromwich, Staffordshire, the only daughter of Elizabeth (née Greig) and Harold Dicken. Her father worked in his family sports retail business and later became a salesman for Dunlop. Ailsa was keen on science in school, but didn't thrive in her local grammar school in Lichfield, disliking the discipline, so her parents sent her to Rocklands, a small, mixed boarding school in Hastings in East Sussex for a year. This school had only around 50 students, and students were encouraged to work at their own pace with a particular focus on mathematics. Students were also taken to institutions around Hastings including a gasworks where they were shown how coking coal was converted into gas to be used in homes. When World War Two broke out her mother moved them to Canada, hoping to spend the war with relatives there. The pair departed in April and by September 3, Britain and Canada were at war with Germany, leaving Ailsa and her mother trapped in Canada. Ailsa's father remained in England and served as a Catering Officer in RAF Bomber Command stations until the end of the War in 1945. Ailsa and her mother eventually settled in Toronto, where Ailsa attended the Malvern Collegiate Institute for three years. In 1943, Ailsa and her mother Elizabeth decided to join the Canadian Women's Army Corps (aged 16, Ailsa had to claim to be 18 to qualify to join up). By 1944 they were both working in clerical jobs in the National Defence Headquarters in Ottawa which was run entirely by female staff to replace male soldiers that were dispatched to England to prepare for the invasion of Nazi-occupied France. Ailsa and her mother ultimately obtained compassionate discharges to return to the UK as Harold Dicken, (serving as a catering officer in the RAF), was undergoing a dangerous operation (which he survived). == Education == Ailsa was able to enter the LSE to study for a degree in economics in 1946, her position as a demobilised servicewoman helping her gain access and a grant. She won the Bowley Prize for a first-year Economics paper. Graduating in 1950, she spent the rest of her career at the institution. Land obtained her PhD from the London School of Economics in 1956, her dissertation was entitled An Application of the Techniques of Linear Programming to the Transportation of Coal, supervised by George Morton. Her PhD work focused on solving a large transport problem without a computer in which the origin to destination costs are unknown and only the rail network distances between junctions are known. == Research == After securing a position as Research Assistantship in the Economics Research Division at LSE in 1950, Land progressed through the ranks of research assistant, lecturer, senior lecturer, reader, and then chaired professor. Her economics background informed her subsequent contributions to OR, beginning with her 1956 dissertation on the application of OR techniques to the transportation of coking coal. Ailsa is most known for her development, along with Alison Doig, of what later came to be called the branch-and-bound method for optimization problems with integer variables. Their work was published in Econometrica in 1960. This work was initially carried out at the London School of Economics under the sponsorship of British Petroleum, with the aim of enhancing existing linear programming models for refinery operations. Ailsa and Alison did not have access to a computer at the time, but they developed an algorithm that could be converted to Fortran by British Petroleum Staff. The method is now the most prevalent solution method for NP-hard optimization problems. Land also worked with Helen Makower and George Morton in the late 1950s on a number of integer programming problems. This included her early investigations of the travelling salesman problem, beginning with a 1955 paper with Morton, and continuing with a 1979 research report on 100 city travelling salesman problems. In addition, Land advanced OR methodology through the publication of notable work on shortest path algorithms, quadratic programming, bicriteria decision problems, and statistical data fitting. Following her retirement from the LSE in 1987, she continued several research projects, resulting in contributions to data envelopment analysis, the quadratic assignment problem, and combinatorial auctions. In addition to her methodological work, Ailsa worked on the development of computational tools. In 1973, Ailsa published her book Fortran Codes for Mathematical Programming: Linear, Quadratic and Discrete, written jointly with Susan Powell. This provided detailed documentation for computer implementations of optimization techniques as well as the underlying mathematical background and a suite of test problems. A subsequent 1979 publication, also with Susan Powell, offered guidance to consumers of mixed-integer programming and combinatorial programming. Her computer codes for data envelopment analysis and for the travelling salesman problem were all made freely available to the optimization community. == Teaching == During Land's teaching career at the LSE, she helped to establish a two-year diploma in OR at the LSE for students from the British Iron and Steel Association. Later she instituted a mathematical programming course at the undergraduate level as well as an advanced graduate course for the MSc program. Land mentored both master's level and PhD students, several of whom have achieved international distinction. == Awards and honours == Land was awarded the Harold Larnder prize by the Canadian Operational Research Society in 1994 for achieving international distinction in operational research. A student award at the London School of Economics, the Ailsa Land Prize, is given annually in her honour. Land was posthumously awarded the EURO Gold Medal, the highest distinction within OR in Europe, at the EURO Conference in 2021. She was inducted into the Hall of Fame of the International Federation of Operational Research Societies in 2023. == Personal life == She met her future husband Frank Land, in her graduating class. He had come to Britain with his parents and twin brother in 1939 as refugees from Nazi Germany and was one of the computing pioneers who developed the Leo computer for J Lyons & Co, and later is a professor at LSE. They married in 1953 and had three children, Frances, Richard and Margi during her PhD studies. Following her retirement from teaching and administration in 1987, Land continued to work on research projects, stating ‘Now I'm retired I can do some research!”, until she and her husband moved to Devon in 2000, where she became a clerk to the parish meeting in Harford, near Ivybridge, between trips abroad, moving to Totnes in 2015. Ailsa Land died on 16 May 2021 at the age of 93. == References == == External links == Media related to Ailsa Land at Wikimedia Commons |
Wikipedia:Aissa Wade#0 | Aissa Wade is a Professor of Mathematics at the Pennsylvania State University. She was the President of the African Institute for Mathematical Sciences centre in Senegal (from 2016 to 2018). == Early life and education == Wade was born in Dakar, Senegal. She studied mathematics at Cheikh Anta Diop University and graduated in 1993. She had to leave Senegal to earn a Ph.D. as there were no opportunities in Africa. Wade earned her Ph.D. at the University of Montpellier in 1996. Her thesis, "Normalisation formelle de structures de Poisson", considered symplectic geometry. Her doctoral advisor was Jean Paul Dufour. == Career == Wade became a postdoctoral researcher at the Abdus Salam International Centre for Theoretical Physics, where she worked on conformal Dirac structures. She held visiting faculty positions at University of North Carolina at Chapel Hill, African University of Science and Technology and Paul Sabatier University. Wade joined Pennsylvania State University and was appointed full professor in 2016. She served as a managing editor of The African Diaspora Journal of Mathematics. She is editor of Afrika Mathematika. She is on the scientific committee of the NextEinstein forum, an initiative to connect science, society and policy in Africa. As the President of the African Institute for Mathematical Sciences, Wade was the first woman to hold this role. She has been awarded funding from the National Science Foundation to support the Senegal Workshop on Geometric Structures. She has been involved with American Association for the Advancement of Science activities to enhance African STEM research, including the provision of evidence-based metrics, case studies and policy recommendations. In 2017 Wade was named a fellow of the African Academy of Sciences. Wade's accomplishments earned her recognition by Mathematically Gifted & Black, where she was featured as a Black History Month 2020 Honoree. == References == |
Wikipedia:Aiyub Khan#0 | Aiyub Khan ( born on 1 July 1967) is a professor of mathematics in Jai Narain Vyas University, and politician in Rajasthan from Indian national congress of Soorsagar constituency Jodhpur candidate in 2018. Khan is member of Rajasthan Public Service Commission, == Education == Khan did his education in Jai Narain Vyas University, he did PhD under supervision of Prem Kumar Bhatia, in 1993. Khan is Fellow of Royal Astronomical Society == Career == Khan worked in SD PG college, Sriganganagar before joining Jai Narain Vyas University . Presently he is active in politics and also member of Rajasthan Public Service Commission, under his supervision 4 students have done PhD. Khan is member of nine professional academies . Khan is member of Academic council of Jai Narain Vyas University == Family == Khan lives with his wife and two sons (Shahbaz Aiyub Khan and Shahjad Aiyub Khan). Shahbaz Aiyub Khan in 2023 assembly elections was candidate for Soorsagar constituency Jodhpur Rajasthan from Indian national congress. == References == == External links == Profile at JNVU |
Wikipedia:Aizik Volpert#0 | Aizik Isaakovich Vol'pert (Russian: Айзик Исаакович Вольперт; 5 June 1923 – January 2006) (the family name is also transliterated as Volpert or Wolpert) was a Soviet and Israeli mathematician and chemical engineer working in partial differential equations, functions of bounded variation and chemical kinetics. == Life and academic career == Vol'pert graduated from Lviv University in 1951, earning the candidate of science degree and the docent title respectively in 1954 and 1956 from the same university: from 1951 on he worked at the Lviv Industrial Forestry Institute. In 1961 he became senior research fellow while 1962 he earned the "doktor nauk" degree from Moscow State University. In the 1970s–1980s A. I. Volpert became one of the leaders of the Russian Mathematical Chemistry scientific community. He finally joined Technion’s Faculty of Mathematics in 1993, doing his Aliyah in 1994. == Work == === Index theory and elliptic boundary problems === Vol'pert developed an effective algorithm for calculating the index of an elliptic problem before the Atiyah-Singer index theorem appeared: He was also the first to show that the index of a singular matrix operator can be different from zero. === Functions of bounded variation === He was one of the leading contributors to the theory of BV-functions: he introduced the concept of functional superposition, which enabled him to construct a calculus for such functions and applying it in the theory of partial differential equations. Precisely, given a continuously differentiable function f : ℝp → ℝ and a function of bounded variation u(x) = (u1(x),...,up(x)) with x ∈ ℝn and n ≥ 1, he proves that f∘u(x) = f(u(x)) is again a function of bounded variation and the following chain rule formula holds: ∂ f ( u ( x ) ) ∂ x i = ∑ k = 1 p ∂ f ¯ ( u ( x ) ) ∂ u k ∂ u k ( x ) ∂ x i ∀ i = 1 , … , n {\displaystyle {\frac {\partial f({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial x_{i}}}=\sum _{k=1}^{p}{\frac {\partial {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial u_{k}}}{\frac {\partial {u_{k}({\boldsymbol {x}})}}{\partial x_{i}}}\qquad \forall i=1,\ldots ,n} where –f(u(x)) is the already cited functional superposition of f and u. By using his results, it is easy to prove that functions of bounded variation form an algebra of discontinuous functions: in particular, using his calculus for n = 1, it is possible to define the product H ⋅ δ of the Heaviside step function H(x) and the Dirac distribution δ(x) in one variable. === Chemical kinetics === His work on chemical kinetics and chemical engineering led him to define and study differential equations on graphs. == Selected publications == Hudjaev, Sergei Ivanovich; Vol'pert, Aizik Isaakovich (1985), Analysis in classes of discontinuous functions and equations of mathematical physics, Mechanics: analysis, vol. 8, Dordrecht–Boston–Lancaster: Martinus Nijhoff Publishers, pp. xviii+678, ISBN 90-247-3109-7, MR 0785938, Zbl 0564.46025. One of the best books about BV-functions and their application to problems of mathematical physics, particularly chemical kinetics. Vol'pert, Aizik Isaakovich (1967), Пространства BV и квазилинейные уравнени, Matematicheskii Sbornik, (N.S.) (in Russian), 73(115) (2): 255–302, MR 0216338, Zbl 0168.07402. A seminal paper where Caccioppoli sets and BV functions are thoroughly studied and the concept of functional superposition is introduced and applied to the theory of partial differential equations: it was also translated as Vol'Pert, A. I. (1967), "Spaces BV and quasi-linear equations", Mathematics of the USSR-Sbornik, 2 (2): 225–267, Bibcode:1967SbMat...2..225V, doi:10.1070/SM1967v002n02ABEH002340, hdl:10338.dmlcz/102500, MR 0216338, Zbl 0168.07402. Vol'pert, Aizik Isaakovich (1972), Дифференциальные уравнения на графах, Matematicheskii Sbornik, (N.S.) (in Russian), 88(130) (4(8)): 578–588, MR 0316796, Zbl 0242.35015, translated in English as Vol'Pert, A. I. (1972), "Differential equations on graphs", Mathematics of the USSR-Sbornik, 17 (4): 571–582, Bibcode:1972SbMat..17..571V, doi:10.1070/SM1972v017n04ABEH001603, Zbl 0255.35013. Vasiliev, V. M.; Volpert, A. I.; Hudiaev, S. I. (1973), "On the method of quasi-stationary concentrations for chemical kinetics equations", Журнал вычислительной математики и математической физики (in Russian), 13 (3): 683–697. Vol'pert, A. I. (1976), "Qualitative methods of investigation of equations of chemical kinetics", Preprint (in Russian), Institute of Chemical Physics, Chernogolovka. Vol'pert, V. A.; Vol'pert, A. I.; Merzhanov, A. G. (1982), "Application of the theory of bifurcations in study of the spinning combustion waves", Doklady Akademii Nauk SSSR (in Russian), 262 (3): 642–645. Vol'pert, V. A.; Vol'pert, A. I.; Merzhanov, A. G. (1982b), "Analysis of nonunidimensional combustion modes by bifurcation theory methods", Doklady Akademii Nauk SSSR (in Russian), 263 (4): 918–921. Vol'pert, V. A.; Vol'pert, A. I.; Merzhanov, A. G. (1983), "Application of the theory of bifurcations to the study of unsteady regimes of combustion", Fizika Goreniya i Vzryva (in Russian), 19: 69–72, translated in English as Vol'Pert, V. A.; Vol'Pert, A. I.; Merzhanov, A. G. (1983), "Application of the theory of bifurcations to the investigation of nonstationary combustion regimes", Combustion, Explosion, and Shock Waves, 19 (4): 435–438, Bibcode:1983CESW...19..435V, doi:10.1007/BF00783642, S2CID 97950149. Vol'pert, V. A.; Vol'pert, A. I. (1989), "Existence and stability of traveling waves in chemical kinetics", Dynamics of Chemical and Biological Systems (in Russian), Novosibirsk: Nauka, pp. 56–131. Vol'pert, Aizik I.; Vol'pert, Vitaly A.; Vol'pert, Vladimir A. (1994), Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140, Providence, R.I.: American Mathematical Society, pp. xii+448, ISBN 0-8218-3393-6, MR 1297766, Zbl 1001.35060. Vol'pert, A. I. (1996), "Propagation of Waves Described by Nonlinear Parabolic Equations (a commentary on article 6)", in Oleinik, O. A. (ed.), I. G. Petrovsky Selected works. Part II: Differential equations and probability theory, Classics of Soviet Mathematics, vol. 5 (part 2), Amsterdam: Gordon and Breach Publishers, pp. 364–399, ISBN 2-88124-979-5, MR 1677648, Zbl 0948.01043. Vol'pert, V. A.; Vol'pert, A. I. (1998), "Convective instability of reaction fronts: linear stability analysis", European Journal of Applied Mathematics, 9 (5): 507–525, doi:10.1017/S095679259800357X, MR 1662311, S2CID 121533943, Zbl 0918.76027. == See also == Atiyah-Singer index theorem Bounded variation Caccioppoli set Differential equation on a graph == Notes == == References == |
Wikipedia:Akhmim wooden tablets#0 | The Akhmim wooden tablets, also known as the Cairo wooden tablets are two wooden writing tablets from ancient Egypt, solving arithmetical problems. They each measure around 18 by 10 inches (460 mm × 250 mm) and are covered with plaster. The tablets are inscribed on both sides. The hieroglyphic inscriptions on the first tablet include a list of servants, which is followed by a mathematical text. The text is dated to year 38 (it was at first thought to be from year 28) of an otherwise unnamed king's reign. The general dating to the early Egyptian Middle Kingdom combined with the high regnal year suggests that the tablets may date to the reign of the 12th Dynasty pharaoh Senusret I, c. 1950 BC. The second tablet also lists several servants and contains further mathematical texts. The tablets are currently housed at the Museum of Egyptian Antiquities in Cairo. The text was reported by Daressy in 1901 and later analyzed and published in 1906. The first half of the tablet details five multiplications of a hekat, a unit of volume made up of 64 dja, by 1/3, 1/7, 1/10, 1/11 and 1/13. The answers were written in binary Eye of Horus quotients and exact Egyptian fraction remainders, scaled to a 1/320 factor named ro. The second half of the document proved the correctness of the five division answers by multiplying the two-part quotient and remainder answer by its respective (3, 7, 10, 11 and 13) dividend that returned the ab initio hekat unity, 64/64. In 2002, Hana Vymazalová obtained a fresh copy of the text from the Cairo Museum, and confirmed that all five two-part answers were correctly checked for accuracy by the scribe that returned a 64/64 hekat unity. Minor typographical errors in Daressy's copy of two problems, the division by 11 and 13 data, were corrected at this time. That all five divisions had been exact was suspected by Daressy but was not proven until 1906. == Mathematical content == === 1/3 case === The first problem divides 1 hekat by writing it as 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + 1 / 32 + 1 / 64 {\displaystyle 1/2+1/4+1/8+1/16+1/32+1/64} + (5 ro) (which equals 1) and dividing that expression by 3. The scribe first divides the remainder of 5 ro by 3, and determines that it is equal to (1 + 2/3) ro. Next, the scribe finds 1/3 of the rest of the equation and determines it is equal to 1 / 4 + 1 / 16 + 1 / 64 {\displaystyle 1/4+1/16+1/64} . The final step in the problem consists of checking that the answer is correct. The scribe multiplies 1 / 4 + 1 / 16 + 1 / 64 + ( 1 + 2 / 3 ) r o {\displaystyle 1/4+1/16+1/64+(1+2/3)ro} by 3 and shows that the answer is (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64) + (5 ro), which he knows is equal to 1. In modern mathematical notation, one might say that the scribe showed that 3 times the hekat fraction (1/4 + 1/16 + 1/64) is equal to 63/64, and that 3 times the remainder part, (1 + 2/3) ro, is equal to 5 ro, which is equal to 1/64 of a hekat, which sums to the initial hekat unity (64/64). === Other fractions === The other problems on the tablets were computed by the same technique. The scribe used the identity 1 hekat = 320 ro and divided 64 by 7, 10, 11 and 13. For instance, in the 1/11 computation, the division of 64 by 11 gave 5 with a remainder 45/11 ro. This was equivalent to (1/16 + 1/64) hekat + (4 + 1/11) ro. Checking the work required the scribe to multiply the two-part number by 11 and showed the result 63/64 + 1/64 = 64/64, as all five proofs reported. === Accuracy === The computations show several minor mistakes. For instance, in the 1/7 computations, 2 × 7 {\displaystyle 2\times 7} was said to be 12 and the double of that 24 in all of the copies of the problem. The mistake takes place in exactly the same place in each of the versions of this problem, but the scribe manages to find the correct answer in spite of this error since the 64/64 hekat unity guided his thinking. The fourth copy of the 1/7 division contains an extra minor error in one of the lines. The 1/11 computation occurs four times and the problems appear right next to one another, leaving the impression that the scribe was practicing the computation procedure. The 1/13 computation appears once in its complete form and twice more with only partial computations. There are errors in the computations, but the scribe does find the correct answer. 1/10 is the only fraction computed only once. There are no mistakes in the computations for this problem. == Hekat problems in other texts == The Rhind Mathematical Papyrus (RMP) contained over 60 examples of hekat multiplication and division in RMP 35, 36, 37, 38, 47, 80, 81, 82, 83 and 84. The problems were different since the hekat unity was changed from the 64/64 binary hekat and ro remainder standard as needed to a second 320/320 standard recorded in 320 ro statements. Some examples include: Problems 35–38 find fractions of the hekat. Problem 38 scaled one hekat to 320 ro and multiplied by 7/22. The answer 101 9/11 ro was proven by multiplying by 22/7, facts not mentioned by Claggett and scholars prior to Vymazalova. Problem 47 scaled 100 hekat to (6400/64) and multiplied (6400/64) by 1/10, 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90 and 1/100 fractions to binary quotient and 1/1320 (ro) remainder unit fraction series. Problem 80 gave 5 Horus eye fractions of the hekat and equivalent fractions as expressions of another unit called the hinu. These were left unclear prior to Vymazalova. Problem 81 generally converted hekat unity binary quotient and ro remainder statements to equivalent 1/10 hinu units making it clear the meaning of the RMP 80 data. The Ebers Papyrus is a famous late Middle Kingdom medical text. Its raw data were written in hekat one-parts suggested by the Akhim wooden tablets, handling divisors greater than 64. == References == == External links == Gardener, Milo, "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, Maryland Publications, New Delhi, pp 157–173. The Arithmetic used to Solve an Ancient Horus-Eye Problem Milo Gardner accessed 22 September 2024 Gillings, R. Mathematics in the Time of the Pharaohs. Boston, Massachusetts: MIT Press, pp. 202–205, 1972. ISBN 0-262-07045-6. (Out of print) Weisstein, Eric W. "Akhmim Wooden Tablet". MathWorld. Scaled AWT Remainders |
Wikipedia:Akivis algebra#0 | In mathematics, and in particular the study of algebra, an Akivis algebra is a nonassociative algebra equipped with a binary operator, the commutator [ x , y ] {\displaystyle [x,y]} and a ternary operator, the associator [ x , y , z ] {\displaystyle [x,y,z]} that satisfy a particular relationship known as the Akivis identity. They are named in honour of Russian mathematician Maks A. Akivis. Formally, if A {\displaystyle A} is a vector space over a field F {\displaystyle \mathbb {F} } of characteristic zero, we say A {\displaystyle A} is an akivis algebra if the operation ( x , y ) ↦ [ x , y ] {\displaystyle \left(x,y\right)\mapsto \left[x,y\right]} is bilinear and anticommutative; and the trilinear operator ( x , y , z ) ↦ [ x , y , z ] {\displaystyle \left(x,y,z\right)\mapsto \left[x,y,z\right]} satisfies the Akivis identity: [ [ x , y ] , z ] + [ [ y , z ] , x ] + [ [ z , x ] , y ] = [ x , y , z ] + [ y , z , x ] + [ z , x , y ] − [ x , z , y ] − [ y , x , z ] − [ z , y , x ] . {\displaystyle \left[\left[x,y\right],z\right]+\left[\left[y,z\right],x\right]+\left[\left[z,x\right],y\right]=\left[x,y,z\right]+\left[y,z,x\right]+\left[z,x,y\right]-\left[x,z,y\right]-\left[y,x,z\right]-\left[z,y,x\right].} An Akivis algebra with [ x , y , z ] = 0 {\displaystyle \left[x,y,z\right]=0} is a Lie algebra, for the Akivis identity reduces to the Jacobi identity. Note that the terms on the right hand side have positive sign for even permutations and negative sign for odd permutations of x , y , z {\displaystyle x,y,z} . Any algebra (even if nonassociative) is an Akivis algebra if we define [ x , y ] = x y − y x {\displaystyle \left[x,y\right]=xy-yx} and [ x , y , z ] = ( x y ) z − x ( y z ) {\displaystyle \left[x,y,z\right]=(xy)z-x(yz)} . It is known that all Akivis algebras may be represented as a subalgebra of a (possibly nonassociative) algebra in this way (for associative algebras, the associator is identically zero, and the Akivis identity reduces to the Jacobi identity). == References == M. R. Bremner, I. R. Hentzel, and L. A. Peresi 2005. "Dimension formulas for the free nonassociative algebra". Communications in Algebra 33:4063-4081. |
Wikipedia:Akra–Bazzi method#0 | In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi. == Formulation == The Akra–Bazzi method applies to recurrence formulas of the form: T ( x ) = g ( x ) + ∑ i = 1 k a i T ( b i x + h i ( x ) ) for x ≥ x 0 . {\displaystyle T(x)=g(x)+\sum _{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x))\qquad {\text{for }}x\geq x_{0}.} The conditions for usage are: sufficient base cases are provided a i {\displaystyle a_{i}} and b i {\displaystyle b_{i}} are constants for all i {\displaystyle i} a i > 0 {\displaystyle a_{i}>0} for all i {\displaystyle i} 0 < b i < 1 {\displaystyle 0<b_{i}<1} for all i {\displaystyle i} | g ′ ( x ) | ∈ O ( x c ) {\displaystyle \left|g'(x)\right|\in O(x^{c})} , where c is a constant and O notates Big O notation | h i ( x ) | ∈ O ( x ( log x ) 2 ) {\displaystyle \left|h_{i}(x)\right|\in O\left({\frac {x}{(\log x)^{2}}}\right)} for all i {\displaystyle i} x 0 {\displaystyle x_{0}} is a constant The asymptotic behavior of T ( x ) {\displaystyle T(x)} is found by determining the value of p {\displaystyle p} for which ∑ i = 1 k a i b i p = 1 {\displaystyle \sum _{i=1}^{k}a_{i}b_{i}^{p}=1} and plugging that value into the equation: T ( x ) ∈ Θ ( x p ( 1 + ∫ 1 x g ( u ) u p + 1 d u ) ) {\displaystyle T(x)\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u)}{u^{p+1}}}du\right)\right)} (see Θ). Intuitively, h i ( x ) {\displaystyle h_{i}(x)} represents a small perturbation in the index of T {\displaystyle T} . By noting that ⌊ b i x ⌋ = b i x + ( ⌊ b i x ⌋ − b i x ) {\displaystyle \lfloor b_{i}x\rfloor =b_{i}x+(\lfloor b_{i}x\rfloor -b_{i}x)} and that the absolute value of ⌊ b i x ⌋ − b i x {\displaystyle \lfloor b_{i}x\rfloor -b_{i}x} is always between 0 and 1, h i ( x ) {\displaystyle h_{i}(x)} can be used to ignore the floor function in the index. Similarly, one can also ignore the ceiling function. For example, T ( n ) = n + T ( 1 2 n ) {\displaystyle T(n)=n+T\left({\frac {1}{2}}n\right)} and T ( n ) = n + T ( ⌊ 1 2 n ⌋ ) {\displaystyle T(n)=n+T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)} will, as per the Akra–Bazzi theorem, have the same asymptotic behavior. == Example == Suppose T ( n ) {\displaystyle T(n)} is defined as 1 for integers 0 ≤ n ≤ 3 {\displaystyle 0\leq n\leq 3} and n 2 + 7 4 T ( ⌊ 1 2 n ⌋ ) + T ( ⌈ 3 4 n ⌉ ) {\displaystyle n^{2}+{\frac {7}{4}}T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {3}{4}}n\right\rceil \right)} for integers n > 3 {\displaystyle n>3} . In applying the Akra–Bazzi method, the first step is to find the value of p {\displaystyle p} for which 7 4 ( 1 2 ) p + ( 3 4 ) p = 1 {\displaystyle {\frac {7}{4}}\left({\frac {1}{2}}\right)^{p}+\left({\frac {3}{4}}\right)^{p}=1} . In this example, p = 2 {\displaystyle p=2} . Then, using the formula, the asymptotic behavior can be determined as follows: T ( x ) ∈ Θ ( x p ( 1 + ∫ 1 x g ( u ) u p + 1 d u ) ) = Θ ( x 2 ( 1 + ∫ 1 x u 2 u 3 d u ) ) = Θ ( x 2 ( 1 + ln x ) ) = Θ ( x 2 log x ) . {\displaystyle {\begin{aligned}T(x)&\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u)}{u^{p+1}}}\,du\right)\right)\\&=\Theta \left(x^{2}\left(1+\int _{1}^{x}{\frac {u^{2}}{u^{3}}}\,du\right)\right)\\&=\Theta (x^{2}(1+\ln x))\\&=\Theta (x^{2}\log x).\end{aligned}}} == Significance == The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort, the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as T ( 1 ) = 0 {\displaystyle T(1)=0} and T ( n ) = T ( ⌊ 1 2 n ⌋ ) + T ( ⌈ 1 2 n ⌉ ) + n − 1 {\displaystyle T(n)=T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {1}{2}}n\right\rceil \right)+n-1} for integers n > 0 {\displaystyle n>0} , and can thus be computed using the Akra–Bazzi method to be Θ ( n log n ) {\displaystyle \Theta (n\log n)} . == See also == Master theorem (analysis of algorithms) Asymptotic complexity == References == == External links == O Método de Akra-Bazzi na Resolução de Equações de Recorrência (in Portuguese) |
Wikipedia:Aksel Frederik Andersen#0 | Aksel Frederik Andersen (10 February 1891 Fodby, Denmark – 1972 Gentofte, Denmark) was a Danish mathematician who worked on infinite series and in particular on Cesàro summation. == References == Sørensen, Henrik Kragh, Aksel Frederik Andersen |
Wikipedia:Aksharapalli#0 | Aksharapalli (Akṣarapallī) is a certain type of alphasyllabic numeration scheme extensively used in the pagination of manuscripts produced in India in pre-modern times. The name Aksharapalli can be translated as the letter system. In this system the letters or the syllables of the script in which the manuscript is written are used to denote the numbers. In contrast to the Aksharapalli system, the ordinary decimal system is called the Ankapalli system. == Examples of syllables used to represent numerals == The following tables give examples of syllables used to represent numerals. The lists are not exhaustive. == Usage == When the Aksharapalli system is used, the various syllables that constitute a number are placed one below the other as in the Chinese language and they are written in the margins of the various leaves of the manuscript. This arrangement may be the consequence an attempt to save space for the contents of the manuscript. This method can be seen in the earliest available manuscript containing the Aksharapalli system which is a manuscript of sixth century CE. == History == Nothing definitely is known about the origin of the system. It is conjectured that the system might have evolved from the ciphered-additive numeral system of Brahmi. This system has been extensively used in Jain manuscripts up to the sixteenth century. The system has also survived for a long time in Nepal. The system was in use as late as the nineteenth century in those regions of India which now constitute the Kerala State. == References == |
Wikipedia:Akshay Venkatesh#0 | Akshay Venkatesh (born 21 November 1981) is an Indian Australian mathematician and a professor (since 15 August 2018) at the School of Mathematics at the Institute for Advanced Study. His research interests are in the fields of counting, equidistribution problems in automorphic forms and number theory, in particular representation theory, locally symmetric spaces, ergodic theory, and algebraic topology. He was the first Australian to have won medals at both the International Physics Olympiad and International Mathematical Olympiad, which he did at the age of 12. In 2018, he was awarded the Fields Medal for his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory. He is the second Australian and the second person of Indian descent to win the Fields Medal. He was on the Mathematical Sciences jury for the Infosys Prize in 2020. == Early years == Akshay Venkatesh was born in Delhi, India, and his family emigrated to Perth in Western Australia when he was two years old. He attended Scotch College. His mother, Svetha, is a computer science professor at Deakin University. A child prodigy, Akshay attended extracurricular training classes for gifted students in the state mathematical olympiad program, and in 1993, whilst aged only 11, he competed at the 24th International Physics Olympiad in Williamsburg, Virginia, winning a bronze medal. The following year, he switched his attention to mathematics and, after placing second in the Australian Mathematical Olympiad, he won a silver medal in the 6th Asian Pacific Mathematics Olympiad, before winning a bronze medal at the 1994 International Mathematical Olympiad held in Hong Kong. He completed his secondary education the same year, turning 13 before entering the University of Western Australia as its youngest ever student. Venkatesh completed the four-year course in three years and became, at 16, the youngest person to earn First Class Honours in pure mathematics from the university. He was awarded the J. A. Woods Memorial Prize as the most outstanding graduate of the year from the Faculties of Science, Engineering, Dentistry, or Medical Science. While at UWA he was also one of the founding members of the Honours Cricket Association. == Research career == Akshay commenced his PhD at Princeton University in 1998 under Peter Sarnak, which he completed in 2002, producing the thesis Limiting forms of the trace formula. He was supported by the Hackett Fellowship for postgraduate study. He was then awarded a postdoctoral position at the Massachusetts Institute of Technology, where he served as a C.L.E. Moore instructor. Venkatesh then held a Clay Research Fellowship from the Clay Mathematics Institute from 2004 to 2006, and was an associate professor at the Courant Institute of Mathematical Sciences at New York University. He was a member of the School of Mathematics at the Institute for Advanced Study (IAS) from 2005 to 2006. He became a full professor at Stanford University on 1 September 2008. After serving as distinguished visiting professor at the IAS in 2017–2018, he became a permanent faculty member of IAS in August 2018. == Recognition == Akshay was awarded the Salem Prize, given to a "young mathematician judged to have done outstanding work in Salem's field of interest—the theory of Fourier series" and the Packard Fellowship in 2007. In 2008, he received the US$10,000 SASTRA Ramanujan Prize, given for "outstanding contributions to areas of mathematics influenced by the great Indian mathematician, Srinivasa Ramanujan" and "only awarded to those under the age of thirty-two (the age of Ramanujan at his time of death)." The prize was presented at the International Conference on Number Theory and Modular Forms, held at SASTRA University in Kumbakonam, Ramanujan's hometown. In 2010, he was an invited speaker at the International Congress of Mathematicians (Hyderabad) and spoke on the topic "Number Theory and Lie Theory and Generalisations." For his exceptionally wide-ranging, foundational and creative contributions to modern number theory, Venkatesh was awarded the Infosys Prize in Mathematical Sciences in 2016. In 2017 he received the Ostrowski Prize, which is awarded every two years for "outstanding achievements in pure mathematics and in the foundations of numerical mathematics." In 2018, he was awarded the Fields Medal, commonly described as the Nobel Prize of mathematics, becoming the second Australian (after Terence Tao) and the second person of Indian descent (after Manjul Bhargava) to be so honoured. The short citation for the medal declared that Venkatesh was being honoured for "his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects." University of Western Australia Professor Michael Giudici said of his former classmate's work that "[i]f it was easy for me to explain, then he wouldn't have received the Fields Medal". Australian mathematician and media personality Adam Spencer said that "[t]his century will be built by mathematicians, whether it's computer coding, algorithms, machine learning, artificial intelligence, app design and the like" and that "we should acknowledge the magnificence of the mathematical mind." Director of the Australian Mathematical Sciences Institute Professor Geoff Prince said "Akshay is an exciting and innovative leader in his field whose work will continue to have wide-ranging implications for mathematics" and a worthy recipient of the Fields medal "given his contribution to improving mathematicians' understanding of analytic number theory, algebraic number theory, and representation theory". The long citation for his Fields Medal describes Venkatesh as having "made profound contributions to an exceptionally broad range of subjects in mathematics" and recognises that he "solved many longstanding problems by combining methods from seemingly unrelated areas, presented novel viewpoints on classical problems, and produced strikingly far-reaching conjectures." Venkatesh's "use of dynamics theory, which studies the equations of moving objects to solve problems in number theory, which is the study of whole numbers, integers and prime numbers" was recognised in the award. "His work uses representation theory, which represents abstract algebra in terms of more easily-understood linear algebra, and topology theory, which studies the properties of structures that are deformed through stretching or twisting, like a Möbius strip." He described his work in 2016 as "looking for new patterns in the arithmetic of numbers". On receiving the award, which is presented every four years, Venkatesh said "A lot of the time when you do math, you're stuck, but at the same time there are all these moments where you feel privileged that you get to work with it. You have this sensation of transcendence, you feel like you've been part of something really meaningful." He was elected as a Fellow of the American Mathematical Society, in the 2025 class of fellows. == Contributions to mathematics == Akshay has made contributions to a wide variety of areas in mathematics, including number theory, automorphic forms, representation theory, locally symmetric spaces and ergodic theory, by himself, and in collaboration with several mathematicians. Using ergodic methods, Venkatesh, jointly with Jordan Ellenberg, made significant progress on the Hasse principle for integral representations of quadratic forms by quadratic forms. In a series of joint works with Manfred Einsiedler, Elon Lindenstrauss and Philippe Michel, Venkatesh revisited the Linnik ergodic method and solved a longstanding conjecture of Yuri Linnik on the distribution of torus orbits attached to cubic number fields. Akshay Venkatesh also provided a novel and more direct way of establishing sub-convexity estimates for L-functions in numerous cases, going beyond the foundational work of Hardy–Littlewood–Weyl, Burgess, and Duke–Friedlander–Iwaniec that dealt with important special cases. This approach eventually resulted in the complete resolution by Venkatesh and Philippe Michel of the sub-convexity problem for GL(1) and GL(2) L-functions over general number fields. == References == == External links == Akshay Venkatesh at the Mathematics Genealogy Project Akshay Venkatesh's results at International Mathematical Olympiad Website at Stanford University Videos of Akshay Venkatesh in the AV-Portal of the German National Library of Science and Technology |
Wikipedia:Al-Jabr#0 | Al-Jabr (Arabic: الجبر), also known as The Compendious Book on Calculation by Completion and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah; or Latin: Liber Algebræ et Almucabola), is an Arabic mathematical treatise on algebra written in Baghdad around 820 by the Persian polymath Al-Khwarizmi. It was a landmark work in the history of mathematics, with its title being the ultimate etymology of the word "algebra" itself, later borrowed into Medieval Latin as algebrāica. Al-Jabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree.: 228 It was the first text to teach elementary algebra, and the first to teach algebra for its own sake. It also introduced the fundamental concept of "reduction" and "balancing" (which the term al-jabr originally referred to), the transposition of subtracted terms to the other side of an equation, i.e. the cancellation of like terms on opposite sides of the equation. The mathematics historian Victor J. Katz regards Al-Jabr as the first true algebra text that is still extant. Translated into Latin by Robert of Chester in 1145, it was used until the sixteenth century as the principal mathematical textbook of European universities. Several authors have also published texts under this name, including Abu Hanifa Dinawari, Abu Kamil, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Ṭūsī. == Legacy == R. Rashed and Angela Armstrong write: Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from the Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems. J. J. O'Connor and E. F. Robertson wrote in the MacTutor History of Mathematics Archive: Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. == The book == The book was a compilation and extension of known rules for solving quadratic equations and for some other problems, and considered to be the foundation of algebra, establishing it as an independent discipline. The word algebra is derived from the name of one of the basic operations with equations described in this book, following its Latin translation by Robert of Chester. == Quadratic equations == The book classifies quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones. Historian Carl Boyer notes the following regarding the lack of modern abstract notations in the book: ... the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation (see History of algebra) found in the Greek Arithmetica or in Brahmagupta's work. Even the numbers were written out in words rather than symbols! Thus the equations are verbally described in terms of "squares" (what would today be "x2"), "roots" (what would today be "x") and "numbers" ("constants": ordinary spelled out numbers, like 'forty-two'). The six types, with modern notations, are: squares equal roots (ax2 = bx) squares equal number (ax2 = c) roots equal number (bx = c) squares and roots equal number (ax2 + bx = c) squares and number equal roots (ax2 + c = bx) roots and number equal squares (bx + c = ax2) Islamic mathematicians, unlike the Hindus, did not deal with negative numbers at all; hence an equation like bx + c = 0 does not appear in the classification, because it has no positive solutions if all the coefficients are positive. Similarly equation types 4, 5 and 6, which look equivalent to the modern eye, were distinguished because the coefficients must all be positive. Al-Jabr ("forcing", "restoring") operation is moving a deficient quantity from one side of the equation to the other side. In an al-Khwarizmi's example (in modern notation), "x2 = 40x − 4x2" is transformed by al-Jabr into "5x2 = 40x". Repeated application of this rule eliminates negative quantities from calculations. Al-Muqābala (المقابله, "balancing" or "corresponding") means subtraction of the same positive quantity from both sides: "x2 + 5 = 40x + 4x2" is turned into "5 = 40x + 3x2". Repeated application of this rule makes quantities of each type ("square"/"root"/"number") appear in the equation at most once, which helps to see that there are only 6 basic solvable types of the problem, when restricted to positive coefficients and solutions. Subsequent parts of the book do not rely on solving quadratic equations. == Area and volume == The second chapter of the book catalogues methods of finding area and volume. These include approximations of pi (π), given three ways, as 3 1/7, √10, and 62832/20000. This latter approximation, equalling 3.1416, earlier appeared in the Indian Āryabhaṭīya (499 CE). == Other topics == Al-Khwārizmī explicates the Jewish calendar and the 19-year cycle described by the convergence of lunar months and solar years. About half of the book deals with Islamic rules of inheritance, which are complex and require skill in first-order algebraic equations. == References == === Notes === === Citations === == Further reading == Hughes, Barnabas B. ed., Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition, (in Latin) Wiesbaden: F. Steiner Verlag, 1989. ISBN 3-515-04589-9 Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7. Rashed, R. The development of Arabic mathematics: between arithmetic and algebra, London, 1994. == External links == 19th Century English Translation at the Internet Archive Al-Khwarizmi Annotated excerpt from a translation of the Compendious Book. University of Duisburg-Essen. The Compendious Book on Calculation by Completion and Balancing – in the Arabic original with an English translation (PDF) Ghani, Mahbub (5 January 2007). "The Science of Restoring and Balancing – The Science of Algebra". Muslim Heritage. |
Wikipedia:Al-Samawal al-Maghribi#0 | Al-Samawʾal ibn Yaḥyā al-Maghribī (Arabic: السموأل بن يحيى المغربي, c. 1130 – c. 1180), commonly known as Samawʾal al-Maghribi, was a mathematician, astronomer and physician. Born to a Jewish family of North African origin, he concealed his conversion to Islam for many years for fear of offending his father, then openly embraced Islam in 1163 after he had a dream telling him to do so. His father was a rabbi from North Africa named Yehuda ibn Abūn. == Mathematics == Al-Samaw'al wrote the mathematical treatise al-Bahir fi'l-jabr, meaning "The brilliant in algebra", at the age of nineteen. He also used the two basic concepts of mathematical induction, though without stating them explicitly. He used this to extend results for the binomial theorem up to n=12 and Pascal's triangle previously given by al-Karaji. == Polemics == He also wrote a famous polemic book in Arabic debating Judaism known as Ifḥām al-Yahūd (Confutation of the Jews). A Latin tract translated from Arabic and later translated into many Western languages, titled Epistola Samuelis Marrocani ad R. Isaacum contra errores Judaeorum, claims to be authored by a certain R. Samuel of Fez "about the year 1072" and is erroneously connected with him. == Notes == == References == Anbouba, Adel (1970). "Al-Samaw'al, Ibn Yaḥyā Al-Maghribī". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 978-0-684-10114-9. O'Connor, John J.; Robertson, Edmund F., "Ibn Yahya al-Maghribi Al-Samawal", MacTutor History of Mathematics Archive, University of St Andrews Samau'al al-Maghribi: Ifham Al-Yahud: Silencing the Jews by Moshe Perlmann, Proceedings of the American Academy for Jewish Research, Vol. 32, Samau'al Al-Maghribi Ifham Al-Yahud: Silencing the Jews (1964) Samaw'al al-Maghribi: Ifham al-yahud, The early recension, by مغربي، السموءل بن يحي، d. ca. 1174. al-Samawʼal ibn Yaḥyá Maghribī; Ibrahim Marazka; Reza Pourjavady; Sabine Schmidtke Publisher: Wiesbaden : Harrassowitz, 2006. OCLC 63514265 Perlmann, Moshe, "Eleventh-Century Andalusian Authors on the Jews of Granada" Proceedings of the American Academy for Jewish Research 18 (1948–49):269-90. == External links == Al-Bahir en Algebre d'As-Samaw'al translation by Salah Ahmad, Roshdi Rashed, Author(s) of Review: David A. King, Isis, Vol. 67, No. 2 (Jun., 1976), pp. 307-308 Al-Asturlabi and as-Samaw'al on Scientific Progress, Osiris, Vol. 9, 1950 (1950), by Franz Rosenthal, pp. 555–566 Naderi, Negar (2007). "Samawʾal: Abū Naṣr Samawʾal ibn Yaḥyā ibn 'Abbās al-Maghribī al-Andalusī". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. p. 1009. ISBN 978-0-387-31022-0. (PDF version) |
Wikipedia:Aladdin Allahverdiyev#0 | Aladdin Allahverdiyev (Aladdin Allahverdiyev Mammadhuseyn; born 29 May 1947) is an Azerbaijani scientist and professor (2001). Soviet, Russian and Azerbaijani scientist in the field of mathematical models development and methods of studying wave and oscillatory processes to create piezoelectric devices and products used in the world's oceans and space studies, in marine seismic exploration, in electronic, defense and medical industries. == Short biography == Aladdin Allahverdiyev was born 29 May 1947 in the village of Kyasaman Vardenis District of the Armenian SSR. In 1964 he finished high school No. 190 in Baku with a medal. The same year he entered the Mechanics and Mathematics Faculty (Department of Mechanics) of Azerbaijan State University named after S.M. Kirov. After the third year he continued his studies at the Mechanics and Mathematics Faculty of Lomonosov Moscow State University. After graduating the university he continued his studies as a postgraduate student at the Department of Wave and Gas Dynamics and later at a closed Dissertation Council defended the thesis "Investigation of electrical parameters of piezoelectric structures with associated fluctuations" and received his PhD in physical and mathematical sciences. After finishing his postgraduate studies at Lomonosov Moscow State University in 1974 he was hired by All-Union Scientific Research Institute "FONON" of the USSR Ministry of Electronic Industry. In 1974–1994 he worked there as a junior researcher, senior researcher, and head of the department of "Mathematical Modeling". In 1994–1995 he worked as Deputy General Director for science and production of joint Russian-American Company "Green Star International". In 1995–1996 he worked as Director of Zelenograd City Association of Small and Medium Business of Moscow. In September 1996 he became the Head of the Department of Higher and Applied Mathematics and Professor of Moscow State Academy of Business Administration (MSABA). Also, in 1997–1998 he was the Dean of the Economics and Management Faculty, and from 2000 he became the First Vice-Rector on educational and methodical work of the academy. From December 2010 till present time he has been working at Moscow Institute of Electronic Technology as Deputy Vice-Rector for education, Head of the Department of Quality Management of educational process, professor. A. M. Allahverdiyev has devoted his life to science and education. In the year 2001 A. M. Allahverdiyev by the decision of the Higher Attestation Commission of the Russian Federation was awarded the academic title of professor. Under his supervision 7 postgraduate students have successfully defended their theses. For several years, A. M. Allahverdiyev was a member of the Board of Vice-Rectors for academic affairs of higher education institutions of the Russian Federation, member of the Coordinating Council of the Department of Education of Moscow, a member of the MSABA Academic Council, and the Chairman of the Scientific and Methodological Council of the academy. == Scientific and Educational Activities == While working at "FONON" Scientific Research Institute as the main theoretician of the Institute he was the scientific director or deputy chief designer of more than 30 scientific research and experimental design works (Research and Development), dedicated to the development of devices and products used in various fields of technology. Many scientific and technical results of his innovative solutions, which is evident from more than 150 scientific articles and 7 inventions published by A. M. Allahverdiyev, have been successfully applied in certain sectors of science and technology: in the study of the world ocean, in marine seismic surveys, implemented by electronic companies (especially in acoustics and microelectronics), defense and medical industries (in particular piezoceramic sensor developed by him and his co-workers have successfully been applied in the development and creation of an artificial heart). In the 1970-80s, scientists and designers working in the field of piezo electronics were tasked to develop a piezo sensor with specified electro-physical parameters and minimal weight and volume. Using Pontryagin maximum principle he was the first to find a theoretical solution to this type of problems. He was also one of the first scientists, who in cooperation with his students solved two- and three-dimensional electroelastic dynamic tasks on the basis of generalized variation principle with the use of methods of finite and boundary elements. He was among the first scientist who could theoretically prove the existence of "edge effect" in piezoceramic disk membranes. During the years of working at MSABA A. M. Allahverdiyev together with co-workers of the Department of Higher and Applied Mathematics headed by him published more than 20 textbooks and manuals for the use of mathematics in economics and management. == Participation in scientific conferences == A.M. Allahverdiyev has presented more than 60 scientific reports (including 10 plenary ones) at 30 international, All-Union, All-Russian industrial conventions, conferences, and symposia, dedicated to the dissemination of surface waves in piezoelectric environments, and the research of electrical parameters of piezoceramic structures under associated fluctuations. Some of those scientific forums include: All-Union Symposium on the propagation of elastic and elastic-plastic waves' (Frunze, 1974, Novosibirsk, 1986), All-Union Conference on continuum mechanics' (Tashkent, 1979), All-Union Conference on ferroelectricity' (Rostov-na-Donu, 1979, Minsk, 1982), All-Union Conference on strength of materials and structural elements over sound and ultrasound loading frequencies' (Kiev, 1979, 1980, 1981, 1982), All-Union Conference on theory of elasticity' (Tbilisi, 1984), All-Union Conference on acoustic electronics and quantum acoustics' (Moscow, 1984), All-Union Conference on the effect of external influences on the real structure of ferroelectric and piezoelectrics' (Chernogolovka, 1984), All-Union Conference on Bounce Modeling and imitation on computer static testing of electronic technology products' (Moscow, 1985), All-Union Congress on Theoretical and Applied Mechanics' (Tashkent, 1986), All-Union Acoustic Conference' (Moscow, 1994), International Conference on 'Crystals: growth, properties, real structure, application' (Alexandrov, 2001, 2002, 2003), The International Jubilee Conference 'Single crystal and their application in the XX century – 2004' and so on. At the last two conferences A. M. Allahverdiyev read 13 including 3 plenary reports. He was a member of either the organizing or program committee for some of these forums. Being a postgraduate student, he participated in the international conference on aeronautics in Baku (1973). == Inventions == Author’s certificate No. 169926 from 4 September 1980 Author's certificate No. 179728 from 6 October 1982 Author's certificate No. 1063257 from 22 August 1983 Author's certificate No. 300747 from 1 September 1989 Author's certificate No. 1534760 from 8 September 1989 Author's certificate No. 308342 from 1 February 1990 Author's certificate No. 308340 from 1 February 1990 == Distinct research (scientific) articles == 'The theory of related oscillations of piezoceramic disks" / "К теории связанных колебаний пьезокерамических дисков" (В кн. "Волновая и газовая динамика", Изд. МГУ, в. 2, М., 1979), (in Russian). "The oscillation of the piezoceramic pulse transducer for medical purposes"/ "О колебании пьезокерамического преобразователя пульса для медицинских целей" ("Военная техника и экономика", No. 3, М., 1978), (in Russian). "Cylindrical piezo receiver of pressure for marine seismic" / "Цилиндрический пьезоприемник давления для морской сейсморазведки" ("Электроника", No. 7, М., 1984), (in Russian). "The theory of unsteady oscillations of piezoceramic disks and wheels" / "К теории неустановившихся колебаний пьезокерамических дисков и колец" (Докл. АН УССР, No. 2 Т.,1980), (in Russian). "Research on orientation characteristics of receiving-emitting transducers"/ "Исследование характеристик направленности приемо-излучающих преобразователей" ("Дефектоскопия", No. 6, Изд. АН СССР, M., 1990), (in Russian). "Method of considering the impact of electrical boundary conditions on the bend in multilayer piezoelectric plates"/ "Метод учета влияния электрических граничных условий на изгиб многослойных пьезоэлектрических пластин", (Материалы докладов международной конференции "Кристаллы: рост, свойства. применение " Александров, 2002), (in Russian), (in Russian). "Calculation of spherical piezoceramic transducers including losses" / "Расчет сферических пьезокерамических преобразователей с учетом потерь" ("Электронная техника", сер. Радиодетали и радиокомпоненты, в. 3(80), М., 1990), (in Russian). "The method of calculating acoustic characteristics of multilayer piezoceramic transducers" / "Метод расчета акустических характеристик многослойных пьезокерамических преобразователей" (Материалы докладов международной конференции "Кристаллы: рост, свойства. применение " Ал-в, 2003), (in Russian). "The study of composite piezoceramic transducers oscillations" / "Исследование колебаний составных пьезокерамических преобразователей" ("Электронная техника", сер. Радиодетали и радиокомпоненты, в. 2(67), М., 1987), (in Russian). "Method for optimizing the shape of oscillating piezoelectric transducer "/ "Метод оптимизации формы пьезоэлектрических преобразователей, совершающих колебания" ("Прикладная механика" К., т. 26, No. 9, 1990"),(in Russian). "The impact of impulse loads on the strength characteristics of piezoceramic elements" / "Влияние импульсных нагрузок на прочностные характеристики пьезокерамических элементов" (В кн.: "Прочность поликристаллических кристаллов", Изд. АН СССР, Л.,1981), (in Russian). "Related unsteady oscillations of piezoceramic cylinders" / "Связанные неустановившиеся колебания пьезокерамических цилиндров" ("Изв. АН УССР, № 2 1982), (in Russian). "The method of calculating the characteristics of piezoceramic transducers within resonant frequencies" / "Метод расчета характеристик пьезокерамических преобразователей в области резонансных частот", ("Электронная техника", сер. Радиодетали и радиокомпоненты, в. 1(82), М., 1991), (in Russian). "Related unsteady oscillations of piezoceramic cylinders" /"Связанные неустановившиеся колебания пьезокерамических цилиндров " (Изв. АН УзССР, No. 2 1982), (in Russian). 'The reliability increase when designing the spherical piezoceramic radiators', 'Spreading of surface acoustic waves in party-homogenous piezoelectric environments' (The International Jubilee Conference 'Single crystal and their application in the XX century – 2004', Aleksandrov, Russia, 2004). 'The acoustical charasteristics of the cylindrical piezoceramic radiator vibrating in infinite liquid medium', 'Spreading of surface acoustic waves in party-homogeneous piezoelectric environments' (The International Jubilee Conference 'Single crystal and their application in the XX century – 2004' Aleksandrov, Russia, 2004) 'Method for determining electroelastic constants of piezoceramic materials" / "Метод определения электроупругих констант пьезокерамических материалов" (В кн.: "Основы физики элементов микроэлектронных приборов", М.,1992), (in Russian). "Method of calculating mechanical stress of piezoceramic rings and disks during alternating electric excitations" / "Метод расчета механических напряжений пьезокерамических колец и дисков при переменном электрическом возбуждении" (В кн.: "Прочность материалов и элементов конструкций при звуковых и ультразвуковых частотах нагружения", Киев, 1980), (in Russian). "Related flexural-shear vibrations of layer-step disc of piezoceramic transducers" / "Связанные изгибно-сдвиговые колебания слойно-ступенчатых дисковых пьезокерамических преобразователей " (" Прикладная механика" Киев, 1987, т. 23, No. 5), (in Russian). "Ultrasonic vibration frequency matching of various composite electroacoustic transducers" / "Согласование частот ультразвуковых колебаний различных составных электроакустических преобразователей". (В сб., "Материалы ХI Всесоюзной конференции по акустоэлектронике и квантовой акустике", Москва, 1984), (in Russian). "Distribution of surface acoustic waves in a piecewise-homogeneous piezoelectric environments" / "Распространение поверхностных акустических волн в кусочно-однородных пьезоэлектрических средах". (В кн., "Материалы VI Всесоюзного сьезда по теоретической и прикладной механике", Изд. "Фан", Ташкент, 1986), (in Russian). "Sustainable development as a paradigm of the philosophy of education" / "Устойчивое развитие как парадигма философии образования". (В кн., "Эпоха глобальных проблем (Опыт философского осмысления)". Изд. МГИДА, М., 2004), (in Russian). "Applying mathematical methods in social sciences"/ "Применение математических методов в социальных науках". (В кн., "Экономика и социальная сфера: человек, город, Россия", Изд. МГИДА, М., 2005), (in Russian). "Rating system as a self-oriented technology and its role in the development of student abilities" / "Рейтинговая система как личностно-ориентированная технология и её роль в развитии способностей студента". (В кн., "На путях к личностно-ориентированному образованию", Изд. МГИДА, М., 2000), (in Russian). "Mathematics and its role in training modern managers" / "Математика и её роль в подготовке современных менеджеров". (В кн., "Философско-педагогический анализ проблемы гуманизации образовательного процесса", Изд. МГИДА, М., 2001), (in Russian). == Some textbooks for universities == The numerical sequences. MSIBA, Moscow, 1998. / Числовые последовательности. Изд. МГИДА, Москва, 1998. (in Russian). Mathematical analysis (for economic students). MSIBA, Moscow, 2001 / Математический анализ (для экономических специальностей). Изд. МГИДА, Москва, 2001(in Russian). Applied Mathematics. Graph theory (for economic students). MSIBA, Moscow, 2001 / Прикладная математика. Теория графов. (для экономических специальностей). Изд. МГИДА, Москва, 2001,(in Russian). Theory of Probability and Mathematical Statistics. MSIBA, Moscow, 2004 / Теория вероятностей и математическая статистика. Изд. МГИДА, Москва, 2004,(in Russian). Linear algebra and elements of analytic geometry. MSIBA, Moscow, 2004 / Линейная алгебра и элементы аналитической геометрии. Изд. МГИДА, Москва, 2004,(in Russian). Differential equations (for economic students). MSIBA, Moscow, 2005 / Дифференциальные уравнения. (для экономических специальностей). Изд. МГИДА, Москва, 2005,(in Russian). Mathematical Statistics (for economic students). MSIBA, Moscow, 2005 / Математическая статистика. (для экономических специальностей). Изд. МГИДА, Москва, 2005,(in Russian). == Honors == For achievements in scientific, industrial and pedagogical activities A. M. Allahverdiyev has received several governmental and industrial awards: the badge "Honorary Worker of Higher Professional Education" the medal "In memory of the 850th anniversary of Moscow" (1997) the medal "All-Russian Exhibition Center Laureat AEC" the medal "250 years of Lomonosov MSU" three badges of distinction for his work, established by the CPSU CC, the Soviet Ministers of the USSR, the Trade Unions CC and the LYCLSU CC in 1975, 1977 and 1980 the Diploma of prize winner of the competition by the Presidium of the Central Board of the All-Union Scientific Medical and Technical Society for his personal contribution to the development of medical technology the laureate of the contest of the Moscow City Government 'Grant of Moscow' in the field of science and technology in education four times (2001–2004) a cash prize; dozens honorary diplomas and certificates of various organizations. By National Committee on Theoretical and Applied Mechanics of Russian Academy of Sciences was awarded the medal of Kh.A. Rakhmatulin, an outstanding mechanical scientist, Hero of Socialist Labor, academician, professor, Doctor of Physical and Mathematical Sciences, laureate of 4 orders of Lenin, several State Prizes of the USSR. By Federation of Cosmonautics of Russia was awarded the medal of G.A.Tyulin, Hero of Socialist Labor, Lieutenant General, Lenin Prize Laureate, Professor, Doctor of Technical Sciences. == Family == Married. Has a son and daughter, and 4 grandchildren. His wife, Makhsati Kengerli Ismail, was born in 1951 in Nakhchivan. She finished school in Nakhchivan with a medal. She graduated from the Faculty of Chemistry at Lomonosov Moscow State University. She has a PhD in Chemistry, and is a leading researcher at the Scientific Research Institute of Materials. She is the author of many scientific articles and holder of certificates of inventions (inventor's certificates). His son, Togrul Allahverdiyev, was born in 1979 in Moscow. He graduated with honor from the Diplomatic Academy of the Ministry of Foreign Affairs of the Russian Federation (specialization 'World Economy'), and with honor from the State University of Management (specialization 'Management'). He has a PhD in economics. He is the author of the monograph 'Corporate Management of Innovative Development', and more than 10 scientific articles. He worked as Head of the Risk Management Department at Svyaz-Invest Company, Head of the Risk Management Department at Cherkizovo Group, Head of the Risk Management Department of the world-famous Russian steel company NLMK, which owns branches and factories in 7 countries of the world: in the United States, Canada, in Western Europe, and also in many regions of the Russian Federation. He is currently working in Amsterdam as executive director of Risk Management & Internal Control at Eurasian Resources Group. He is married and has a daughter. His daughter, Allahverdiyeva-Rafibeyli Ayten was born in 1982 in Moscow. She finished Moscow general high school No. 842 with a medal. She graduated with honor from I.M. Sechenov First Moscow State Medical University. She finished residency studies and works as a surgeon in the city of Doha, the capital of the state of Qatar in the field of plastic surgery. She is married and has 3 children. == Sources == Propagation of elastic and elastic-plastic waves. Frunze 1979, (in Russian). The strength of materials and structural elements over sound and ultrasound loading frequencies. Kiev, 1979, (in Russian). The effect of external influences on the real structure of ferroelectric and piezoelectrics. Chernogolovka, 1981,(in Russian). Physical phenomena in polycrystalline ferroelectrics. Ed. AS USSR Leningrad, 1984, (in Russian). The strength of polycrystalline ferroelectrics. Ed. AS USSR Leningrad, 1984, (in Russian). Bounce/Failure Modeling and imitation on computer static testing of electronic technology products, Moscow, 1986? (in Russian)/ Physical foundations of microelectronics. Ed. MIET, Moscow, 1986, (in Russian). Physical fundamentals of microelectronic devices. Ed. MIET, Moscow, 1987, (in Russian). Mathematical modeling of physical processes in the microchip elements. Ed. MIET, Moscow, 1988, (in Russian). Theoretical Foundations of functional electronics. Ed. MIET, Moscow, 1992, (in Russian). Crystals, growth, properties, application. Ed. RAS, Aleksandrov, 2001,(in Russian). Philosophic-pedagogical analyze of the problem humanization of educational process. Moscow, 2001, (in Russian). The International Jubilee Conference 'Single crystal and their application in the XX century −2004' Aleksandrov, Russia, 2004 The era of global issues (experience of philosophical understanding). Ed. MSIBA, Moscow, 2004 The numerical sequence. Ed. MSIBA, Moscow, 1998, (in Russian). Mathematical Analysis. Ed. MSIBA, Moscow, 2001, (in Russian). Applied Mathematics. Graph theory. Ed. MSIBA, Moscow, 2001, (in Russian). Differential equations. Ed. MSIBA, Moscow, 2005, (in Russian). Theory of Probability and Mathematical Statistics. Ed. MSIBA, Moscow, 2004, (in Russian). Linear algebra and elements of analytical geometry. Ed. MSIBA, Moscow, 2004, (in Russian). Mathematical statistics. Ed. MSIBA, Moscow, 2005, (in Russian). == References == == External links == Библиографическое описание издания: Аллавердиев, Аладдин Мамедович. Московский городской педагогический университет (in Russian). resources.mgpu.ru. 2016. Archived from the original on 30 June 2016. Retrieved 13 July 2016. АЛЛАВЕРДИЕВ АЛАДДИН МАМЕД ГУСЕЙН ОГЛЫ (in Russian). nomerorg.com. 2016. Archived from the original on 13 July 2016. Retrieved 13 July 2016. Allaverdiev, A. M.; Akhmedov, N. B.; Shermergor, T. D. (1987). "Coupled flexural-shear oscillations of stepwise-layered piezoceramic disk transducers". Soviet Applied Mechanics. 23 (5): 465–471. Bibcode:1987SvApM..23..465A. doi:10.1007/BF00888059. S2CID 121809851. |
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