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De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna
De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna is a 38-page mathematical treatise written in the early 17th century by Thomas Harriot, lost for many years, and finally published in facsimile form in 2009 in the book Thomas Harriot's Doctrine of Triangular Numbers: the "Magisteria Magna". Harriot's work dates from before the invention of calculus, and uses finite differences to accomplish many of the tasks that would later be made more easy by calculus.
De numeris triangularibus
Thomas Harriot wrote De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna in the early 1600s, and showed it to his friends.[1] By 1618 it was complete,[2] but in 1621 Harriot died before publishing it. Some of its material was published posthumously, in 1631, as Artis analyticae praxis, but the rest languished in the British Library among many other pages of Harriot's works,[1] and became forgotten until its rediscovery in the late 1700s.[2] It was finally published in its entirety, as part of the 2009 book Thomas Harriot’s Doctrine of Triangular Numbers: the "Magisteria Magna".[1]
The title can be translated as "The Great Doctrine of triangular numbers and, through them, of arithmetic progressions".[1] Harriot's work concerns finite differences, and their uses in interpolation for calculating mathematical tables for navigation.[2] Harriot forms the triangular numbers through the inverse process to finite differencing, partial summation, starting from a sequence of constant value one. Repeating this process produces the higher-order binomial coefficients, which in this way can be thought of as generalized triangular numbers, and which give the first part of Harriot's title.[3]
Harriot's results were only improved 50 years later by Isaac Newton, and prefigure Newton's use of Newton polynomials for interpolation.[1][4] As reviewer Matthias Schemmel writes, this work "shows what was possible in dealing with functional relations before the advent of the calculus".[3]
The work was written as a 38-page manuscript in Latin, and Harriot wrote it up as if for publication, with a title page. However, much of its content consists of calculations and formulas with very little explanatory text,[1][4] leading at least some of Harriot's contemporaries such as Sir Charles Cavendish to complain of the difficulty of understanding it.[1]
Thomas Harriot’s Doctrine
The monograph Thomas Harriot’s Doctrine of Triangular Numbers: the "Magisteria Magna", edited by Janet Beery and Jackie Stedall, was published in 2009 by the European Mathematical Society in their newly created Heritage of European Mathematics series. Its subject is De numeris triangularibus, and the third of its three sections consists of a facsimile reproduction of Harriot's manuscript, with each page facing a page of commentary by the editors,[1][2] including translations of its Latin passages.[3] The earlier parts of Beery and Stedall's book survey the material in Harriot's work, the context for this work, the chronology of its loss and recovery, and the effect of this work on the 17th-century mathematicians who read it.[2][4]
Although reviewer Matthias Schemmel suggests that the 2009 monograph is primarily aimed at historians of mathematics, who "will welcome this book as providing new insights into the development of mathematics", he suggests that it may also be of interest to other mathematicians and could perk their interest in the history of mathematics.[3]
References
1. Gouvêa, Fernando Q. (March 2009), "Review of Thomas Harriot's Doctrine of Triangular Numbers", MAA Reviews, Mathematical Association of America
2. mbec (May 2011), "Review of Thomas Harriot's Doctrine of Triangular Numbers", EMS Reviews, European Mathematical Society
3. Schemmel, Matthias (September 2010), "Before calculus (review of Thomas Harriot's Doctrine of Triangular Numbers)", Notes and Records of the Royal Society of London, 64 (3): 303–304, doi:10.1098/rsnr.2010.0016, JSTOR 20753908, S2CID 202575019
4. Shea, William R. (2010), "Review of Thomas Harriot's Doctrine of Triangular Numbers", Mathematical Reviews, MR 2516550
| Wikipedia |
Thomas Archer Hirst
Thomas Archer Hirst FRS (22 April 1830 – 16 February 1892) was a 19th-century English mathematician, specialising in geometry. He was awarded the Royal Society's Royal Medal in 1883.
Life
Thomas Hirst was born in Heckmondwike, Yorkshire, England, where both his parents came from families in the wool trade. He was the youngest of four sons. The family moved to Wakefield so that the boys could attend a better school. Thomas attended Wakefield Proprietary School for four years from 1841.[1] Of these days, he said[2] "... I could obtain the most rudimentary and necessary instruction. I remember, however, that here mathematics was my favourite study ..." He left the school at fifteen to work as an apprentice engineer in Halifax, surveying for proposed railway lines. It was there that he met John Tyndall, ten years older than Hirst and working as an engineer in the same firm.
In his late teens, at the instigation of Tyndall, Hirst decided to go to Germany for education, initially in chemistry. He eventually received a doctorate in mathematics from the University of Marburg in 1852 (tutor: Friedrich Ludwig Stegmann). In 1853, he attended geometry lectures by Jakob Steiner at University of Berlin. Hirst married Anna Martin in 1854, and spent much of the decade of the 1850s on the European continent, where he socialised with many mathematicians, and used his inherited wealth to support himself.
From 1860 to 1864, Hirst taught at University College School, but resigned because he wanted more time for his mathematical research. He was appointed Professor of Physics at University College London in 1865, and he succeeded Augustus De Morgan to the Chair of Mathematics at UCL in 1867. In 1873 he was appointed as the first Director of Studies at the new Royal Naval College, Greenwich. He retired from that post in 1882, to be succeeded by William Davidson Niven.
From the 1860s onwards, Hirst also allocated much of his time in England to the administrative committees of British science. He was an active member of the governing councils of the Royal Society, the British Association for the Advancement of Science, and the London Mathematical Society. He was the founding president of an association to reform school mathematics curricula and also worked to promote the education of women. Alongside his old friend Tyndall, Hirst was a member of T. H. Huxley's London X-Club. He died in London in 1892, four weeks after he had made the last entry in his journal, and was buried on the eastern side of Highgate Cemetery.[3][4]
Vestiges
In his early days, Hirst wrote extensively in his notebooks (sometimes called the Journal), recording everything he read and much of what he was thinking about. This extraordinary record of about fifty years is preserved in the library of the Royal Institution. As a result, we know much about the development of his mind before he became a professional mathematician. We know, for example, what the effect was of his reading the Vestiges of the Natural History of Creation, that epoch-making book authored anonymously by Robert Chambers which promoted the idea of evolution in 1844.
"Almost no-one reads like this anymore. It is the reading practice of a self-improving autodidact, shaped by Bible-reading amongst denominations of learned liberal Dissent... Hirst copied large chunks into his journal... the journal shows that Hirst moved between Vestiges and other related works such as Paley's Natural Theology and John Arthur Phillips' Geology of Yorkshire..."[5]
Both Hirst and Tyndall left in their journals and letters evidence that Vestiges (especially its geological evidence) made a good case against the story of Genesis and the case for divine intervention; yet they were not atheists. They simply came to the conclusion that parts of the Old Testament were allegorical.
Mathematics
Hirst was a projective geometer in the style of Poncelet and Steiner. He was not an adherent of the algebraic geometry approach of Cayley and Sylvester, despite being a friend of theirs. His speciality was Cremona transformations.
Notes
1. "Hirst biography". Archived from the original on 3 March 2016. Retrieved 21 February 2016.
2. H J Gardner and R J Wilson, Thomas Archer Hirst – Mathematician Xtravagant I. A Yorkshire surveyor, Amer. Math. Monthly 100 (1993), 435–441.
3. Brock W. H. and MacLeod R. M. 1974. The life and journals of Thomas Archer Hirst. Historia Mathematica 1, 181–3.
4. Lee, Sidney, ed. (1901). "Hirst, Thomas Archer" . Dictionary of National Biography (1st supplement). Vol. 2. London: Smith, Elder & Co. pp. 426–7. ; and Proc Roy Soc 52, (1892–93) 12–18
5. Secord, James A. 2000. Victorian sensation: the extraordinary publication, reception and secret authorship of the Vestiges of the Natural History of Creation. Chicago. p343 et seq.
References
• Ueber conjugirte Diameter im dreiaxigen Ellipsoid. Inaugural-Dissertation, welche mit Genehmigung der philosophischen Facultät zu Marburg zur Erlangung der Doctorwürde einreicht Thomas Archer Hirst aus England. Marburg, Druck und Papier von Joh. Aug. Koch. 1852. [20 pages].
External links
• O'Connor, John J.; Robertson, Edmund F., "Thomas Archer Hirst", MacTutor History of Mathematics Archive, University of St Andrews
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Thomas Jech
Thomas J. Jech (Czech: Tomáš Jech, pronounced [ˈtomaːʃ ˈjɛx]; born January 29, 1944, in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years.
Life
He was educated at Charles University (his advisor was Petr Vopěnka) and from 2000 is at the Institute of Mathematics of the Academy of Sciences of the Czech Republic.
Work
Jech's research also includes mathematical logic, algebra, analysis, topology, and measure theory.
Jech gave the first published proof of the consistency of the existence of a Suslin line. With Karel Prikry, he introduced the notion of precipitous ideal. He gave several models where the axiom of choice failed, for example one with ω1 measurable. The concept of a Jech–Kunen tree is named after him and Kenneth Kunen.
Bibliography
• "Non-provability of Souslin's hypothesis", Comment. Math. Univ. Carolinae, 8: 291–305, 1967, MR 0215729
• Lectures in set theory, with particular emphasis on the method of forcing, Springer-Verlag Lecture Notes in Mathematics 217 (1971) (ISBN 978-3540055648)
• The axiom of choice, North-Holland 1973 (Dover paperback edition ISBN 978-0-486-46624-8)
• (with K. Hrbáček) Introduction to set theory, Marcel Dekker, 3rd edition 1999 (ISBN 978-0824779153)
• Multiple forcing, Cambridge University Press 1986 (ISBN 978-0521266598)[1]
• Set Theory: The Third Millennium Edition, revised and expanded, 2006, Springer Science & Business Media, ISBN 3-540-44085-2. 1st ed. 1978;[2] 2nd (corrected) ed. 1997
References
1. Baumgartner, James (1989). "Review: Multiple forcing by Thomas Jech" (PDF). Bull. Amer. Math. Soc. (N.S.). 20 (1): 103–107. doi:10.1090/s0273-0979-1989-15716-9.
2. Kunen, Kenneth (1980). "Review: Set theory by Thomas Jech" (PDF). Bull. Amer. Math. Soc. (N.S.). 3, Part 1 (1): 775–777. doi:10.1090/S0273-0979-1980-14818-1.
External links
• Home page, with a copy at Penn state.
• Thomas Jech at the Mathematics Genealogy Project
Set theory
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Thomas Jones (mathematician)
Thomas Jones (23 June 1756 – 18 July 1807) was Head Tutor at Trinity College, Cambridge, for twenty years and an outstanding teacher of mathematics. He is notable as a mentor of Adam Sedgwick.
Thomas Jones
Portrait of Thomas Jones (1756–1807) by D. Gardner
Born(1756-06-23)23 June 1756
Berriew, Montgomeryshire, Wales
Died18 July 1807(1807-07-18) (aged 51)
Edgware Road, London, England
Alma materSt John's College, Cambridge
Trinity College, Cambridge
AwardsFirst Smith's Prize
Scientific career
FieldsMathematician
InstitutionsTrinity College, Cambridge
Academic advisorsThomas Postlethwaite
John Cranke
Notable studentsAdam Sedgwick
John Hudson
Biography
Jones was born at Berriew, Montgomeryshire, in Wales.
On completing his studies at Shrewsbury School, Jones was admitted to St John's College, Cambridge, on 28 May 1774, as a 'pensioner' (i.e. a fee-paying student, as opposed to a scholar or sizar).[1] He was believed to be an illegitimate son of Mr Owen Owen, of Tyncoed, and his housekeeper, who afterwards married a Mr Jones, of Traffin, County Kerry, Thomas then being brought up as his son.[1]
On 27 June 1776, Jones migrated from St John's College to Trinity College. He became a scholar in 1777 and obtained his BA in 1779, winning the First Smith's Prize and becoming Senior Wrangler. In 1782, he obtained his MA and became a Fellow of Trinity College in 1781. He became a Junior Dean, 1787–1789 and a Tutor, 1787–1807. He was ordained a deacon at the Peterborough parish on 18 June 1780. Then he was ordained priest, at the Ely parish on 6 June 1784, canon of Fen Ditton, Cambridgeshire, in 1784, and then canon of Swaffham Prior, also 1784. On 11 December 1791, he preached before the university, at Great St Mary's, a sermon against duelling (from Exodus XX. 13), which was prompted by a duel that had lately taken place near Newmarket between Henry Applewhaite and Richard Ryecroft, undergraduates of Pembroke, in which the latter was fatally wounded. Jones died on 18 July 1807, in lodgings in Edgware Road, London. He is buried in the cemetery of Dulwich College. A bust and a memorial tablet are in the ante-chapel of Trinity College.
His academic mentor was John Cranke (1746–1816). His Cambridge tutor was Thomas Postlethwaite.
Notes
1. "Thomas Jones (JNS774T)". A Cambridge Alumni Database. University of Cambridge.
References
• Dictionary of National Biography, Smith, Elder & Co., 1908–1986, vol. 10, pp. 1055–1056.
• J. Wilkes, Encyclopedia Londinensis, Eds. J. Jones and J. Adlard, 1810–1829, vol. 11, pp. 256–258.
• J.W. Clark and T.M. Hughes, The Life and Letters of the Reverend Adam Sedgwick, Cambridge University Press: 1890; vol. 1, pp. 73–75.
• J. Gascoigne, Cambridge in the Age of Enlightenment, 1989, pp. 226–227, p. 232, p. 234, p. 243.
• P. Searby, A History of the University of Cambridge, vol. 3 (1750–1870), ed. C.N.L. Brooke et al., 1997. pp. 309–310.
• Oxford Dictionary of National Biography, vol. 30, eds. H. C. G. Matthew and B. Harrison, 2004, p. 645.
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Thomas Jones Enright
Thomas Jones Enright (August 15, 1947 – January 27, 2019) was an American mathematician known for his work in the algebraic theory of representations of real reductive Lie groups.
Thomas Jones Enright
BornAugust 15, 1947
Concord, New Hampshire, United States
DiedJanuary 27, 2019(2019-01-27) (aged 71)
San Diego, California, United States
Alma materHarvard University B.S. & University of Washington Ph.D.
AwardsAlfred P. Sloan Research Fellowship
Scientific career
FieldsMathematics
InstitutionsUCSD
Doctoral advisorRamesh A. Gangolli
Biography
Enright received a B.S. from Harvard University in 1969 and a Ph.D. in 1973 from the University of Washington under the direction of Ramesh A. Gangolli. From 1973 to 1975 he was the Hedrick Assistant Professor in UCLA working with Veeravalli S. Varadarajan, and spent the 1976-1977 year after in the Institute for Advanced Study at Princeton, N. J. before starting at University of California at San Diego in 1977. He was chair of the mathematics department of UCSD from 1986 to 1990.[1] In 2010 he retired due symptoms of Parkinson's disease.
Contributions
In the mid-1970s, Enright introduced new methods that led him to an algebraic way of looking at discrete series (which were fundamental representations constructed by Harish-Chandra in the early 1960s), and to an algebraic proof of the Blattner multiplicity formula.
He was known for Enright–Varadarajan modules,[2][3] Enright resolutions, and the Enright completion functor,[4][5][6][7] which has had a lasting influence in algebra.
Recognition
• Recipient of Alfred P. Sloan Fellowship, 1978[8]
• Enright's work was the subject of a Bourbaki Seminar by Michel Duflo[9]
Bibliography
• Enright, Thomas J (1979). "On the Fundamental Series of a Real Semisimple Lie Algebra: Their Irreducibility, Resolutions and Multiplicity Formulae". Annals of Mathematics. 110 (1): 1–82. doi:10.2307/1971244. JSTOR 1971244.
• Enright, Thomas J.; Varadarajan, V. S. (1975). "On an Infinitesimal Characterization of the Discrete Series". Annals of Mathematics. 102 (1): 1–15. doi:10.2307/1970970. JSTOR 1970970.
• Enright, Thomas J (1979). "On the Fundamental Series of a Real Semisimple Lie Algebra: Their Irreducibility, Resolutions and Multiplicity Formulae". Annals of Mathematics. 110 (1): 1–82. doi:10.2307/1971244. JSTOR 1971244.
• Enright, Thomas; Howe, Roger; Wallach, Nolan (1983-01-01). Trombi, P. C., ed. A Classification of Unitary Highest Weight Modules. Progress in Mathematics. Birkhäuser Boston. pp. 97–143. doi:10.1007/978-1-4684-6730-7_7. ISBN 9780817631352.
• Enright, T. J.; Wallach, N. R. (1980). "Notes on homological algebra and representations of Lie algebras". Duke Mathematical Journal. 47 (1): 1–15. doi:10.1215/S0012-7094-80-04701-8.
• Davidson, Mark G.; Enright, Thomas J.; Stanke, Ronald J. (1991). "Differential operators and highest weight representations". Memoirs of the American Mathematical Society. 94 (455): 0. doi:10.1090/memo/0455.
• Enright, Thomas J.; Hunziker, Markus; Pruett, W. Andrew (2014-01-01). Howe, Roger; Hunziker, Markus; Willenbring, Jeb F., eds. Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties. Progress in Mathematics. Springer New York. pp. 121–184. doi:10.1007/978-1-4939-1590-3_6. ISBN 9781493915897.
• Enright, Thomas J (1978). "On the algebraic construction and classification of Harish-Chandra modules". Proceedings of the National Academy of Sciences. 75 (3): 1063–1065. Bibcode:1978PNAS...75.1063E. doi:10.1073/pnas.75.3.1063. PMC 411407. PMID 16592507.
References
1. Support, Math Computing. "UCSD Math | Department History". www.math.ucsd.edu. Retrieved 2016-03-15.
2. Wallach, Nolan R. (1976-01-01). "On the Enright–Varadarajan modules: A construction of the discrete series". Annales Scientifiques de l'École Normale Supérieure. Série 4. 9: 81–101. doi:10.24033/asens.1304. ISSN 0012-9593.
3. "Parthasarathy: A generalization of the Enright–Varadarajan modules". www.numdam.org. Retrieved 2016-04-23.
4. König, Steffen; Mazorchuk, Volodymyr (2002-01-01). "Enright's completions and injectively copresented modules". Transactions of the American Mathematical Society. 354 (7): 2725–2743. doi:10.1090/S0002-9947-02-02958-6. ISSN 0002-9947.
5. Jakelić, Dijana (2007-06-01). "On crystal bases and Enright's completions". Journal of Algebra. 312 (1): 111–131. arXiv:math/0512506. doi:10.1016/j.jalgebra.2006.11.045. S2CID 120487.
6. Khomenko, Oleksandr; Mazorchuk, Volodymyr (2004-10-15). "On Arkhipov's and Enright's functors". Mathematische Zeitschrift. 249 (2): 357–386. doi:10.1007/s00209-004-0702-8. ISSN 0025-5874. S2CID 121941150.
7. Deodhar, Vinay V. (1980-06-01). "On a construction of representations and a problem of Enright". Inventiones Mathematicae. 57 (2): 101–118. Bibcode:1980InMat..57..101D. doi:10.1007/BF01390091. ISSN 0020-9910. S2CID 121286075.
8. "Past Fellows". www.sloan.org. Retrieved 2016-03-29.
9. Duflo, Michel (1979-01-01). "Représentations de carré intégrable des groupes semi-simples réels". Séminaire Bourbaki vol. 1977/78 Exposés 507–524. Lecture Notes in Mathematics (in French). Vol. 710. Springer Berlin Heidelberg. pp. 22–40. doi:10.1007/bfb0069971. ISBN 9783540092438.
External links
Wikimedia Commons has media related to Thomas Jones Enright.
• Thomas Jones Enright at the Mathematics Genealogy Project
• Homepage of Thomas Jones Enright
• Thomas Jones Enright's obituary
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Thomas Kappeler
Thomas Kappeler (12 February 1953 – 30 May 2022) was a Swiss mathematician and professor at the University of Zurich.[1][2] Kappeler's main research was in global analysis, partial differential equations and dynamical systems in infinite dimensions.
Kappeler co-founded the Zurich Graduate School in Mathematics, a joint doctoral program of the Mathematics departments of ETH Zurich and University of Zurich.[3] He also actively supported young kids with talent in mathematics. He was the co-leader of the children's math club Junior Euler Society.[4]
Life
Kappeler studied mathematics at ETH Zurich, where he did his Ph.D. in 1981 under the supervision of Corneliu Constantinescu.[5] He was a visiting professor at the University of California, Berkeley, the University of Pennsylvania, Brandeis University and Brown University. He was a professor at Ohio State University from 1990 till 1996.[6]After that he became a mathematics professor at University of Zurich.[7]
Selected publications
Kappeler published more than 150 research articles.[8] He also published two books on PDEs.
• Craig, Walter; Kappeler, Thomas; Strauss, Walter (1995). "Microlocal dispersive smoothing for the Schrödinger equation". Communications on Pure and Applied Mathematics. 48 (8): 769–860. doi:10.1002/cpa.3160480802.
• Burghelea, Dan; Friedlander, Leonid; Kappeler, Thomas (1992). "Mayer-Vietoris type formula for determinants of elliptic differential operators". Journal of Functional Analysis. 107: 34–65. doi:10.1016/0022-1236(92)90099-5.
• Kappeler, Thomas; Perry, Peter A.; Shubin, Mikhail A.; Topalov, Peter (2005). "The Miura map on the line". International Mathematics Research Notices. 2005 (50): 3091–3133. doi:10.1155/IMRN.2005.3091.
• Kappeler, Thomas; Topalov, Peter (1 November 2006). "Global wellposedness of KdV in H−1(T,R)". Duke Mathematical Journal. 135 (2): 327–360. doi:10.1215/S0012-7094-06-13524-X.
• Kappeler, Thomas; Pöschel, Jürgen (2009). "On the periodic KdV equation in weighted Sobolev spaces". Annales de l'Institut Henri Poincaré C. 26 (3): 841–853. Bibcode:2009AIHPC..26..841K. doi:10.1016/j.anihpc.2008.03.004.
• Henrici, Andreas; Kappeler, Thomas (2009). "Nekhoroshev theorem for the periodic Toda lattice". Chaos. 19 (3): 033120. arXiv:0812.4912. Bibcode:2009Chaos..19c3120H. doi:10.1063/1.3196783. PMID 19792000. S2CID 18290712.
• Cohen, Daniel C.; Costa, Armindo; Farber, Michael; Kappeler, Thomas (2012). "Topology of Random 2-Complexes". Discrete & Computational Geometry. 47 (1): 117–149. arXiv:1006.4229. doi:10.1007/s00454-011-9378-0. MR 2886093. S2CID 254038254.
Books
• Kappeler, Thomas; Pöschel, Jürgen (2003). KdV & KAM. doi:10.1007/978-3-662-08054-2. ISBN 978-3-540-02234-3.
• Grébert, Benoît; Kappeler, Thomas (2014). The Defocusing NLS Equation and Its Normal Form. doi:10.4171/131. ISBN 978-3-03719-131-6.
References
1. Thomas Kappeler. Obituary in Neue Zürcher Zeitung. 4. Juni 2022, P. 12, page accessed 24 February 2023.
2. News, and Preview of significant forthcoming events: Obituary in math.ch
3. "Zurich Graduate School in Mathematics - Announcement". Zurich Graduate School in Mathematics. Retrieved 24 February 2023.
4. "Masslösung für ein Mathe-Genie". Neue Zürcher Zeitung. 3 October 2013. p. 15. Retrieved 24 February 2022.
5. Thomas Kappeler at the Mathematics Genealogy Project
6. "Thomas Kappeler". typeset.io. Retrieved 1 March 2023.
7. "Prof. Dr. Thomas Kappeler". Universität Zürich. 18 November 2021. Retrieved 4 June 2022.
8. "Kappeler, Thomas". mathscinet.ams.org. American Mathematical Society. Retrieved 27 February 2023.
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Thomas Kirkman
Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane as one of ten leading 19th-century British mathematicians.[1][2][3] In the 1840s, he obtained an existence theorem for Steiner triple systems that founded the field of combinatorial design theory, while the related Kirkman's schoolgirl problem is named after him.[4][5]
Thomas Penyngton Kirkman
Born(1806-03-31)31 March 1806
Bolton, Lancashire, England
Died3 February 1895(1895-02-03) (aged 88)
Bowdon near Manchester, England
Occupation(s)Mathematician, Minister
Known forKirkman's schoolgirl problem
Early life and education
Kirkman was born 31 March 1806 in Bolton, in the north west of England, the son of a local cotton dealer. In his schooling at the Bolton Grammar School, he studied classics, but no mathematics was taught in the school. He was recognised as the best scholar at the school, and the local vicar guaranteed him a scholarship at Cambridge, but his father would not allow him to go. Instead, he left school at age 14 to work in his father's office.[1][2][3]
Nine years later, defying his father, he went to Trinity College Dublin, working as a private tutor to support himself during his studies. There, among other subjects, he first began learning mathematics. He earned a B.A. in 1833 and returned to England in 1835.[1][2][3]
Ordination and ministry
On his return to England, Kirkman was ordained into the ministry of the Church of England and became the curate in Bury and then in Lymm. In 1839 he was invited to become rector of Croft with Southworth, a newly founded parish in Lancashire, where he would stay for 52 years until his retirement in 1892. Theologically, Kirkman supported the anti-literalist position of John William Colenso, and was also strongly opposed to materialism. He published many tracts and pamphlets on theology, as well as a book Philosophy Without Assumptions (1876).[1][2][3]
Kirkman married Eliza Wright in 1841; they had seven children. To support them, Kirkman supplemented his income with tutoring, until Eliza inherited enough property to secure their living. The rectorship itself did not demand much from Kirkman, so from this point forward he had time to devote to mathematics.[1][2]
Kirkman died 4 February 1895 in Bowdon. His wife died ten days later.[1][3]
Mathematics
Kirkman's first mathematical publication was in the Cambridge and Dublin Mathematical Journal in 1846, on a problem involving Steiner triple systems that had been published two years earlier in The Lady's and Gentleman's Diary by Wesley S. B. Woolhouse.[1][2][3] Despite Kirkman's and Woolhouse's contributions to the problem, Steiner triple systems were named after Jakob Steiner who wrote a later paper in 1853.[1] Kirkman's second research paper, in 1848, concerned pluquaternions.
In 1848, Kirkman published First Mnemonical Lessons, a book on mathematical mnemonics for schoolchildren. It was not successful, and Augustus De Morgan criticised it as "the most curious crochet I ever saw".[1][2][3]
Kirkman's schoolgirl problem
Next, in 1849, Kirkman studied the Pascal lines determined by the intersection points of opposite sides of a hexagon inscribed within a conic section. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. That is, these lines and points form a projective configuration of type 603603.[1]
In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in The Lady's and Gentleman's Diary:
Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.
This problem became known as Kirkman's schoolgirl problem, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years.[1][2][3]
Pluquaternions
In 1848 Kirkman wrote "On Pluquaternions and Homoid Products of n Squares".[6] Generalizing the quaternions and octonions, Kirkman called a pluquaternion Qa a representative of a system with a imaginary units, a > 3. Kirkman's paper was dedicated to confirming Cayley's assertions concerning two equations among triple-products of units as sufficient to determine the system in case a = 3 but not a = 4.[7] By 1900 these number systems were called hypercomplex numbers, and later treated as part of the theory of associative algebras.
Polyhedral combinatorics
Beginning in 1853, Kirkman began working on combinatorial enumeration problems concerning polyhedra, beginning with a proof of Euler's formula and concentrating on simple polyhedra (the polyhedra in which each vertex has three incident edges). He also studied Hamiltonian cycles in polyhedra, and provided an example of a polyhedron with no Hamiltonian cycle, prior to the work of William Rowan Hamilton on the Icosian game. He enumerated cubic Halin graphs, over a century before the work of Halin on these graphs.[8] He showed that every polyhedron can be generated from a pyramid by face-splitting and vertex-splitting operations, and he studied self-dual polyhedra.[1][3]
Late work
Kirkman was inspired to work in group theory by a prize offered beginning in 1858 (but in the end never awarded) by the French Academy of Sciences. His contributions in this area include an enumeration of the transitive group actions on sets of up to ten elements. However, as with much of his work on polyhedra, Kirkman's work in this area was weighed down by newly invented terminology and, perhaps because of this, did not significantly influence later researchers.[1][3]
In the early 1860s, Kirkman fell out with the mathematical establishment and in particular with Arthur Cayley and James Joseph Sylvester, over the poor reception of his works on polyhedra and groups and over issues of priority. Much of his later mathematical work was published (often in doggerel) in the problem section of the Educational Times and in the obscure Proceedings of the Literary and Philosophical Society of Liverpool.[1] However, in 1884 he began serious work on knot theory, and with Peter Guthrie Tait published an enumeration of the knots with up to ten crossings.[3] He remained active in mathematics even after retirement, until his death in 1895.[3]
Awards and honours
In 1857, Kirkman was elected as a fellow of the Royal Society for his research on pluquaternions and partitions.[1] He was also an honorary member of the Literary and Philosophical Society of Manchester and the Literary and Philosophical Society of Liverpool, and a foreign member of the Dutch Society of Science.[2]
Since 1994, the Institute of Combinatorics and its Applications has handed out an annual Kirkman medal, named after Kirkman, to recognise outstanding combinatorial research by a mathematician within four years of receiving a doctorate.
Notes
1. Biggs, N. L. (1981), "T. P. Kirkman, mathematician", The Bulletin of the London Mathematical Society, 13 (2): 97–120, doi:10.1112/blms/13.2.97, MR 0608093.
2. Macfarlane, Alexander (1916), Lectures on Ten British Mathematicians of the Nineteenth Century, New York: John Wiley & Sons, Inc..
3. O'Connor, John J.; Robertson, Edmund F. (1996), "Thomas Penyngton Kirkman", MacTutor History of Mathematics Archive, University of St Andrews
4. Tahta, Dick (2006), The Fifteen Schoolgirls, Black Apollo Press, ISBN 1-900355-48-5.
5. Cameron, Peter J. (2002), "Steiner triple systems", Encyclopaedia of Design Theory.
6. London and Edinburgh Philosophical Magazine 1848, p 447 Google books link Archived 17 June 2014 at the Wayback Machine
7. A. J. Crilly (2006) Arthur Cayley: Mathematician Laureate of the Victorian Era, Johns Hopkins University Press, p. 143 on Kirkman's collaboration with Cayley
8. Kirkman, Th. P. (1856), "On the enumeration of x-edra having triedral summits and an (x − 1)-gonal base", Philosophical Transactions of the Royal Society of London: 399–411, doi:10.1098/rstl.1856.0018, JSTOR 108592.
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Thomas M. Liggett
Thomas Milton Liggett (March 29, 1944 – May 12, 2020) was a mathematician at the University of California, Los Angeles. He worked in probability theory, specializing in interacting particle systems.
Thomas Milton Liggett
Born(1944-03-29)March 29, 1944
Danville, Kentucky, U.S.
DiedMay 12, 2020(2020-05-12) (aged 76)
Los Angeles, California, U.S.
Alma materOberlin College (BA)
Stanford University (MS, PhD)
Spouse
Christina Marie Goodale
(m. 1972)
Children2
Scientific career
ThesisWeak Convergence of Conditioned Sums of Independent Random Vectors (1969)
Doctoral advisorSamuel Karlin
Doctoral studentsNorman Matloff
Early life
Thomas Milton Liggett was born on March 29, 1944, in Danville, Kentucky.[1] Liggett moved at the age of two with his missionary parents to Latin America, where he was educated in Bueno Aires, Argentina and San Juan, Puerto Rico. He graduated from Oberlin College with a Bachelor of Arts in 1965, where he was influenced towards probability by Samuel Goldberg (b. 1925), an ex-student of William Feller. He moved to Stanford, taking classes with Kai Lai Chung, and writing his thesis, Weak Convergence of Conditioned Sums of Independent Random Vectors, in 1969 with advisor Samuel Karlin on problems associated with the invariance principle. He graduated with a Master of Science in 1966 and a Doctor of Philosophy in 1969.[1][2][3]
Career
Liggett joined the faculty at UCLA in 1969, where he spent his entire career. He became a professor in the mathematics department in 1976, and served as department chair from 1991 to 1994. He retired in 2011, but remained active within the department.[4] He was the advisor of Norman Matloff.[3]
Liggett had contributed to numerous areas of probability theory, including subadditive ergodic theory, random graphs, renewal theory, and was best known for his pioneering work on interacting particle systems, including the contact process, the voter model, and the exclusion process.[5][6] His two books in this field have been influential.[7][8]
Liggett was the managing editor of the Annals of Probability from 1985–1987. He held a Sloan Research Fellowship from 1973–1977, and a Guggenheim Fellowship from 1997–1998. He was the Wald Memorial Lecturer of the Institute of Mathematical Statistics in 1996, and was elected to the National Academy of Sciences in 2008.[2][9] He had been elected to the American Academy of Arts & Sciences in 2012,[10] and in 2012 he also became a fellow of the American Mathematical Society.[11]
Personal life
Liggett married Christina Marie Goodale on August 19, 1972. They had two children, Timothy and Amy.[1] Liggett died on May 12, 2020, in Los Angeles.[4][12]
Notes
1. Vitale, Sarah A. (December 1992). Who's who in California. ISBN 978-1-880142-01-1.
2. "Tom Liggett's curriculum vitae".
3. Thomas M. Liggett at the Mathematics Genealogy Project
4. "In Memoriam: Thomas M. Liggett". www.math.ucla.edu. May 28, 2020. Archived from the original on March 24, 2022. Retrieved March 24, 2021.
5. "Tom Liggett's publications on Google Scholar".
6. "Publications of Tom Liggett since 2000".
7. Liggett, T.M. (1985). Interacting Particle Systems. Springer. ISBN 0-387-96069-4.
8. Liggett, T.M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer. ISBN 3-540-65995-1.
9. "UCLA Newsroom, 13 June 2009".
10. "Six UCLA professors elected to American Academy of Arts and Sciences".
11. List of Fellows of the American Mathematical Society, retrieved 2013-01-27.
12. Durrett, Rick (July 16, 2020). "Obituary: Thomas M. Liggett 1944–2020". imstat.org. Retrieved March 24, 2022.
External links
• Thomas M. Liggett at the Mathematics Genealogy Project
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Thomas J. Osler
Thomas Joseph Osler (April 26, 1940 – March 26, 2023) was an American mathematician, national champion distance runner, and author.
Thomas J. Osler
Osler at whiteboard in 2020
Born(1940-04-26)April 26, 1940
Camden, New Jersey, U.S.
DiedMarch 26, 2023(2023-03-26) (aged 82)
Alma mater
• Drexel University (BS)
New York University (PhD)
Scientific career
FieldsMathematics
InstitutionsRowan University
Early life and education
Born in 1940 in Camden, New Jersey,[1] Osler was a graduate of Camden High School in 1957 and then studied physics at Drexel University, graduating in 1962.[2][3] He completed his PhD at the Courant Institute of Mathematical Sciences of New York University,[4] in 1970. His dissertation, Leibniz Rule, the Chain Rule, and Taylor's Theorem for Fractional Derivatives, was supervised by Samuel Karp.[5]
Career
Osler taught at Saint Joseph's University and the Rensselaer Polytechnic Institute[6] before joining the mathematics department at Rowan University in New Jersey in 1972;[7] he was a full professor at Rowan University until his death.[4]
In mathematics, Osler is best known for his work on fractional calculus.[8][9][10] He also gave a series of product formulas for $\pi $ that interpolate between the formula of Viète and that of Wallis.[11]
In 2009, the New Jersey Section of the Mathematical Association of America gave him their Distinguished Teaching Award.[12][13] A mathematics conference was held at Rowan University in honor of his 70th birthday in 2010.[6]
Running
Osler won three national Amateur Athletic Union championships at 25 km (1965), 30 km and 50 mi (1967).[14][15] Osler won the 1965 Philadelphia Marathon, finishing the race in freezing-cold weather in a time of 2:34:07.[16] In the course of his career he has won races of nearly every length from one mile to 100 miles.
Osler was involved in the creation of the Road Runners Club of America with Olympian Browning Ross; together they were elected as co-secretaries in 1959[17] and were among the four first official elected officers of the newly formed club.[18] He served on the Amateur Athletic Union Standards Committee in 1979.[19] He has been credited with helping to popularize the idea of walk breaks among US marathon runners.[1][3]
In 1980, Osler was inducted into the Road Runners Club of America Hall of fame.[17][20]
Running publications
Osler was the author of several books and booklets on running:
• Guide to Long Distance Running (a 20-page booklet coauthored with Edward Dodd) was published in 1965 by the South Jersey Track Club.[21]
• The Conditioning of Distance Runners (a 29-page booklet) was published in 1967 by the Long Distance Log.[1][3][21] It was reprinted in 1984–1985 in Runner's World magazine[22][23] and reprinted with a new foreword by Amby Burfoot in 2019.[24]
• Serious Runner's Handbook: Answers to Hundreds of your Running Questions (187 pages) was published by World Publications in 1978.[25]
• Ultramarathoning: The Next Challenge (299 pages, coauthored with Edward Dodd) was also published by World Publications, in 1979.[26]
Personal life and death
Osler was a resident of Glassboro, New Jersey.[12]
Osler died on March 26, 2023, at the age of 82.[27]
References
1. Benyo, Richard; Henderson, Joe (2002). ""Tom Osler"". Running Encyclopedia. Human Kinetics. ISBN 0736037349.
2. "It All Adds Up: Running, teaching and math". Rowan Today. Rowan University. September 16, 2009.
3. Englehart, Richard (September 2008). "Like a Cat Chases Mice". Marathon & Beyond.
4. "Tom Osler, PhD". Faculty and Staff. Rowan University Mathematics Department. Retrieved March 27, 2023.
5. Thomas J. Osler at the Mathematics Genealogy Project
6. "Oslerfest: Prominent mathematicians to pay tribute to legendary Rowan prof". Rowan Today. Rowan University. April 12, 2010.
7. "Osler honored for distinguished teaching by Mathematical Association of America". Rowan Today. Rowan University. April 17, 2009.
8. Yang, Xiao-Jun; Gao, Feng; Ju, Yang (2020). "Section 2.3: Osler fractional calculus". General Fractional Derivatives with Applications in Viscoelasticity. Academic Press. pp. 107–111. ISBN 9780128172094.
9. Almeida, Ricardo (2019). "Further properties of Osler's generalized fractional integrals and derivatives with respect to another function". The Rocky Mountain Journal of Mathematics. 49 (8): 2459–2493. doi:10.1216/RMJ-2019-49-8-2459. hdl:10773/27488. MR 4058333. S2CID 214139065.
10. Nishimoto, Katsuyuki (1977). "Osler's cut and Nishimoto's cut". Journal of the College of Engineering of Nihon University, Series B. 18: 9–13. MR 0486359.
11. Arndt, Jörg; Haenel, Christoph (2001). "12.8 Viète ✕ Wallis = Osler". π Unleashed. Berlin Heidelberg New York: Springer-Verlag. pp. 160–162. ISBN 3-540-66572-2.
12. Shryock, Bob (May 7, 2009). "Running Man". South Jersey Times.
13. "New Jersey Section Archives". Mathematical Association of America. Retrieved November 23, 2020.
14. "United States Champions (with Local Connections)". Retrieved November 23, 2020.
15. United States Championships (Men). GBR Athletics. Retrieved November 25, 2020.
16. "Osler Captures Phila. Marathon", Asbury Park Press, December 27, 1965. Accessed November 24, 2020. "Philadelphia – Tom Osler of the South Jersey Track Club, 25-year-old New York University graduate student from Camden, N.J., scored an easy victory in the Ruthrauff Marathon race yesterday through Fairmount Park. Osier braved sub-freezing temperatures and stiff winds to cover the 26 miles, 385 yards in two hours, 34 minutes and seven seconds."
17. "History of Road Runners Club of America" (PDF). Road Runners Club of America. Retrieved November 24, 2020.
18. "50th Anniversary Report". Road Runners Club of America. Retrieved November 24, 2020.
19. "Pertinent Trivia" (PDF). Measurement News (88): 14. March 1988.
20. "Distance Running History". Road Runner's Club of America. Retrieved November 24, 2020.
21. Morison, Ray Leon (June 1975). An Annotated Bibliography of Track and Field Books Published in the United States Between 1960–1974 (PDF) (Master's thesis). San Jose State University. pp. 23, 33 – via Education Resources Information Center.
22. Osler, Tom (December 1984). "The Conditioning of Distance Runners (part 1)". Runner's World: 52–57, 87.
23. Osler, Tom (January 1985). "The Conditioning of Distance Runners (part 2)". Runner's World: 44–47, 80.
24. Osler, Thomas J. (1967). The Conditioning of Distance Runners (2019 ed.). Y42K Publishing. ISBN 9781710036725.
25. Osler, Tom (1978). Serious Runner's Handbook: Answers to Hundreds of Your Running Questions. Mountain View, California, USA: World Publications, Inc. ISBN 0-89037-126-1. Briefly reviewed in "Books". The Marine Corps Gazette. 1978. pp. 57–60; see in particular p. 59.
26. Osler, Tom; Dodd, Ed (1979). Ultramarathoning: The Next Challenge. Mountain View, California, USA: World Publications, Inc. ISBN 0-89037-169-5. See also Edwards, Sally (September 1983). "Ultramarathoning—A Dying Sport". UltraRunning Magazine. The book Ultramarathoning by Tom Osler and Ed Dodd had a shelf life of about 2 years, with 6,000 copies printed before the publisher (World Publications) discontinued it.
27. "Thomas J. Osler". South Jersey Times. March 27, 2023. Retrieved March 27, 2023 – via Legacy.com.
External links
• Thomas J. Osler publications indexed by Google Scholar
USA Championship winners in the men's 25K run
• 1925: Albert Michelsen
• 1926: Frank Titterton
• 1927: Jacob Kaysing
• 1928: Frank Titterton
• 1929–31: William Agee
• 1932: Juan Carlos Zabala (ARG)
• 1933: Dave Komonen (CAN)
• 1934: Eino Pentti
• 1935: Ellison Brown
• 1936: Lou Gregory
• 1937: Johnny A. Kelley
• 1938: Ellison Brown
• 1939: Lou Gregory
• 1940: Ellison Brown
• 1941–44: Johnny A. Kelley
• 1945: Charles Robbins
• 1946–47: Thomas Crane
• 1948: Victor Dyrgall
• 1949: Charles Robbins
• 1950: Jesse Van Zant
• 1951: Thomas Crane
• 1952: Browning Ross
• 1953: John DiComandrea
• 1954: Nick Costes
• 1955: Browning Ross
• 1956–59: Johnny J. Kelley
• 1960–61: Pete McArdle
• 1962: Orville Flynn
• 1963: Pete McArdle
• 1964: Johnny J. Kelley
• 1965: Thomas J. Osler
• 1966: Ed Winrow
• 1967–68: Kerry Pearce
• 1969: Skip Houk
• 1970: Moses Mayfield
• 1971: Robert Fitts
• 1972–73: Paul Talkington
• 1974: Ed Mendoza
• 1975: Steve Hoag
• 1976: Frank Shorter
• 1977–78: Duncan MacDonald
• 1979: Stan Vernon
• 1980: Gary Tuttle
• 1981: ?
• 1982: Frank Plasso
• 1983: Vincent Fleming
• 1984–86: ?
• 1987: Greg Meyer
• 1988–94: Not held
• 1995: Keith Brantly
• 1996: Alfredo Vigueras
• 1997–98: John Sence
• 1999: Todd Williams
• 2000: Shawn Found
• 2001: Chad Johnson
• 2002: David Morris
• 2003–04: Dan Browne
• 2005: Brian Sell
• 2006: Fernando Cabada
• 2007–08: Brian Sell
• 2009: Dan Browne
• 2010: Andrew Carlson
• 2011: Fernando Cabada
• 2012: Joseph Chirlee
• 2013: Josphat Boit
• 2014: Christo Landry
• 2015: Jared Ward
• 2016: Christo Landry
• 2017: Dathan Ritzenhein
• 2018: Sam Chelanga
• 2019: Parker Stinson
• 2020: Not held
Distance was 15 miles from 1925 to 1932
USA Championship winners in the men's 30K run
• 1930–31: Fred Ward
• 1932: William Steiner
• 1933: Juan Carlos Zabala (ARG)
• 1934: Lou Gregory
• 1935: Les Pawson
• 1936: Pat Dengis
• 1937: Mel Porter
• 1938: Les Pawson
• 1939: Lou Gregory
• 1940: Barney Gedwillas
• 1941: Lou Gregory
• 1942: Don Heinecke
• 1943: John Connolly
• 1944: Fred Kline
• 1945: Charles Robbins
• 1946–47: William Steiner
• 1948: Victor Dyrgall
• 1949: Jesse Van Zant
• 1950: Kim Valentine
• 1951: Victor Dyrgall
• 1952: Browning Ross
• 1953: Not held
• 1954–55: Browning Ross
• 1956–57: Ted Corbitt
• 1958: Robert Carman
• 1959: Alex Breckenridge
• 1960–62: Pete McArdle
• 1963: Norm Higgins
• 1964: Hal Higdon
• 1965: Lou Castagnola
• 1966: Ed Winrow
• 1967: Thomas J. Osler
• 1968–69: ?
• 1970: Eamon O'Reilly
• 1971: Michael Kimball
• 1972: Paul Talkington
• 1973: Robert Fitts
• 1974: Reid Harter
• 1975: John Vitale
• 1976: Bill Rodgers
• 1977: Tom Fleming
• 1978: John Vitale
• 1979: Barry Brown
• 1980: John Ziegler
• 1981: Pete Pfitzinger
• 1982: Phil Coppess
• 1983: Barry Brown
• 1984: Don Norman
• 1985: Marty Froelick
• 1986–87: ?
• 1988: Herb Wills
Distance was 20 miles from 1930 to 1932
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Thomas Gerald Room
Thomas Gerald Room FRS FAA (10 November 1902 – 2 April 1986) was an Australian mathematician who is best known for Room squares. He was a Foundation Fellow of the Australian Academy of Science.[1][2]
FRS FAA
Thomas Gerald Room
Born(1902-11-02)2 November 1902
Died2 April 1986(1986-04-02) (aged 83)
NationalityAustralian
Alma materSt John's College, Cambridge
Scientific career
FieldsMathematics
InstitutionsUniversity of Sydney
Biography
Thomas Room was born on 10 November 1902, near London, England. He studied mathematics in St John's College, Cambridge, and was a wrangler in 1923. He continued at Cambridge as a graduate student, and was elected as a fellow in 1925, but instead took a position at the University of Liverpool. He returned to Cambridge in 1927, at which time he completed his PhD, with a thesis supervised by H. F. Baker.[3][4] Room remained at Cambridge until 1935, when he moved to the University of Sydney, where he accepted the position of Chair of the Mathematics Department, a position he held until his retirement in 1968.[5]
During World War II he worked for the Australian government, helping to decrypt Japanese communications. In January 1940, with the encouragement of the Australian Army, he, together with some colleagues at the University of Sydney, began to study Japanese codes. The others were the mathematician Richard Lyons and the classicists Arthur Dale Trendall and Athanasius Treweek. By this time Room had already begun learning Japanese under Margaret Ethel Lake (1883-?) at the University of Sydney. In May 1941 Room and Treweek attended a meeting at the Victoria Barracks in Melbourne with the Director of Naval Intelligence of the Royal Australian Navy, several Australian Army intelligence officers and Eric Nave, an expert Japanese cryptographer with the Royal Australian Navy. As a result it was agreed that Room's group, with the agreement of the University of Sydney, would move in August 1941 to work under Nave at the Special Intelligence Bureau in Melbourne. On 1 September 1941, Room was sent to the Far East Combined Bureau in Singapore to study the codebreaking techniques used there. After the outbreak of war they were working for FRUMEL (Fleet Radio Unit Melbourne), a joint American-Australian intelligence unit, but when Lieutenant Rudolph Fabian took over command of FRUMEL and particularly when, in October 1942, FRUMEL was placed under direct control of the US Navy, civilians such as the member of Room's group were found surplus to requirements and returned to their academic posts. [6][7]
After the war, Room served as dean of the faculty of science at the University of Sydney from 1952 to 1956 and again from 1960 to 1965.[8] He also held visiting positions at the University of Washington in 1948, and the Institute for Advanced Study and Princeton University in 1957.[9][10][11] He retired from Sydney in 1968 but took short-term positions afterwards at Westfield College in London and the Open University before returning to Australia in 1974. He died on 2 April 1986.
Room married Jessica Bannerman, whom he met in Sydney, in 1937; they had one son and two daughters.[12][13]
Research
Room's PhD work concerned generalizations of the Schläfli double six, a configuration formed by the 27 lines on a cubic algebraic surface.[1][4]
In 1938 he published the book The geometry of determinantal loci through the Cambridge University Press.[1] Nearly 500 pages long, the book combines methods of synthetic geometry and algebraic geometry to study higher-dimensional generalizations of quartic surfaces and cubic surfaces. It describes many infinite families of algebraic varieties, and individual varieties in these families, following a unifying principle that nearly all loci arising in algebraic geometry can be expressed as the solution to an equation involving the determinant of an appropriate matrix.[1][14]
In the postwar period, Room shifted the focus of his work to Clifford algebra and spinor groups.[1] Later, in the 1960s, he also began investigating finite geometry, and wrote a textbook on the foundations of geometry.[1]
Room invented Room squares in a brief note published in 1955.[15] A Room square is an n × n grid in which some of the cells are filled by sets of two of the numbers from 0 to n in such a way that each number appears once in each row or column and each two-element set occupies exactly one cell of the grid. Although Room squares had previously been studied by Robert Richard Anstice,[16] Anstice's work had become forgotten and Room squares were named after Room. In his initial work on the subject, Room showed that, for a Room square to exist, n must be odd and cannot equal 3 or 5. It was later shown by W. D. Wallis in 1973 that these are necessary and sufficient conditions: every other odd value of n has an associated Room square. The nonexistence of a Room square for n = 5 and its existence for n = 7 can both be explained in terms of configurations in projective geometry.[1]
Despite retiring in 1968, Room remained active mathematically for several more years, and published the book Miniquaternion geometry: An introduction to the study of projective planes in 1971 with his student Philip B. Kirkpatrick.[1]
Awards and honours
In 1941, Room won the Thomas Ranken Lyle Medal of the Australian National Research Council and was elected as a Fellow of the Royal Society.[1][17][18] He was one of the Foundation Fellows of the Australian Academy of Science, chartered in 1954.[1][2] From 1960 to 1962, he served as president of the Australian Mathematical Society and he later became the first editor of its journal.[1]
The T. G. Room award of the Mathematical Association of New South Wales, awarded to the student with the best score in the NSW Higher School Certificate Mathematics Extension 2 examination, is named in Room's honour.[1][19]
References
1. Hirschfeld, J. W. P.; Wall, G. E. (1987). "Thomas Gerald Room. 10 November 1902 – 2 April 1986". Biographical Memoirs of Fellows of the Royal Society. 33: 575–601. doi:10.1098/rsbm.1987.0020. JSTOR 769963. S2CID 73328766.. Also published in Historical Records of Australian Science 7 (1): 109–122, doi:10.1071/HR9870710109. An abridged version is online at the web site of the Australian Academy of Science.
2. John Mack. Room, Thomas Gerald (1902–1986). {{cite book}}: |work= ignored (help) First published in Australian Dictionary of Biography, Volume 18, (MUP), 2012.
3. Hirschfeld, J. W. P.; Wall, G. E. (1987). "Thomas Gerald Room. 10 November 1902 – 2 April 1986", Biographical Memoirs of Fellows of the Royal Society 33: 574. doi:10.1098/rsbm.1987.0020. JSTOR 769963.. Also published in Historical Records of Australian Science 7 (1): 109–122, doi:10.1071/HR9870710109. An abridged version is online at the web site of the Australian Academy of Science.
4. Thomas Gerald Room at the Mathematics Genealogy Project
5. "The University. Chair of Mathematics. Professor T. G. Room", The Sydney Morning Herald, 21 December 1934.
6. Peter Kornicki, Eavesdropping on the Emperor: Interrogators and Codebreakers in Britain's War with Japan (London: Hurst & Co., 2021), pp. 209-211, 216-7.
7. Peter Donovan, and John Mack, ‘Sydney University, T. G. Room and codebreaking in WW II’, Australian mathematical society gazette 29 (2002): 76-85, 141-8.
8. Hirschfeld, J. W. P.; Wall, G. E. (1987), "Thomas Gerald Room. 10 November 1902 – 2 April 1986". Biographical Memoirs of Fellows of the Royal Society 33: 574. doi:10.1098/rsbm.1987.0020. JSTOR 769963. Also published in Historical Records of Australian Science 7 (1): 109–122, doi:10.1071/HR9870710109. An abridged version is online at the web site of the Australian Academy of Science.
9. Hirschfeld, J. W. P.; Wall, G. E. (1987). "Thomas Gerald Room. 10 November 1902 – 2 April 1986". Biographical Memoirs of Fellows of the Royal Society 33: 574. doi:10.1098/rsbm.1987.0020. JSTOR 769963.. Also published in Historical Records of Australian Science 7 (1): 109–122, doi:10.1071/HR9870710109. An abridged version is online at the web site of the Australian Academy of Science.
10. "Princeton Appoints 17 Guest Professors", The New York Times, 4 September 1957.
11. "Institute Names 128 For Research; Scholars Will Do Advanced Study on Historical Topics And in Mathematics", The New York Times, 15 September 1957.
12. Hirschfeld, J. W. P.; Wall, G. E. (1987). "Thomas Gerald Room. 10 November 1902 – 2 April 1986". Biographical Memoirs of Fellows of the Royal Society 33: 574. doi:10.1098/rsbm.1987.0020. JSTOR 769963.. Also published in Historical Records of Australian Science 7 (1): 109–122, doi:10.1071/HR9870710109. An abridged version is online at the web site of the Australian Academy of Science.
13. "Professor and Bride Dodge Rice", The Sydney Morning Herald, 8 November 1937.
14. Review of The geometry of determinantal loci by Virgil Snyder (1939), Bulletin of the AMS 45: 499–501, doi:10.1090/S0002-9904-1939-07011-0.
15. Room, T. G. (1955), "A new type of magic square", Mathematical Gazette, 39: 307, doi:10.2307/3608578, JSTOR 3608578, S2CID 125711658.
16. O'Connor, John J.; Robertson, Edmund F., "Robert Anstice", MacTutor History of Mathematics Archive, University of St Andrews.
17. "Lyle Medals Awarded", The Sydney Morning Herald, 10 July 1941.
18. Thomas Ranken Lyle Medal Archived 28 November 2010 at the Wayback Machine, Australian Academy of Science. Retrieved 6 June 2010.
19. The T G Room Award Archived 4 August 2012 at archive.today, Mathematical Association of New South Wales. Retrieved 1 June 2010.
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Thomas Royen
Thomas Royen (born 6 July 1947) is a retired German professor of statistics who has been affiliated with the University of Applied Sciences Bingen. Royen came to prominence in the spring of 2017 for a relatively simple proof for the Gaussian Correlation Inequality (GCI), a conjecture that originated in the 1950s, which he had published three years earlier without much recognition.[1] A proof of this conjecture, which lies at the intersection of geometry, probability theory and statistics, had eluded top experts for decades.[2]
Thomas Royen
Born
Thomas Royen
(1947-07-06) 6 July 1947
Frankfurt am Main, Germany
CitizenshipGermany
Alma materGoethe University Frankfurt
University of Freiburg
Technical University Dortmund (PhD)
Known forProof of Gaussian correlation inequality
Scientific career
FieldsMathematics, Statistics
ThesisOn Convergence Against Stable Laws (1975)
Life
Youth, studies and private life
Royen was born in 1947 to Paul Royen, a professor with the institute for inorganic chemistry at the Goethe University Frankfurt and Elisabeth Royen, also a chemist. From 1966 to 1971, he studied mathematics and physics at his father's university and the University of Freiburg. After graduating, he worked as a tutor at the University of Freiburg, before transferring to the Technical University of Dortmund for his doctoral thesis. After attaining his PhD in 1975 with a thesis called Über die Konvergenz gegen stabile Gesetze (On Convergence Against Stable Laws), he worked as a Wissenschaftlicher Assistent at Dortmund University's institute for statistics. Married with children, Royen lives in Schwalbach am Taunus.
Career
In 1977, Royen started working as a statistician for the pharmaceutical company Hoechst AG. From 1979 to 1985, he worked at the company's own educational facility teaching mathematics and statistics. Starting in 1985 until becoming an emeritus in 2010, he taught statistics and mathematics at the University of Applied Sciences Bingen in Rhineland-Palatinate.[3]
Royen worked mainly on probability distributions, in particular multivariate chi-squares and gamma distributions, to improve some frequently used statistical test procedures. Nearly half of his circa 30 publications were written when he was aged over sixty. Because he was annoyed over some contradictory reviews and in a few cases also over the incompetence of a referee, he decided in his later years, when his actions had no influence anymore on his further career, to publish his papers on the online platform arXiv.org and sometimes in a less renowned Indian journal to fulfill, at least formally, the condition of a peer review.[4]
Proof of the Gaussian correlation inequality
On 17 July 2014, a few years after his retirement, when brushing his teeth, Royen had a flash of insight: how to use the Laplace transform of the multivariate gamma distribution to achieve a relatively simple proof for the Gaussian correlation inequality, a conjecture on the intersection of geometry, probability theory and statistics, formulated after work by Dunnett and Sobel (1955) and the American statistician Olive Jean Dunn (1958),[5] that had remained unsolved since then. He sent a copy of his proof to Donald Richards, an acquainted American mathematician, who had worked on a proof of the GCI for 30 years. Richards immediately saw the validity of Royen's proof and subsequently helped him to transform the mathematical formulas into LaTeX. When Royen contacted other reputed mathematicians, though, they didn't bother to investigate his proof, because Royen was relatively unknown, and these mathematicians therefore estimated the chance that Royen's proof would be false as very high.[2]
Royen published this proof in an article with the title A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions on arXiv[6] and subsequently in the Far East Journal of Theoretical Statistics,[7] a relatively unknown periodical based in Allahabad, India, for which Royen was at the time voluntarily working as a referee himself. Due to this, his proof went at first largely unnoticed by the scientific community,[8] until in late 2015 two Polish mathematicians, Rafał Latała and Dariusz Matlak, wrote a paper in which they reorganized Royen's proof in a way that was intended to be easier to follow.[1] In July 2015, Royen supplemented his proof with a further paper in arXiv Some probability inequalities for multivariate gamma and normal distributions.[9]
A 2017 article by Natalie Wolchover about Royen's proof in Quanta Magazine resulted in greater academic and public recognition for his achievement.[10][11]
References
1. Rafal Latala and Dariusz Matlak, Royen's proof of the Gaussian correlation inequality, arXiv:1512.08776
2. "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Natalie Wolchover. 28 March 2017. Retrieved 1 May 2017.
3. "Curriculum Vitae Thomas Royen". Forschungsplattform des Landes Rheinland-Pfalz. Landesregierung Rheinland-Pfalz.
4. "Der Beweis" (in German). Sibylle Anderl. 7 April 2017.
5. Dunn, Olive Jean (March 1959). "Estimation of the medians for dependent variables". The Annals of Mathematical Statistics. 30 (1): 192–197. doi:10.1214/aoms/1177706374. JSTOR 2237135.
6. Royen, T. (2014). "A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions". arXiv:1408.1028 [math.PR]. Supplemented by Royen, Thomas (2015). "Some probability inequalities for multivariate gamma and normal distributions". arXiv:1507.00528 [math.PR].
7. Thomas Royen: A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, in: Far East Journal of Theoretical Statistics, Part 48 Nr. 2, Pushpa Publishing House, Allahabad 2014, p.139–145
8. In the Quanta magazine article, for instance, Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved.
9. Royen, Thomas (2 July 2015). "Some probability inequalities for multivariate gamma and normal distributions". arXiv:1507.00528 [math.PR].
10. Crew, Bec (4 April 2017). "This German Retiree Solved One of World's Most Complex Maths Problems - And No One Noticed". ScienceAlert. Retrieved 26 December 2020. I know of people who worked on it for 40 years," Donald Richards, a statistician from Pennsylvania State University, told Natalie Wolchover at Quanta Magazine. "I myself worked on it for 30 years.
11. Dambeck, Holger (4 April 2017). "67-Jähriger löst altes Statistikproblem" [67-year-old solves old statistics problem]. Der Spiegel (in German). Retrieved 26 December 2020. "Ich kenne Leute, die daran 40 Jahre gearbeitet haben", sagte er dem "Quanta Magazine".
External links
• Profile of Thomas Royen on Science Portal Rheinland-Pfalz (RLP)
• Thomas Royen, "A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions", arXiv:1408.1028
• Thomas Royen, "Some probability inequalities for multivariate gamma and normal distributions", arXiv:1507.00528
• Thomas Royen, "A note on the existence of the multivariate gamma distribution", arXiv:1606.04747
• (in French) Bourbaki, Frank Barthe on YouTube. January 14, 2017
• Informal interview with Thomas Royen at University of Applied Sciences Bingen on YouTube (in German)
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Thomas Rudd
Thomas Rudd (c.1583–1656) was an English military engineer and mathematician.
Life
The eldest son of Thomas Rudd of Higham Ferrers, Northamptonshire, he was born in 1583 or 1584. He served during his earlier years as a military engineer in the Low Countries. On 10 July 1627, King Charles I appointed him ‘chief engineer of all castles, forts, and fortifications within Wales,’ at a salary of £240 per annum. Subsequently, he was appointed the King's principal engineer for fortifications, and in 1635 he visited Portsmouth in this capacity to settle a question between the governor and the admiralty as to the removal of some naval buildings which interfered with proposed fortifications. In 1638, he visited Guernsey and Jersey at the request of the governors, Charles Danvers, Earl of Danby and Sir Thomas Jermyn, to survey the castles on those islands and report upon them to the board of ordnance.
In February of the following year, Rudd petitioned the board of ordnance for the payment of arrears of salary; in June, the board recommended the petition to the council, mentioning Rudd's services. In April, having been employed in making a survey of the Portsmouth defences, he recommended that they should be reconstructed at an estimated cost of £4,956. In June, Rudd went to Dover to superintend the repairs to the harbour and to the Archcliffe bulwark or fort, and in October he reported to the council that the works were delayed for want of funds, and suggested that the revenues of the harbour, as well as the dues, should be devoted to the maintenance of the harbour and fort. To this, the council assented on 29 May 1640, and on 31 December following directed all mayors, sheriffs, and justices to impress workmen in and about London and elsewhere for the works at Dover, which had been entrusted to Rudd.
In October 1640, Rudd went to Portsmouth to finish the fortifications, on the special application of Colonel Goring, the governor, and he divided his attention during 1641 between Portsmouth and Dover. The work at Portsmouth was retarded for want of funds, and in January 1642 the governor demanded stores, and leave to use materials for fortification, according to Rudd's survey of the previous year.
Rudd served as chief engineer on the Royalist side throughout the First English Civil War, and in 1655, his estate at Higham Ferrers was decimated on an assessment for the payment of the militia, as a punishment for his adherence to the Royalist cause. He died in 1656, aged 72, and was buried in Higham Ferrers church, where several epitaphs composed by himself were inscribed on his tomb.
Works
Rudd put his name to two texts on geometry, Practical Geometry, in two parts (London, 1650), and an edition of Euclid's Elements under the title Euclides Elements of Geometry, the first six Books in a compendious form contrasted and demonstrated, whereunto is added the Mathematical Preface of Mr. John Dee (London, 1651), but both works show extensive appropriation (without attribution) from Dutch sources of the early 1600s. In particular, Rudd's selection of a hundred questions is largely, but not exclusively, culled from the compilation of Sybrandt Hanszoon van Harlingen (Cardinael) (1578–1647), Hondert Geometrische Questien [A Hundred Geometrical Problems], published c. 1612.[1]
He wrote the supplement to The Compleat Body of the Art Military, by Lieutenant-colonel Richard Elton, London, 1650; 2nd edit. 1659. This supplement consists of six chapters, dealing with the duties of officers, the marching of troops and the art of gunnery. Sir James Turner, in his Pallas Armata (1683), refers to another work by Rudd on sieges; but this cannot now be traced.
• T. Rudd, Practical Geometry, in Two Parts: The first, Shewing how to perform four Species of Arithmetick, (viz. Addition, Subtraction, Multiplication, and Division,) together with Reduction, and the Rule of Proportion in figures. The Second, Containing A Hundred Geometrical Questions, with their Solutions and Demonstrations, some of them being performed Arithmetically, and others Geometrically, yet all without the help of algebra. A Worke very necessary for all Men, but principally for Surveyors of Land, Engineers, and all other Students in the Mathematicks. (Printed by Robert Leybourn for Robert Boydell and Samuel Satterthwaite, London, 1650); available at Early English Books.
Occult
Rudd has been claimed as an occultist.[2] Peter J. French writes that he was "steeped in hermeticism" and an admirer of Dee's Monas Hieroglyphica. Among the Harleian manuscripts is a hermetic treatise that has been attributed to Rudd.[3] According to The Goetia of Dr. Rudd by occult author Stephen Skinner and David Rankine, Rudd was at the centre of a group of angel magicians.
Family
Rudd was married three times:
• first, to Elizabeth, daughter of Robert Castle of Glatton, Huntingdonshire;
• secondly, to Margaret, daughter of Edward Doyley of Overbury Hall, Suffolk;
• and thirdly, to Sarah, daughter of John Rolt of Milton Ernes, Bedfordshire.
He left an only daughter, Judith, by his third wife; she married, first a kinsman, Anthony Rudd, and secondly, Goddard Pemberton, and died on 23 March 1680.
References
• "Rudd, Thomas" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
• M. H. Sitters, Sybrandt Hansz. Cardinael (1578-1647). Een eigenzinnig meetkundige tussen Stevin en Huygens
Notes
1. R. G. Archibald, 'Thomas Rudd and Sybrandt Cardinael's Hondert geometrische Questien', Nieuw Archief voor Wiskunde 11 (1915), 191–5.
2. Keys to the Gateway of Magic, Stephen Skinner & David Rankine, Golden Hoard Press, 2005
3. Peter J. French, John Dee: The World of an Elizabethan Magus (1984), note p. 172. Cf. British Library Harley MSS 6482-6483
External links
• Angel Magic by Vincent Bridges
• "Rudd, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/24247. (Subscription or UK public library membership required.)
Attribution
This article incorporates text from a publication now in the public domain: "Rudd, Thomas". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
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Thomas Simpson Evans
Thomas Simpson Evans (1777–1818) was a British mathematician.
Life
Evans, eldest son of the Rev. Lewis Evans (1755–1827), by his wife, Ann Norman, was baptised in August 1777. He was named after Thomas Simpson, the mathematician.[1]
In or about 1797 Evans appears to have taken charge of a private observatory at Blackheath belonging to William Larkins, formerly accountant-general to the East India Company in Bengal. After the death of Larkins, 24 April 1800, he was taken on as an assistant by Nevil Maskelyne at the Royal Observatory, Greenwich, but resigned the post in 1805.[1]
In that year, or perhaps in 1803, Evans was appointed mathematical master under his father at the Royal Military Academy, Woolwich. Here he continued until 1810, when he accepted the mastership of the mathematical school at New Charlton, near Woolwich, which office he vacated in 1813 to become master of the mathematics at Christ's Hospital, London. His attainments won for him the degree of LL.D. (from what university is not known) and the fellowship of the Linnean Society.[1]
Evans died 28 October 1818, aged 41.[1]
Works
Evans left a completed translation of Antonio Cagnoli's Trigonometria piana e sferica, besides other translations of scientific works and a collection of unfinished papers in several branches of philosophy. He also contributed some articles to the Philosophical Magazine, among which were:[1]
• "Problems on the Reduction of Angles" (vol. xxviii.);
• "An Abridgment of the Life of Julien Le Roy, the Watchmaker, by his Son" (vol. xxxi.);
• "A Short Account of Improvements gradually made in determining the Astronomic Refraction" (vol. xxxvi.);
• "Historical Memoranda respecting Experiments intended to ascertain the Calorific Powers of the different Prismatic Rays" (vol. xlv.);
• "On the Laws of Terrestrial Magnetism in different Latitudes" (vol. xlix.).
Evans's library was considered a valuable collection of mathematical and philosophical works.[1]
Family
By his marriage in 1797 to Deborah, daughter of John Mascall of Ashford, Kent, Evans had five children:
• Thomas Simpson Evans (1798–1880), vicar of St Leonard's, Shoreditch;
• Aspasia Evans (1799–1876), a spinster;
• Herbert Norman Evans, M.D. (1802–1877), book collector;
• Arthur Benoni Evans (d. 1838); and
• Lewis Evans (1815–1869), head-master of Sandbach Free Grammar School, Cheshire.[1]
References
1. "Evans, Thomas Simpson" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
Attribution
This article incorporates text from a publication now in the public domain: "Evans, Thomas Simpson". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
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Thomas W. Tucker
Thomas William Tucker (born July 15, 1945) is an American mathematician, the Charles Hetherington Professor of Mathematics at Colgate University,[1] and an expert in the area of topological graph theory.[2][3]
Tucker did his undergraduate studies at Harvard University, graduating in 1967,[1] and obtained his Ph.D. from Dartmouth College in 1971, under the supervision of Edward Martin Brown.[4]
Tucker's father, Albert W. Tucker, was also a professional mathematician, and his brother, Alan Tucker, and son, Thomas J. Tucker, are also professional mathematicians.
References
1. Faculty web page, retrieved 2014-10-29.
2. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley Interscience, 1987
3. Thomassen, Carsten (1988). "Review: Topological Graph Theory, by Jonathan L. Gross and Thomas W. Tucker" (PDF). Bull. Amer. Math. Soc. (N.S.). 19 (2): 560–561. doi:10.1090/s0273-0979-1988-15742-4.
4. Thomas William Tucker at the Mathematics Genealogy Project
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Thomas W. Hungerford
Thomas William Hungerford (March 21, 1936 – November 28, 2014)[1] was an American mathematician who worked in algebra and mathematics education. He is the author or coauthor of several widely used[2] and widely cited[3] textbooks covering high-school to graduate-level mathematics. From 1963 until 1980 he taught at the University of Washington and then at Cleveland State University until 2003. From 2003–2014 he was at Saint Louis University. Hungerford had a special interest in promoting the use of technology to teach mathematics.
Thomas W. Hungerford
Born(1936-03-21)March 21, 1936
Oak Park, Illinois, US
DiedNovember 28, 2014(2014-11-28) (aged 78)
St. Louis, Missouri, US
NationalityAmerican
Alma materCollege of the Holy Cross
University of Chicago
Scientific career
FieldsAlgebra
Education
InstitutionsUniversity of Washington,
Cleveland State University,
St. Louis University
Doctoral advisorSaunders Mac Lane
Hungerford did his undergraduate work at the College of the Holy Cross and defended his Ph.D. thesis at the University of Chicago in 1963 (advised by Saunders Mac Lane). Throughout his career he wrote more than a dozen widely used mathematics textbooks, ranging from high school to graduate level.[1]
Bibliography
Graduate
• 1974 Algebra (Graduate Texts in Mathematics #73). Springer Verlag. ISBN 3-540-90518-9
Undergraduate
• 1997 Abstract Algebra: An Introduction, 2nd Edition. Cengage. ISBN 0-03-010559-5
• 2005 Contemporary College Algebra and Trigonometry, 2nd Edition. Cengage. ISBN 0-534-46665-6
• 2005 Contemporary College Algebra, 2nd Edition. Cengage. ISBN 0-534-46656-7
• 2006 Contemporary Trigonometry. Cengage. ISBN 0-534-46638-9
• 2009 Contemporary Precalculus, 5th Edition (with Douglas J. Shaw). Cengage. ISBN 0-495-55441-3
• 2011 Mathematics with Applications, 10th Edition (with Margaret L. Lial and John P. Holcomb, Jr). Pearson. ISBN 0-321-64632-0
• 2011 Finite Mathematics with Applications, 10th Edition (with Margaret L. Lial and John P. Holcomb, Jr). Pearson. ISBN 0-321-64554-5
• 2013 Abstract Algebra: An Introduction, 3rd Edition, Cengage. ISBN 1-111-56962-2
High school
• 2002 Precalculus: A Graphing Approach (with Irene Jovell and Betty Mayberry). Holt, Rinehart & Winston. ISBN 0-03-056511-1
References
1. Batesville, Inc. "Obituary for Thomas W. Hungerford at Collier's Funeral Home". www.colliersfuneralhome.com. {{cite web}}: |first= has generic name (help)
2. Algebra (Graduate Texts in Mathematics) at goodreads
3. Citations at docjax.net the document search engine
External links
• Thomas W. Hungerford at the Mathematics Genealogy Project
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Thomas Willwacher
Thomas Hans Willwacher (born 12 April 1983) is a German mathematician and mathematical physicist working as a Professor at the Institute of Mathematics, ETH Zurich.[1]
Thomas H. Willwacher
Born (1983-04-12) 12 April 1983
Freiburg im Breisgau, West Germany
NationalityGerman
Alma materETH Zurich
AwardsAndré Lichnerowicz Prize (2012)
EMS Prize (2016)
Scientific career
FieldsMathematical physics
InstitutionsETH Zurich
ThesisCyclic formality (2009)
Doctoral advisorGiovanni Felder
Biography
Willwacher completed his PhD at ETH Zurich in 2009 with a thesis on "Cyclic Formality", under the supervision of Giovanni Felder, Alberto Cattaneo, and Anton Alekseev.[2] He was later a Junior member of the Harvard Society of Fellows.
In July 2016 Willwacher was awarded a prize from the European Mathematical Society for "his striking and important research in a variety of mathematical fields: homotopical algebra, geometry, topology and mathematical physics, including deep results related to Kontsevich's formality theorem and the relation between Kontsevich's graph complex and the Grothendieck-Teichmüller Lie algebra".[3][4]
Notable results of Willwacher include the proof of Maxim Kontsevich's cyclic formality conjecture and the proof that the Grothendieck–Teichmüller Lie algebra is isomorphic to the degree zero cohomology of Kontsevich's graph complex.
References
1. "Prof. Dr. Thomas Willwacher". ETH Zurich. ETH Zurich. Retrieved 25 July 2016.
2. Thomas Willwacher at the Mathematics Genealogy Project
3. "7 ECM Berlin: Twelve prizes awarded". European Mathematical Society. 18 July 2016. Retrieved 31 August 2016.
4. "7ECM — Laureates". 7th European Congress of Mathematics. European Mathematical Society. July 2016. Retrieved 31 August 2016.
External links
• "Homepage at ETH Zurich".
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Thomas Allen (mathematician)
Thomas Allen (or Alleyn) (21 December 1542 – 30 September 1632) was an English mathematician and astrologer. Highly reputed in his lifetime, he published little, but was an active private teacher of mathematics. He was also well connected in the English intellectual networks of the period.[1]
Early life
He was born in Uttoxeter, Staffordshire. He was admitted scholar of Trinity College, Oxford, in 1561; and graduated as M.A. in 1567. In 1571 he left his college and fellowship, and moved to Gloucester Hall.[1] He became known for his knowledge of antiquity, philosophy, and mathematics.[2]
At Gloucester Hall
Gloucester Hall suited Allen, a sympathiser at least with Catholicism, because there was no stringent religious observance required there; indeed there was no chapel in the Hall.[3] Allen's beliefs have been classified as "church papist", but also his posture as "crypto-Catholic": a Catholic faith combined with outward conformity to the Church of England.[4][5] He joined there his friends Edmund Reynolds, Miles Windsor, and George Napper, who had also left their colleges at a time of increasing religious tensions on Oxford; Napper was to be a Catholic martyr. Trinity shed six more of its Fellows within a few years.[6][7]
Allen encouraged other scholars to migrate there, such as John Budden[8] and William Burton.[9] He had a wide range of pupils and followers: Kenelm Digby[3] and Brian Twyne[10] in natural philosophy, with Theodore Haak coming later.[11] The mathematical school of Allen included Thomas Harriot and Walter Warner,[12] and Sir John Davies (to whom Allen taught Catholic doctrine).[13]
Mathematical geography was an important topical subject in which Allen was reputed, pursued by several groups in England, including another around Henry Briggs: Allen may have taught the geographer Richard Hakluyt. He did teach Robert Fludd and Sir Thomas Aylesbury.[14] In the humanities there were Robert Hegge,[15] and William Fulbecke.[16] When the Camden Chair of Ancient History was being set up in the early 1620s, Allen successfully supported the candidacy of Degory Wheare with William Camden; and a few years later, in 1626, Wheare came to Gloucester Hall as Principal.[17]
Allen died at Gloucester Hall.
Other associations
Allen corresponded with Henry Percy, 9th Earl of Northumberland.[14] Northumberland invited Allen to visit, and he spent some time with the Syon House group around the Earl; he became acquainted with Thomas Harriot, John Dee, and other mathematicians. He also knew well Sir Robert Cotton, William Camden, and their antiquarian associates.[2] He pointed out the historian Æthelweard (Fabius Quaestor) to Camden.[18]
Astrologer
Allen was noted as astrologer to Robert Dudley, 1st Earl of Leicester, as Dee was for Queen Elizabeth.[19] Foster believes Allen probably met Leicester through Lady Paulet, wife of Hugh Paulet, née Elizabeth Blount, who was the widow of Sir Thomas Pope, the founder of Trinity College, Oxford.[20]
There is a surviving 62-page horoscope cast for the teenage Philip Sidney in the Ashmole manuscripts in the Bodleian Library, in the period 1570–2 when he was studying at Oxford, where Leicester was Chancellor, and it has been attributed to Allen; the case has also been made that it was by Dee. A link between the two is that Edward Kelley is said to have worked briefly for Allen.[21][22] Allen definitely cast a natal horoscope for Robert Pierrepont (1584), and cast also for William Herbert, 3rd Earl of Pembroke, a later Chancellor of Oxford, in 1626.[23]
Reputation
Allen's skill in mathematics and astrology earned him the credit of being a magician. In an incident related in John Aubrey's Brief Lives, it was in a visit to Holme Lacy as the guest of Sir John Scudamore that the servants threw his ticking watch into the moat, thinking it the Devil.[24] The author of Leicester's Commonwealth accuses him of employing the art of "figuring" to further the earl of Leicester's unlawful designs, and of endeavouring by the "black art" to bring about a match between his patron and the Queen.[2] There Allen's name is coupled with Dee's as atheists, in a series of claims that Leicester found physicians and other lackeys for his evil-doing at Oxford and elsewhere.[25]
After his death, funeral orations praising Allen were given by William Burton and George Bathurst (1610–1644).[26][27] Burton's retailed the story of how Leicester had offered a bishopric to Allen, who declined the offer. Allen in fact was, by choice, not in holy orders.[28]
Works
He wrote a Latin commentary on the second and third books of Claudius Ptolemy of Pelusium, Concerning the Judgment of the Stars, or, as it is commonly called, Of the Quadripartite Construction, with an Exposition. He also wrote notes on John Bale's De Scriptoribus M. Britanniae.[2]
Library and legacy
Allen collected manuscripts relating to history, antiquity, astronomy and astrology, philosophy, and mathematics. At least 250 items from his library can still be traced.[1] He also acquired manuscripts from dissolved monasteries, such as Reading Abbey, for which his sources may have been Gerbrand Harkes, the Protestant dealer, and Clement Burdett.[29] While in Allen's possession, most of his manuscripts were unbound or had simple covers.[30]
A considerable part of Allen's collection was presented to the Bodleian Library by Sir Kenelm Digby, to whom it had been left:[2] over 200 manuscripts, which were rebound in calf. This bequest was strong in works by early English scientists, including Roger Bacon, Simon Bredon, John Eschenden, Robert Grosseteste, John Sharp, and Richard Wallingford. But Allen's library was in flux during his lifetime, as he lent or gave items, and was consulted by others. He was a significant supporter of Sir Thomas Bodley's effort to found the Library; and gave it a number of works. Some went to the Cottonian Library, presumably via Richard James. Sir Thomas Aylesbury, another former pupil,[14] was another one of Allen's major legatees.[31] The Cuthbert Gospel of St John, seen in his library by James Ussher, appears to have left his possession by 1622, as it is not in a catalogue of that date. Ussher wrote to Camden in 1606 of the help he had had from Allen's collection, consulting William of Malmesbury, and a papal bull from Giraldus Cambrensis via Johannes Rossus.[32]
Notes
1. Turner, A. J. "Allen, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/387. (Subscription or UK public library membership required.)
2. One or more of the preceding sentences incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Allen, Thomas". Encyclopædia Britannica. Vol. 1 (11th ed.). Cambridge University Press. p. 692.
3. Foster, Michael. "Digby, Kenelm". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/7629. (Subscription or UK public library membership required.)
4. Foster, p. 99.
5. Library, Bodleian (1999). The Bodleian Library record. University Press. p. 381. Retrieved 17 April 2012.
6. Foster, p. 104.
7. Herbermann, Charles, ed. (1913). "Ven. George Napper" . Catholic Encyclopedia. New York: Robert Appleton Company.
8. Levack, Brian P. "Budden, John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/3882. (Subscription or UK public library membership required.)
9. Boran, Elizabethanne. "Burton, William". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/4142. (Subscription or UK public library membership required.)
10. Hegarty, A. J. "Twyne, Brian". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/27924. (Subscription or UK public library membership required.)
11. Keller, A. G. "Haak, Theodore". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/11827. (Subscription or UK public library membership required.)
12. Maxwell, Susan M. "Hues, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/14045. (Subscription or UK public library membership required.)
13. McGurk, J. J. N. "Davies, Sir John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/7242. (Subscription or UK public library membership required.)
14. Cormack, Lesley B. (22 December 1997). Charting an Empire: Geography at the English Universities, 1580–1620. University of Chicago Press. p. 124. ISBN 978-0-226-11606-8. Retrieved 17 April 2012.
15. Larminie, Vivienne. "Hegge, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/12865. (Subscription or UK public library membership required.)
16. Woolf, D. R. "Fulbecke, William". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/10221. (Subscription or UK public library membership required.)
17. Salmon, J. H. M. "Wheare, Diagory". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/29180. (Subscription or UK public library membership required.)
18. philological.bham.ac.uk, Camden, English-Saxons.
19. A. L. Rowse, Simon Forman: Sex and Society in Shakespeare's Age (1974), p. 7.
20. Foster, p. 120.
21. Katherine Duncan-Jones, Sir Philip Sidney, Courtier Poet (1991), pp. 50–1.
22. Peter J. French, John Dee: The World of an Elizabethan Magus, p. 129 note 4, and p. 113 note 2.
23. Foster, p. 124.
24. Hugh Trevor-Roper, Archbishop Laud (2000), p. 62.
25. Frank J. Burgoyne, History of Queen Elizabeth, Amy Robsart and the Earl of Leicester being a reprint of "Leycesters commonwealth", 1641 (1904), pp. 99–100; archive.org.
26. "Allen, Thomas (1542–1632)" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
27. Society, Royal (1973). Notes and records of the Royal Society. Royal Society of London. p. 194. Retrieved 16 April 2012.
28. Forster, p. 127.
29. Coates, Alan (22 April 1999). English Medieval Books: The Reading Abbey Collections from Foundation to Dispersal. Oxford University Press. p. 138. ISBN 978-0-19-820756-6. Retrieved 17 April 2012.
30. "Thomas ALLEN 1540?-1632". Book Owners Online. Retrieved 29 September 2022.{{cite web}}: CS1 maint: url-status (link)
31. Horner, Patrick J. (16 August 2007). The Index of Middle English Prose: Handlist III: A Handlist of Manuscripts Containing Middle English Prose in the Digby Collection, Bodleian Library. Boydell & Brewer Ltd. pp. x–xi. ISBN 978-1-84384-151-7. Retrieved 17 April 2012.
32. Charles Richard Elrington The Whole Works of the Most. Rev. James Ussher D.D. vol. 15 (1864), pp. 5–18; archive.org.
References
• Foster, Michael (1981), "Thomas Allen (1540–1632), Gloucester Hall and the Survival of Catholicism in Post-Reformation Oxford'" (PDF), Oxoniensia, XLVI: 99–128
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Thomas Banchoff
Thomas Francis Banchoff (born April 7, 1938) is an American mathematician specializing in geometry. He is a professor at Brown University, where he has taught since 1967. He is known for his research in differential geometry in three and four dimensions, for his efforts to develop methods of computer graphics in the early 1990s, and most recently for his pioneering work in methods of undergraduate education utilizing online resources.
Banchoff graduated from the University of Notre Dame in 1960, receiving his B.A. in Mathematics, and received his Masters and Ph.D. from UC Berkeley in 1962 and 1964, where he was a student of Shiing-Shen Chern.[1] Before going to Brown he taught at Harvard University and the University of Amsterdam. In 2012 he became a fellow of the American Mathematical Society.[2] In addition, he was a president of the Mathematical Association of America.[3]
Selected works
• with Stephen Lovett: Differential Geometry of Curves and Surfaces (2nd edition), A. K. Peters 2010
• with Terence Gaffney, Clint McCrory: Cusps of Gauss Mappings, Pitman 1982
• with John Wermer: Linear Algebra through Geometry, Springer Verlag 1983
• Beyond the third dimension: geometry, computer graphics, and higher dimensions, Scientific American Library, Freeman 1990
• Triple points and surgery of immersed surfaces. Proc. Amer. Math. Soc. 46 (1974), 407–413. (concerning the number of triple points of immersed surfaces in $R^{3}$.)
• Critical points and curvature for embedded polyhedra. Journal of Differential Geometry 1 (1967), 245–256. (Theorem of Gauß-Bonnet for Polyhedra)
Teaching Experience
• Benjamin Peirce Instructor, Harvard, 1964 - 1966
• Research Associate, Universiteit van Amsterdam, 1966 - 1967;
• Brown University:
• Asst Professor, 1967
• Associate Professor 1970
• Professor 1973 - 2014
• G. Leonard Baker Visiting Professor of Mathematics, Yale, 1998
• Visiting Professor, University of Notre Dame, 2001
• Visiting Professor, UCLA, 2002
• Visiting Professor, University of Georgia, 2006
• Visiting Professor, Stanford University, 2010
• Visiting Professor, Technical University of Berlin, 2012
• Visiting Professor, Sewanee: the University of the South, 2015
• Visiting Professor, Carnegie Mellon University, 2015
• Visiting Professor, Baylor University, 2016
• Paul Halmos Visiting Professor, Santa Clara University, 2018[4]
Further reading
• Donald J. Albers & Gerald L. Alexanderson (2011) Fascinating Mathematical People: interviews and memoirs, "Tom Banchoff", pp 57–78, Princeton University Press, ISBN 978-0-691-14829-8.
• Illustrating Beyond the Third Dimension by Thomas Banchoff & Davide P. Cervone
References
1. Thomas Banchoff at the Mathematics Genealogy Project
2. List of Fellows of the American Mathematical Society, retrieved 2012-11-03.
3. MAA presidents: Thomas Banchoff
4. "Curriculum Vitae". www.math.brown.edu. Retrieved 2023-02-02.
External links
• Personal web page
• biography as president of MAA
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Thomas Fantet de Lagny
Thomas Fantet de Lagny (7 November 1660 – 11 April 1734) was a French mathematician, well known for his contributions to computational mathematics, and for calculating π to 112 correct decimal places.
Thomas Fantet de Lagny
Born(1660-11-07)7 November 1660
Lyon, France
Died11 April 1734(1734-04-11) (aged 73)
Paris, France
NationalityFrench
Known forCalculating π
Scientific career
FieldsMathematics
InstitutionsFrench Academy of Sciences
Notable studentsAdrien Maurice de Noailles
Biography
Thomas Fantet de Lagny was son of Pierre Fantet, a royal official in Grenoble, and Jeanne d'Azy, the daughter of a physician from Montpellier.
He entered a Jesuit College in Lyon, where he became passionate about mathematics, as he studied some mathematical texts such as Euclid by Georges Fournier and an algebra text by Jacques Pelletier du Mans. Then he studied three years in the Faculty of Law in Toulouse.
In 1686, he went to Paris and became a mathematics tutor to the Noailles family. He collaborated with de l'Hospital under the name of de Lagny, and at that time he started publishing his first mathematical papers.
He came back to Lyon when, on 11 December 1695, he was named an associate of the Académie Royale des Sciences. Then, in 1697, he became professor of hydrography at Rochefort for 16 years.
De Lagny returned to Paris in 1714, and became a librarian at the Bibliothèque du roi, and a deputy director of the Banque Générale between 1716 and 1718. On 7 July 1719, he was awarded a pension by the Académie Royale des Sciences, finally earning his living from science. In 1723, he became a pensionnaire at the academy, replacing Pierre Varignon who died in 1722, but had to retire in 1733.
De Lagny died on 11 April 1734. While he was dying, someone asked him: "What is the square of 12?" and he answered immediately: "144."[1]
Computing π
In 1719, de Lagny calculated π to 127 decimal places, using Gregory's series for arctangent, but only 112 decimals were correct. This remained the record until 1789, when Jurij Vega calculated 126 correct digits of π.
Bibliography
• Méthode nouvelle infiniment générale et infiniment abrégée pour l’extraction des racines quarrées, cubiques... (Paris, 1691)
• Méthodes nouvelles et abrégées pour l’extraction et l’approximation des racines (Paris, 1692)
• Nouveaux élémens d’arithmétique et d’algébre ou introduction aux mathématiques (Paris, 1697)
• Trignonmétrie française ou reformée (Rochefort, 1703)
• De la cubature de la sphére où l’on démontr une infinité de portions de sphére égales à des pyramides rectilignes (La Rochelle, 1705)
• Analyse générale ou Méthodes nouvelles pour résoudre les probémes de tous les genres et de tous degrés à l’infini, M. Richer, ed. (Paris, 1733)
References
• O'Connor, John J.; Robertson, Edmund F., "Thomas Fantet de Lagny", MacTutor History of Mathematics Archive, University of St Andrews
• Lagny, Thomas Fantet de, Encyclopedia.com
1. Chalmers, Alexander. "Lagny, Thomas Fantet De". General Biographical Dictionary. Retrieved 11 January 2021.
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Thomas Joannes Stieltjes
Thomas Joannes Stieltjes (Dutch: [ˈstilcəs], 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral.
Thomas Joannes Stieltjes
Born(1856-12-29)29 December 1856
Zwolle, Netherlands
Died31 December 1894(1894-12-31) (aged 38)
Toulouse, France
NationalityDutch
Alma materÉcole Normale Supérieure
Known forRiemann–Stieltjes integral
Lebesgue–Stieltjes integration
Stieltjes constants
Stieltjes matrix
Stieltjes transformation
Stieltjes polynomials
Laplace–Stieltjes transform
Stieltjes–Wigert polynomials
Chebyshev–Markov–Stieltjes inequalities
Heine–Stieltjes polynomials
Stieltjes moment problem
Fourier-Stieltjes algebra
Henstock-Kurzweil-Stieltjes integral
Mertens conjecture
Stieltjes–Osgood theorem
Scientific career
FieldsMathematics
InstitutionsTU Delft, University of Leiden
Doctoral advisorCharles Hermite
Jean Gaston Darboux
Biography
Stieltjes was born in Zwolle on 29 December 1856. His father (who had the same first names) was a civil engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam, and also seated in the Dutch parliament. Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading the works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were 2 further failures (in 1875 and 1876), and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen (who was the director of Leiden University), and Stieltjes Jr. was able to get a job as an assistant at Leiden Observatory.
Soon afterwards, Stieltjes began a correspondence with Charles Hermite which lasted for the rest of his life. He originally wrote to Hermite concerning celestial mechanics, but the subject quickly turned to mathematics and he began to devote his spare time to mathematical research.
The director of Leiden Observatory, van de Sande-Bakhuyzen, responded quickly to Stieltjes' request on 1 January 1883 to stop his observational work to allow him to work more on mathematical topics. In 1883, he also married Elizabeth Intveld in May. She also encouraged him to move from astronomy to mathematics. And in September, Stieltjes was asked to substitute at University of Delft for F.J. van den Berg. From then until December of that year, he lectured on analytical geometry and on descriptive geometry. He resigned his post at the observatory at the end of that year.
In 1884, Stieltjes applied for a chair in Groningen. He was initially accepted, but in the end turned down by the Department of Education, since he lacked the required diplomas. In 1884, Hermite and professor David Bierens de Haan arranged for an honorary doctorate to be granted to Stieltjes by Leiden University, enabling him to become a professor. In 1885, he was appointed as member of the Royal Dutch Academy of Sciences (Koninklijke Nederlandse Akademie van Wetenschappen, KNAW), the next year he became foreign member.[1] In 1889, he was appointed professor of differential and integral calculus at Toulouse University.
Research
Stieltjes worked on almost all branches of analysis, continued fractions and number theory, and for his work, he is sometimes called "the father of the analytic theory of continued fractions".
His work is also seen as important as a first step towards the theory of Hilbert spaces. Other important contributions to mathematics that he made involved discontinuous functions and divergent series, differential equations, interpolation, the gamma function and elliptic functions. He became known internationally because of the Riemann–Stieltjes integral.
Awards
Stieltjes' work on continued fractions earned him the Ormoy Prize of the Académie des Sciences.
See also
• Chebyshev–Markov–Stieltjes inequalities
• Lebesgue–Stieltjes integral
• Laplace–Stieltjes transform
• Riemann–Stieltjes integral
• Heine–Stieltjes polynomials
• Stieltjes–Wigert polynomials
• Stieltjes polynomials
• Stieltjes constants
• Stieltjes matrix
• Stieltjes moment problem
• Stieltjes transformation (and Stieltjes inversion formula)
• Annales de la Faculté des Sciences de Toulouse co-founded by Stieltjes
• Montel's theorem
References
1. "Thomas Jan Stieltjes Jr. (1856–1894)". Royal Netherlands Academy of Arts and Sciences. Retrieved 30 July 2015.
External links
• Media related to Thomas Joannes Stieltjes jr. at Wikimedia Commons
• O'Connor, John J.; Robertson, Edmund F., "Thomas Joannes Stieltjes", MacTutor History of Mathematics Archive, University of St Andrews
• Thomas Joannes Stieltjes at the Mathematics Genealogy Project
• Œuvres complètes de Thomas Jan Stieltjes, pub. par les soins de la Société mathématique d'Amsterdam. (Groningen: P. Noordhoff, 1914–18) (PDF copy at UMDL, text in Dutch, French and German)
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| Wikipedia |
Thomas–Fermi equation
In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi,[1][2] which can be derived by applying the Thomas–Fermi model to atoms. The equation reads
${\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}$
subject to the boundary conditions
$y(0)=1\quad ;\quad y(+\infty )=0$ ;\quad y(+\infty )=0}
If $y$ approaches zero as $x$ becomes large, this equation models the charge distribution of a neutral atom as a function of radius $x$. Solutions where $y$ becomes zero at finite $x$ model positive ions.[3] For solutions where $y$ becomes large and positive as $x$ becomes large, it can be interpreted as a model of a compressed atom, where the charge is squeezed into a smaller space. In this case the atom ends at the value of $x$ for which $dy/dx=y/x$.[4][5]
Transformations
Introducing the transformation $z=y/x$ converts the equation to
${\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dz}{dx}}\right)-z^{3/2}=0$
This equation is similar to Lane–Emden equation with polytropic index $3/2$ except the sign difference. The original equation is invariant under the transformation $x\rightarrow cx,\ y\rightarrow c^{-3}y$. Hence, the equation can be made equidimensional by introducing $y=x^{-3}u$ into the equation, leading to
$x^{2}{\frac {d^{2}u}{dx^{2}}}-6x{\frac {du}{dx}}+12u=u^{3/2}$
so that the substitution $u=e^{t}$ reduces the equation to
${\frac {d^{2}u}{dt^{2}}}-7{\frac {du}{dt}}+12u=u^{3/2}.$
If $w(u)={\frac {du}{dt}}$ then the above equation becomes
$w{\frac {dw}{du}}-7w+12u=u^{3/2}.$
But this first order equation has no known explicit solution, hence, the approach turns to either numerical or approximate methods.
Sommerfeld's approximation
The equation has a particular solution $y_{p}(x)$, which satisfies the boundary condition that $y\rightarrow 0$ as $x\rightarrow \infty $, but not the boundary condition y(0)=1. This particular solution is
$y_{p}(x)={\frac {144}{x^{3}}}.$
Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932.[6] If the transformation $x=1/t,\ w=yt$ is introduced, the equation becomes
$t^{4}{\frac {d^{2}w}{dt^{2}}}=w^{3/2},\quad w(0)=0,\ w(\infty )\sim t.$
The particular solution in the transformed variable is then $w_{p}(t)=144t^{4}$. So one assumes a solution of the form $w=w_{p}(1+\alpha t^{\lambda })$ and if this is substituted in the above equation and the coefficients of $\alpha $ are equated, one obtains the value for $\lambda $, which is given by the roots of the equation $\lambda ^{2}+7\lambda -6=0$. The two roots are $\lambda _{1}=0.772,\ \lambda _{2}=-7.772$, where we need to take the positive root to avoid the singularity at the origin. This solution already satisfies the first boundary condition ($w(0)=0$), so, to satisfy the second boundary condition, one writes to the same level of accuracy for an arbitrary $n$
$W=w_{p}(1+\beta t^{\lambda })^{n}=[144t^{3}(1+\beta t^{\lambda })^{n}]t.$
The second boundary condition will be satisfied if $144t^{3}(1+\beta t^{\lambda })^{n}=144t^{3}\beta ^{n}t^{\lambda n}(1+\beta ^{-1}t^{-\lambda })^{n}\sim 1$ as $t\rightarrow \infty $. This condition is satisfied if $\lambda n+3=0,\ 144\beta ^{n}=1$ and since $\lambda _{1}\lambda _{2}=-6$, Sommerfeld found the approximation as $\lambda =\lambda _{1},\ n=-3/\lambda _{1}=\lambda _{2}/2$. Therefore, the approximate solution is
$y(x)=y_{p}(x)\{1+[y_{p}(x)]^{\lambda _{1}/3}\}^{\lambda _{2}/2}.$
This solution predicts the correct solution accurately for large $x$, but still fails near the origin.
Solution near origin
Enrico Fermi[7] provided the solution for $x\ll 1$ and later extended by Edward B. Baker.[8] Hence for $x\ll 1$,
${\begin{aligned}y(x)={}&1-Bx+{\frac {1}{3}}x^{3}-{\frac {2B}{15}}x^{4}+\cdots {}\\[6pt]&\cdots +x^{3/2}\left[{\frac {4}{3}}-{\frac {2B}{5}}x+{\frac {3B^{2}}{70}}x^{2}+\left({\frac {2}{27}}+{\frac {B^{3}}{252}}\right)x^{3}+\cdots \right]\end{aligned}}$
where $B\approx 1.588071$.[9][10]
It has been reported by Salvatore Esposito[11] that the Italian physicist Ettore Majorana found in 1928 a semi-analytical series solution to the Thomas–Fermi equation for the neutral atom, which however remained unpublished until 2001.
Using this approach it is possible to compute the constant B mentioned above to practically arbitrarily high accuracy; for example, its value to 100 digits is $B=1.588071022611375312718684509423950109452746621674825616765677418166551961154309262332033970138428665$.
References
1. Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
2. Bender, Carl M., and Steven A. Orszag. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 2013.
3. pp. 9-12, N. H. March (1983). "1. Origins – The Thomas–Fermi Theory". In S. Lundqvist and N. H. March. Theory of The Inhomogeneous Electron Gas. Plenum Press. ISBN 978-0-306-41207-3.
4. March 1983, p. 10, Figure 1.
5. p. 1562,Feynman, R. P.; Metropolis, N.; Teller, E. (1949-05-15). "Equations of State of Elements Based on the Generalized Fermi-Thomas Theory" (PDF). Physical Review. American Physical Society (APS). 75 (10): 1561–1573. Bibcode:1949PhRv...75.1561F. doi:10.1103/physrev.75.1561. ISSN 0031-899X.
6. Sommerfeld, A. "Integrazione asintotica dell’equazione differenziale di Thomas–Fermi." Rend. R. Accademia dei Lincei 15 (1932): 293.
7. Fermi, E. (1928). "Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente". Zeitschrift für Physik (in German). Springer Science and Business Media LLC. 48 (1–2): 73–79. Bibcode:1928ZPhy...48...73F. doi:10.1007/bf01351576. ISSN 1434-6001. S2CID 122644389.
8. Baker, Edward B. (1930-08-15). "The Application of the Fermi-Thomas Statistical Model to the Calculation of Potential Distribution in Positive Ions". Physical Review. American Physical Society (APS). 36 (4): 630–647. Bibcode:1930PhRv...36..630B. doi:10.1103/physrev.36.630. ISSN 0031-899X.
9. Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111], Francisco M.Fernández, Physics Letters A 372, 28 July 2008, 5258-5260, doi:10.1016/j.physleta.2008.05.071.
10. The analytical solution of the Thomas-Fermi equation for a neutral atom, G I Plindov and S K Pogrebnya, Journal of Physics B: Atomic and Molecular Physics 20 (1987), L547, doi:10.1088/0022-3700/20/17/001.
11. Esposito, Salvatore (2002). "Majorana solution of the Thomas-Fermi equation". American Journal of Physics. 70 (8): 852–856. arXiv:physics/0111167. Bibcode:2002AmJPh..70..852E. doi:10.1119/1.1484144. S2CID 119063230.
| Wikipedia |
Thomas–Yau conjecture
In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the existence of a solution to the special Lagrangian equation inside a Hamiltonian isotopy class of Lagrangian submanifolds. In particular the conjecture contains two difficulties: first it asks what a suitable stability condition might be, and secondly if one can prove stability of an isotopy class if and only if it contains a special Lagrangian representative.
The Thomas–Yau conjecture was proposed by Richard Thomas and Shing-Tung Yau in 2001,[1][2] and was motivated by similar theorems in algebraic geometry relating existence of solutions to geometric partial differential equations and stability conditions, especially the Kobayashi–Hitchin correspondence relating slope stable vector bundles to Hermitian Yang–Mills metrics.
The conjecture is intimately related to mirror symmetry, a conjecture in string theory and mathematical physics which predicts that mirror to a symplectic manifold (which is a Calabi–Yau manifold) there should be another Calabi–Yau manifold for which the symplectic structure is interchanged with the complex structure.[3] In particular mirror symmetry predicts that special Lagrangians, which are the Type IIA string theory model of BPS D-branes, should be interchanged with the same structures in the Type IIB model, which are given either by stable vector bundles or vector bundles admitting Hermitian Yang–Mills or possibly deformed Hermitian Yang–Mills metrics. Motivated by this, Dominic Joyce rephrased the Thomas–Yau conjecture in 2014, predicting that the stability condition may be understood using the theory of Bridgeland stability conditions defined on the Fukaya category of the Calabi–Yau manifold, which is a triangulated category appearing in Kontsevich's homological mirror symmetry conjecture.[4]
Statement
The statement of the Thomas–Yau conjecture is not completely precise, as the particular stability condition is not yet known. In the work of Thomas and Thomas–Yau, the stability condition was given in terms of the Lagrangian mean curvature flow inside the Hamiltonian isotopy class of the Lagrangian, but Joyce's reinterpretation of the conjecture predicts that this stability condition can be given a categorical or algebraic form in terms of Bridgeland stability conditions.
Special Lagrangian submanifolds
Consider a Calabi–Yau manifold $(X,\omega ,\Omega )$ of complex dimension $n$, which is in particular a real symplectic manifold of dimension $2n$. Then a Lagrangian submanifold is a real $n$-dimensional submanifold $L\subset X$ such that the symplectic form is identically zero when restricted to $L$, that is $\left.\omega \right|_{L}=0$. The holomorphic volume form $\Omega \in \Omega ^{n,0}(X)$, when restricted to a Lagrangian submanifold, becomes a top degree differential form. If the Lagrangian is oriented, then there exists a volume form $dV_{L}$ on $L$ and one may compare this volume form to the restriction of the holomorphic volume form: $\left.\Omega \right|_{L}=fdV_{L}$ for some complex-valued function $f:L\to \mathbb {C} $. The condition that $X$ is a Calabi–Yau manifold implies that the function $f$ has norm 1, so we have $f=e^{i\Theta }$ where $\Theta :L\to [0,2\pi )$ is the phase angle of the function $f$. In principle this phase function is only locally continuous, and its value may jump. A graded Lagrangian is a Lagrangian together with a lifting $\vartheta :L\to \mathbb {R} $ of the phase angle to $\mathbb {R} $, which satisfies $\Theta =\vartheta \mod 2\pi $ everywhere on $L$.
An oriented,graded Lagrangian $L$ is said to be a special Lagrangian submanifold if the phase angle function $\vartheta $ is constant on $L$. The average value of this function, denoted $\theta $, may be computed using the volume form as
$\theta =\arg \int _{L}\Omega ,$
and only depends on the Hamiltonian isotopy class of $L$. Using this average value, the condition that $\Theta $ is constant may be written in the following form, which commonly occurs in the literature. This is the definition of a special Lagrangian submanifold:
$\mathrm {Im} (e^{-i\theta }\left.\Omega \right|_{L})=0.$
Hamiltonian isotopy classes
The condition of being a special Lagrangian is not satisfied for all Lagrangians, but the geometric and especially physical properties of Lagrangian submanifolds in string theory are predicted to only depend on the Hamiltonian isotopy class of the Lagrangian submanifold. An isotopy is a transformation of a submanifold inside an ambient manifold which is a homotopy by embeddings. On a symplectic manifold, a symplectic isotopy requires that these embeddings are by symplectomorphisms, and a Hamiltonian isotopy is a symplectic isotopy for which the symplectomorphisms are generated by Hamiltonian functions. Given a Lagrangian submanifold $L$, the condition of being a Lagrangian is preserved under Hamiltonian (in fact symplectic) isotopies, and the collection of all Lagrangian submanifolds which are Hamiltonian isotopic to $L$ is denoted $[L]$, called the Hamiltonian isotopy class of $L$.
Lagrangian mean curvature flow and stability condition
Given a Riemannian manifold $M$ and a submanifold $\iota :N\hookrightarrow M$, the mean curvature flow is a differential equation satisfied for a one-parameter family $\iota _{t}$ of embeddings defined for in $t$ some interval $[0,T)$ with images denoted $N^{t}$, where $N^{0}=N$. Namely, the family satisfies mean curvature flow if
${\frac {d\iota _{t}}{dt}}=H_{\iota _{t}}$
where $H_{\iota _{t}}$ is the mean curvature of the submanifold $N^{t}\subset M$. This flow is the gradient flow of the volume functional on submanifolds of the Riemannian manifold $M$, and there always exists short time existence of solutions starting from a given submanifold $N$.
On a Calabi–Yau manifold, if $L$ is a Lagrangian, the condition of being a Lagrangian is preserved when studying the mean curvature flow of $L$ with respect to the Calabi–Yau metric. This is therefore called the Lagrangian mean curvature flow (Lmcf). Furthermore, for a graded Lagrangian $(L,\vartheta )$, Lmcf preserves Hamiltonian isotopy class, so $L^{t}\in [L]$ for all $t\in [0,T)$ where the Lmcf is defined.
Thomas introduced a conjectural stability condition[1] defined in terms of gradings when splitting into Lagrangian connected sums. Namely a graded Lagrangian $(L,\vartheta )$ is called stable if whenever it may be written as a graded Lagrangian connected sum
$(L,\vartheta )=(L_{1},\vartheta _{1})\#(L_{2},\vartheta _{2})$
the average phases satisfy the inequality
$\theta _{1}<\theta _{2}.$
In the later language of Joyce using the notion of a Bridgeland stability condition, this was further explained as follows. An almost-calibrated Lagrangian (which means the lifted phase is taken to lie in the interval $(-\pi /2,\pi /2)$ or some integer shift of this interval) which splits as a graded connected sum of almost-calibrated Lagrangians corresponds to a distinguished triangle
$L_{1}\to L_{1}\#L_{2}\to L_{2}\to L_{1}[1]$
in the Fukaya category. The Lagrangian $(L,\vartheta )$ is stable if for any such distinguished triangle, the above angle inequality holds.
Statement of the conjecture
The conjecture as originally proposed by Thomas is as follows:
Conjecture:[1] An oriented, graded, almost-calibrated Lagrangian $L$ admits a special Lagrangian representative in its Hamiltonian isotopy class $[L]$ if and only if it is stable in the above sense.
Following this, in the work of Thomas–Yau, the behaviour of the Lagrangian mean curvature flow was also predicted.
Conjecture (Thomas–Yau):[1][2] If an oriented, graded, almost-calibrated Lagrangian $L$ is stable, then the Lagrangian mean curvature flow exists for all time and converges to a special Lagrangian representative in the Hamiltonian isotopy class $[L]$.
This conjecture was enhanced by Joyce, who provided a more subtle analysis of what behaviour is expected of the Lagrangian mean curvature flow. In particular Joyce described the types of finite-time singularity formation which are expected to occur in the Lagrangian mean curvature flow, and proposed expanding the class of Lagrangians studied to include singular or immersed Lagrangian submanifolds, which should appear in the full Fukaya category of the Calabi–Yau.
Conjecture (Thomas–Yau–Joyce):[4] An oriented, graded, almost-calibrated Lagrangian $L$ splits as a graded Lagrangian connected sum $L=L_{1}\#\cdots \#L_{k}$ of special Lagrangian submanifolds $L_{i}$ with phase angles $\theta _{1}>\cdots >\theta _{k}$ given by the convergence of the Lagrangian mean curvature flow with surgeries to remove singularities at a sequence of finite times $0<T_{1}<\cdots <T_{k}$. At these surgery points, the Lagrangian may change its Hamiltonian isotopy class but preserves its class in the Fukaya category.
In the language of Joyce's formulation of the conjecture, the decomposition $L=L_{1}\#\cdots \#L_{k}$ is a symplectic analogue of the Harder-Narasimhan filtration of a vector bundle, and using Joyce's interpretation of the conjecture in the Fukaya category with respect to a Bridgeland stability condition, the central charge is given by
$Z(L)=\int _{L}\Omega $,
the heart ${\mathcal {A}}$ of the t-structure defining the stability condition is conjectured to be given by those Lagrangians in the Fukaya category with phase $\theta \in (-\pi /2,\pi /2)$, and the Thomas–Yau–Joyce conjecture predicts that the Lagrangian mean curvature flow produces the Harder–Narasimhan filtration condition which is required to prove that the data $(Z,{\mathcal {A}})$ defines a genuine Bridgeland stability condition on the Fukaya category.
References
1. Thomas, R. P. (2001). "Moment Maps, Monodromy and Mirror Manifolds". Symplectic Geometry and Mirror Symmetry. pp. 467–498. arXiv:math/0104196. doi:10.1142/9789812799821_0013. ISBN 978-981-02-4714-0. S2CID 15284349.
2. Thomas, R. P.; Yau, S.-T. (2002). "Special Lagrangians, stable bundles and mean curvature flow". Communications in Analysis and Geometry. 10 (5): 1075–1113. doi:10.4310/CAG.2002.V10.N5.A8. S2CID 2153403.
3. Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Ravi, Vakil; Zaslow, Eric (2003). Mirror Symmetry (PDF). Clay Mathematics Monographs. Vol. 1. AMS and Clay Mathematics Institute. p. 929. ISBN 978-0-8218-2955-4.
4. Joyce, Dominic (2015). "Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow". EMS Surveys in Mathematical Sciences. 2: 1–62. arXiv:1401.4949. doi:10.4171/EMSS/8. S2CID 118102887.
| Wikipedia |
Thompson factorization
In mathematical finite group theory, a Thompson factorization, introduced by Thompson (1966), is an expression of some finite groups as a product of two subgroups, usually normalizers or centralizers of p-subgroups for some prime p.
References
• Aschbacher, Michael (2000), Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10 (2nd ed.), Cambridge University Press, ISBN 0-521-78145-0, MR 1777008
• Aschbacher, Michael (1981), "On the failure of the Thompson factorization in 2-constrained groups", Proceedings of the London Mathematical Society, Third Series, 43 (3): 425–449, doi:10.1112/plms/s3-43.3.425, ISSN 0024-6115, MR 0635564
• Thompson, John G. (1966), "Factorizations of p-solvable groups", Pacific Journal of Mathematics, 16 (2): 371–372, doi:10.2140/pjm.1966.16.371, ISSN 0030-8730, MR 0188296
| Wikipedia |
Thompson sporadic group
In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order
215 · 310 · 53 · 72 · 13 · 19 · 31
= 90745943887872000
≈ 9×1016.
This article is about the sporadic simple group. For the three unusual infinite groups F, T and V found by Thompson, see Thompson groups.
Algebraic structure → Group theory
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History
Th is one of the 26 sporadic groups and was found by John G. Thompson (1976) and constructed by Smith (1976). They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E8. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the Dempwolff group (which unlike the Thompson group is a subgroup of the compact Lie group E8).
Representations
The centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson group acts on a vertex operator algebra over the field with 3 elements. This vertex operator algebra contains the E8 Lie algebra over F3, giving the embedding of Th into E8(3).
The Schur multiplier and the outer automorphism group of the Thompson group are both trivial.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Th, the relevant McKay-Thompson series is $T_{3C}(\tau )$ (OEIS: A007245),
$T_{3C}(\tau )={\Big (}j(3\tau ){\Big )}^{1/3}={\frac {1}{q}}\,+\,248q^{2}\,+\,4124q^{5}\,+\,34752q^{8}\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots \,$
and j(τ) is the j-invariant.
Maximal subgroups
Linton (1989) found the 16 conjugacy classes of maximal subgroups of Th as follows:
• 2+1+8 · A9
• 25 · L5(2) This is the Dempwolff group
• (3 x G2(3)) : 2
• (33 × 3+1+2) · 3+1+2 : 2S4
• 32 · 37 : 2S4
• (3 × 34 : 2 · A6) : 2
• 5+1+2 : 4S4
• 52 : GL2(5)
• 72 : (3 × 2S4)
• 31 : 15
• 3D4(2) : 3
• U3(8) : 6
• L2(19)
• L3(3)
• M10
• S5
References
• Linton, Stephen A. (1989), "The maximal subgroups of the Thompson group", Journal of the London Mathematical Society, Second Series, 39 (1): 79–88, doi:10.1112/jlms/s2-39.1.79, ISSN 0024-6107, MR 0989921
• Smith, P. E. (1976), "A simple subgroup of M? and E8(3)", The Bulletin of the London Mathematical Society, 8 (2): 161–165, doi:10.1112/blms/8.2.161, ISSN 0024-6093, MR 0409630
• Thompson, John G. (1976), "A conjugacy theorem for E8", Journal of Algebra, 38 (2): 525–530, doi:10.1016/0021-8693(76)90235-0, ISSN 0021-8693, MR 0399193
External links
• MathWorld: Thompson group
• Atlas of Finite Group Representations: Thompson group
| Wikipedia |
Thompson groups
In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted $F\subseteq T\subseteq V$, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.
This article is about the three unusual infinite groups F, T and V found by Thompson. For the sporadic simple group, see Thompson sporadic group.
The Thompson groups, and F in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.
It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.
Higman (1974) introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.
Presentations
A finite presentation of F is given by the following expression:
$\langle A,B\mid \ [AB^{-1},A^{-1}BA]=[AB^{-1},A^{-2}BA^{2}]=\mathrm {id} \rangle $
where [x,y] is the usual group theory commutator, xyx−1y−1.
Although F has a finite presentation with 2 generators and 2 relations, it is most easily and intuitively described by the infinite presentation:
$\langle x_{0},x_{1},x_{2},\dots \ \mid \ x_{k}^{-1}x_{n}x_{k}=x_{n+1}\ \mathrm {for} \ k<n\rangle .$
The two presentations are related by x0=A, xn = A1−nBAn−1 for n>0.
Other representations
The group F also has realizations in terms of operations on ordered rooted binary trees, and as a subgroup of the piecewise linear homeomorphisms of the unit interval that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2.
The group F can also be considered as acting on the unit circle by identifying the two endpoints of the unit interval, and the group T is then the group of automorphisms of the unit circle obtained by adding the homeomorphism x→x+1/2 mod 1 to F. On binary trees this corresponds to exchanging the two trees below the root. The group V is obtained from T by adding the discontinuous map that fixes the points of the half-open interval [0,1/2) and exchanges [1/2,3/4) and [3/4,1) in the obvious way. On binary trees this corresponds to exchanging the two trees below the right-hand descendant of the root (if it exists).
The Thompson group F is the group of order-preserving automorphisms of the free Jónsson–Tarski algebra on one generator.
Amenability
The conjecture of Thompson that F is not amenable was further popularized by R. Geoghegan—see also the Cannon–Floyd–Parry article cited in the references below. Its current status is open: E. Shavgulidze[1] published a paper in 2009 in which he claimed to prove that F is amenable, but an error was found, as is explained in the MR review.
It is known that F is not elementary amenable, see Theorem 4.10 in Cannon–Floyd–Parry.
If F is not amenable, then it would be another counterexample to the now disproved von Neumann conjecture for finitely-presented groups, which states that a finitely-presented group is amenable if and only if it does not contain a copy of the free group of rank 2.
Connections with topology
The group F was rediscovered at least twice by topologists during the 1970s. In a paper that was only published much later but was in circulation as a preprint at that time, P. Freyd and A. Heller [2] showed that the shift map on F induces an unsplittable homotopy idempotent on the Eilenberg–MacLane space K(F,1) and that this is universal in an interesting sense. This is explained in detail in Geoghegan's book (see references below). Independently, J. Dydak and P. Minc [3] created a less well-known model of F in connection with a problem in shape theory.
In 1979, R. Geoghegan made four conjectures about F: (1) F has type FP∞; (2) All homotopy groups of F at infinity are trivial; (3) F has no non-abelian free subgroups; (4) F is non-amenable. (1) was proved by K. S. Brown and R. Geoghegan in a strong form: there is a K(F,1) with two cells in each positive dimension.[4] (2) was also proved by Brown and Geoghegan [5] in the sense that the cohomology H*(F,ZF) was shown to be trivial; since a previous theorem of M. Mihalik [6] implies that F is simply connected at infinity, and the stated result implies that all homology at infinity vanishes, the claim about homotopy groups follows. (3) was proved by M. Brin and C. Squier.[7] The status of (4) is discussed above.
It is unknown if F satisfies the Farrell–Jones conjecture. It is even unknown if the Whitehead group of F (see Whitehead torsion) or the projective class group of F (see Wall's finiteness obstruction) is trivial, though it easily shown that F satisfies the strong Bass conjecture.
D. Farley [8] has shown that F acts as deck transformations on a locally finite CAT(0) cubical complex (necessarily of infinite dimension). A consequence is that F satisfies the Baum–Connes conjecture.
See also
• Higman group
• Non-commutative cryptography
References
1. Shavgulidze, E. (2009), "The Thompson group F is amenable", Infinite Dimensional Analysis, Quantum Probability and Related Topics, 12 (2): 173–191, doi:10.1142/s0219025709003719, MR 2541392
2. Freyd, Peter; Heller, Alex (1993), "Splitting homotopy idempotents", Journal of Pure and Applied Algebra, 89 (1–2): 93–106, doi:10.1016/0022-4049(93)90088-b, MR 1239554
3. Dydak, Jerzy; Minc, Piotr (1977), "A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR's", Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, 25: 55–62, MR 0442918
4. Brown, K.S.; Geoghegan, Ross (1984), An infinite-dimensional torsion-free FP_infinity group, vol. 77, pp. 367–381, Bibcode:1984InMat..77..367B, doi:10.1007/bf01388451, MR 0752825
5. Brown, K.S.; Geoghegan, Ross (1985), "Cohomology with free coefficients of the fundamental group of a graph of groups", Commentarii Mathematici Helvetici, 60: 31–45, doi:10.1007/bf02567398, MR 0787660
6. Mihalik, M. (1985), "Ends of groups with the integers as quotient", Journal of Pure and Applied Algebra, 35: 305–320, doi:10.1016/0022-4049(85)90048-9, MR 0777262
7. Brin, Matthew.; Squier, Craig (1985), "Groups of piecewise linear homeomorphisms of the real line", Inventiones Mathematicae, 79 (3): 485–498, Bibcode:1985InMat..79..485B, doi:10.1007/bf01388519, MR 0782231
8. Farley, D. (2003), "Finiteness and CAT(0) properties of diagram groups", Topology, 42 (5): 1065–1082, doi:10.1016/s0040-9383(02)00029-0, MR 1978047
• Cannon, J. W.; Floyd, W. J.; Parry, W. R. (1996), "Introductory notes on Richard Thompson's groups" (PDF), L'Enseignement Mathématique, IIe Série, 42 (3): 215–256, ISSN 0013-8584, MR 1426438
• Cannon, J.W.; Floyd, W.J. (September 2011). "WHAT IS...Thompson's Group?" (PDF). Notices of the American Mathematical Society. 58 (8): 1112–1113. ISSN 0002-9920. Retrieved December 27, 2011.
• Geoghegan, Ross (2008), Topological Methods in Group Theory, Graduate Texts in Mathematics, vol. 243, Springer Verlag, arXiv:math/0601683, doi:10.1142/S0129167X07004072, ISBN 978-0-387-74611-1, MR 2325352
• Higman, Graham (1974), Finitely presented infinite simple groups, Notes on Pure Mathematics, vol. 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874
| Wikipedia |
Thompson uniqueness theorem
In mathematical finite group theory, Thompson's original uniqueness theorem (Feit & Thompson 1963, theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1970) gave a shorter proof of the uniqueness theorem.
References
• Bender, Helmut (1970), "On the uniqueness theorem", Illinois Journal of Mathematics, 14 (3): 376–384, doi:10.1215/ijm/1256053074, ISSN 0019-2082, MR 0262351
• Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 188, Cambridge University Press, ISBN 978-0-521-45716-3, MR 1311244
• Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics, 13: 775–1029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, MR 0166261
| Wikipedia |
Thornton Carle Fry
Thornton Carle Fry (7 January 1892, Findlay, Ohio – 1 January 1991) was an applied mathematician, known for his two widely-used textbooks, Probability and its engineering uses (1928)[1][2] and Elementary differential equations (1929).[3][4]
Career
Thornton C. Fry received his bachelor's degree from Findlay College in 1912 and then pursued graduate study in Wisconsin in mathematics, physics, and astronomy. He received his M.A. in 1913[5] and his Ph.D. in 1920 in applied mathematics from the University of Wisconsin-Madison with thesis under the supervision of Charles S. Slichter.[6][7]
Fry was employed as an industrial mathematician by Western Electric Company from 1916 to 1924 and then by Bell Telephone Laboratories (Bell Labs), which was half-owned by Western Electric. He headed a corporate division for industrial applications of mathematics and statistics and was involved in research and development for the U.S. federal government in both world wars.
After retiring (due to reaching the mandatory retirement age) from Bell Labs, he worked as a consultant with Boeing Scientific Research Labs and also, during the 1960s, with Walter Orr Roberts, director of the National Center for Atmospheric Research.[8]
In 1924 Fry was an Invited Speaker of the International Congress of Mathematicians in Toronto.[9] In 1982 the Mathematical Association of America (MAA) gave him the MAA's distinguished service award.[10]
Selected publications
• Fry, Thornton C. (1916). "The graphical computation of transit factors". Popular Astronomy. 24: 17–21. Bibcode:1916PA.....24...17F.
• Fry, Thornton C. (1916). "Graphical solution of the position of a body in an elliptic orbit". The Astronomical Journal. 29: 141–146. Bibcode:1916AJ.....29..141F. doi:10.1086/104144.
• Fry, Thornton C. (1919). "The solution of circuit problems. Mathematical methods of investigation resulting from the application of Fourier's integral". Physical Review. 14 (2): 115–136. Bibcode:1919PhRv...14..115F. doi:10.1103/physrev.14.115.
• with R. V. L. Hartley: Hartley, R. V. L.; Fry, Thornton C. (1921). "The binaural location of pure tones". Physical Review. 18 (6): 431–442. Bibcode:1921PhRv...18..431H. doi:10.1103/PhysRev.18.431.
• with R. V. L. Hartley: Hartley, R. V. L.; Fry, Thornton C. (1922). "The binaural location of complex sounds". Bell Labs Technical Journal. 1 (2): 33–42. doi:10.1002/j.1538-7305.1922.tb00387.x.
• Fry, Thornton C. (1929). "The theory of the Schroteffekt". Journal of the Franklin Institute. 199 (2): 203–220. doi:10.1016/S0016-0032(25)91045-9.
• Fry, T. C. (1929). "The use of continued fractions in the design of electrical networks". Bull. Amer. Math. Soc. 35 (4): 463–498. doi:10.1090/S0002-9904-1929-04747-5.
• Fry, Thornton C. (1929). "Differential equations as a foundation for electrical circuit theory". The American Mathematical Monthly. 36 (10): 499–504. doi:10.1080/00029890.1929.11987013. JSTOR 2299959.
• Fry, Thornton C. (1932). "Two problems in potential theory". The American Mathematical Monthly. 39 (4): 199–209. doi:10.2307/2300663. JSTOR 2300663.
• with John R. Carson: Carson, John R.; Fry, Thornton C. (1937). "Variable frequency electric circuit theory with an application to the theory of frequency-modulation". Bell Labs Technical Journal. 16 (4): 513–540. doi:10.1002/j.1538-7305.1937.tb00766.x.
• Fry, Thornton C. (1937). "The semicentennial celebration of the American Mathematical Society—September 6–9, 1938". Bulletin of the American Mathematical Society. 43 (9): 593–594. doi:10.1090/S0002-9904-1937-06580-3.
• Fry, Thornton C. (1938). "The χ2-test of significance". Journal of the American Statistical Association. 33 (203): 513–525. doi:10.1080/01621459.1938.10502328.
• "Industrial Mathematics by Thornton C. Fry, Mathematical Research Director, Bell Telephone Laboratories, New York, NY". In: Research—A National Resource. II. Industrial Research, Report of the National Resource Council to the National Resources Planning Board, December 1940. Washington, DC: U. S. Government Printing Office. 1941. pp. 268–288.
• Fry, Thornton C. (1941). "Industrial mathematics". Bell Labs Technical Journal. 20 (3): 255–292. doi:10.1002/j.1538-7305.1941.tb03138.x.
• Fry, Thornton C. (1945). "Some numerical methods for locating roots of polynomials" (PDF). Quarterly of Applied Mathematics. 3 (2): 89–105. doi:10.1090/qam/12910.
• Fry, Thornton C. (1956). "Mathematics as a profession today in industry". The American Mathematical Monthly. 63 (2): 71–80. doi:10.2307/2306427. JSTOR 2306427.
• Fry, T. C. (1964). "Mathematicians in industry—the first 75 years". Science. 143 (3609): 934–938. Bibcode:1964Sci...143..934F. doi:10.1126/science.143.3609.934. JSTOR 1712826. PMID 17743925.
Patents
• "System for determining the direction of propagation of wave energy." U.S. Patent 1,502,243, issued July 22, 1924.
• "Harmonic analyzer." U.S. Patent 1,503,824, issued August 5, 1924.
• "Filtering circuit." U.S. Patent 1,559,864, issued November 3, 1925.
References
1. Struik, D. J. (1930). "Review of Probability and its Engineering Uses by T. C. Fry". Bull. Amer. Math. Soc. 36: 19–21. doi:10.1090/S0002-9904-1930-04858-2.
2. Fry, Thornton (1928). Probability and its Engineering Uses. D. Van Nostrand Company, Inc.
3. Longley, W. R. (1930). "Review of Elementary Differential Equations by Thornton C. Fry". Bull. Amer. Math. Soc. 36 (3): 173–174. doi:10.1090/S0002-9904-1930-04918-6.
4. Fry, Thornton (1929). Elementary Differential Equations. D. Van Nostrand Company, Inc.
5. Fry, Thornton Carle (1913). A contact transformation. University of Wisconsin-Madison.
6. Thornton Carle Fry at the Mathematics Genealogy Project
7. Fry, Thornton Carle (1921). The Application of Modern Theories of Integration to the Solution of Differential Equations. University of Wisconsin-Madison.
8. Firor, John; Trimble, Virginia (1997). "Obituary. Thornton Carl (sic) Fry, 1892–1991s". Bulletin of the American Astronomical Society. 29 (4): 1470–1471. Bibcode:1997BAAS...29.1470F. This obituary and several other sources have the name "Thornton Carl Fry" instead of the correct "Thornton Carle Fry".
9. Fry, T. C. "The use of the integraph in the practical solution of differential equations by Picard's method of successive approximations". In: Proc. Intern. Mathematical Congress held in Toronto, 1924. Vol. 2. pp. 407–428. (See integraph.)
10. Price, G. Baley. "Award for distinguished Service to Dr. Thornton Carl (sic) Fry". American Mathematical Monthly. 89 (2): 81–83. doi:10.1080/00029890.1982.11995388.
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| Wikipedia |
John A. Thorpe
John Alden Thorpe (born February 29, 1936 in Lewiston, Maine) is an American mathematician, known for contributions to the field of differential geometry.[1]
Thorpe obtained his Bachelor's degree in 1958 from the Massachusetts Institute of Technology. His Ph.D. was done at Columbia University, under the direction of James Eells (Higher Order Sectional Curvature).[2] From 1963 to 1965, he was Moore Instructor at MIT and Assistant Professor at Haverford College in 1965. In 1967 and 1968 he was a visitor at the Institute for Advanced Study. From 1968, he was Associate Professor and then Professor at the State University of New York at Stony Brook (SUNY). From 1987 he was Professor and Dean at the State University of New York in Buffalo, and from 1993 at Queens College of City University of New York, where he also served as Provost. From 1984 to 1987 he served on the Board of Governors of the Mathematical Association of America. From 1998 to 2001 he was Executive Director of the National Council of Teachers of Mathematics.
He and Nigel Hitchin independently found an inequality between topological invariants, which provides a necessary condition for the existence of Einstein metrics on four-dimensional smooth compact manifolds. It is now known as the Hitchin-Thorpe inequality.[3]
Books
• Elementary Topics in Differential Geometry, Springer Verlag, Undergraduate Texts in Mathematics, 1979
• Lecture Notes on Elementary Topology and Geometry (with I.M. Singer), Springer Verlag, Undergraduate Texts in Mathematics, 1967
References
1. American Men and Women of Science, Thomson Gale 2004
2. Mathematics Genealogy Project
3. Thorpe Some remarks on the Gauss-Bonnet formula, J. Math. Mech. 18 (1969), S. 779--786
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| Wikipedia |
Thorvald N. Thiele
Thorvald Nicolai Thiele (24 December 1838 – 26 September 1910) was a Danish astronomer and director of the Copenhagen Observatory.[1] He was also an actuary and mathematician, most notable for his work in statistics, interpolation and the three-body problem.
Thorvald N. Thiele
Born(1838-12-24)24 December 1838
Copenhagen
Died26 September 1910(1910-09-26) (aged 71)
NationalityDanish
Alma materUniversity of Copenhagen
Scientific career
FieldsAstronomy, Statistics
InstitutionsCopenhagen Observatory
Doctoral advisorHeinrich Louis d'Arrest
Thiele made notable contributions to the statistical study of random time series and introduced the cumulants and likelihood functions, and was considered to be one of the greatest statisticians of all time by Ronald Fisher.[2] In the early 1900s he also developed and proposed a generalisation of approval voting to multiple winner elections called sequential proportional approval voting,[3] which was briefly used for party lists in Sweden when proportional representation was introduced in 1909.
Thiele also was a founder and Mathematical Director of the Hafnia Insurance Company and led the founding of the Danish Society of Actuaries. It was through his insurance work that he came into contact with fellow mathematician Jørgen Pedersen Gram.
Thiele was the father of astronomer Holger Thiele.
The main-belt asteroids 843 Nicolaia (discovered by his son Holger) and 1586 Thiele are named in his honour.[1]
Selected publications
• Sur la compensation de quelques erreurs quasi-systématiques par la méthode des moindres carrés. 1880.
• Theory of observations. 1903.
• Interpolationsrechnung. 1909.[4]
See also
• Founders of statistics
• Gram–Charlier series
• Kalman filter
• Least squares
• Thiele's interpolation formula
• Time series
Notes and references
Notes
1. Schmadel, Lutz D. (2007). "(1586) Thiele". Dictionary of Minor Planet Names – (1586) Thiele. Springer Berlin Heidelberg. pp. 125–126. doi:10.1007/978-3-540-29925-7_1587. ISBN 978-3-540-00238-3.
2. Jerzy Neyman, “Note on an Article by Sir Ronald Fisher,” Journal of the Royal Statistical Society, Series B (Methodological), 18, 2 (July 1956): 288–294, doi:10.1111/j.2517-6161.1956.tb00236.x.
3. Reuterskiöld, C. A.; Phragmén, E.; Svensén, Emil; Huss, E. G.; Fahlbeck, Pontus E.; Alin, Oscar (1899). "1899 Vol. 2 no. 2". Statsvetenskaplig Tidskrift. 2. Archived from the original on 2015-06-18. {{cite journal}}: |last1= has generic name (help)
4. Rietz, H. L. (1911). "Book Review: Interpolationsrechnung". Bulletin of the American Mathematical Society. 17 (7): 372–374. doi:10.1090/S0002-9904-1911-02086-9. ISSN 0002-9904.
References
• Steffen L. Lauritzen (2002). Thiele: Pioneer in Statistics. Oxford University Press. p. 288. ISBN 978-0-19-850972-1.
• 1. Introduction to Thiele, S. L. Lauritzen
• 2. On the application of the method of least squares to some cases, in which a combination of certain types of inhomogeneous random sources of errors gives these a 'systematic' character, T. N. Thiele
• 3. Time series analysis in 1880: a discussion of contributions made by T. N. Thiele, S. L. Lauritzen
• 4. The general theory of observations: calculus of probability and the method of least squares, T. N. Thiele
• 5. T. N. Thiele's contributions to statistics, A. Hald
• 6. On the halfinvariants in the theory of observations, T. N. Thiele
• 7. The early history of cumulants and the Gram–Charlier series, A. Hald
• 8. Epilogue, S. L. Lauritzen
• Anders Hald (1998). A History of Mathematical Statistics from 1750 to 1930. New York: Wiley. ISBN 978-0-471-17912-2.
• Anders Hald. "T. N. Thiele's contributions to statistics" International Statistical Review volume 49, (1981), number 1: 1—20.
• Anders Hald. "The early history of the cumulants and the Gram–Charlier series" International Statistical Review volume 68 (2000), number 2,´: 137—153.
• Steffen L. Lauritzen. "Time series analysis in 1880. A discussion of contributions made by T.N. Thiele". International Statistical Review 49, 1981, 319–333.
• Steffen L. Lauritzen, Aspects of T. N. Thiele’s Contributions to Statistics. Bulletin of the International Statistical Institute, 58, 27–30, 1999.
External links
• O'Connor, John J.; Robertson, Edmund F., "Thorvald N. Thiele", MacTutor History of Mathematics Archive, University of St Andrews
• Steffen L. Lauritzen, "Aspects of T. N. Thiele’s Contributions to Statistics," Bulletin of the International Statistical Institute, 58 (1999): 27–30.
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| Wikipedia |
Thrackle
A thrackle is an embedding of a graph in the plane in which each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, the crossing must be transverse.[1]
Linear thrackles
A linear thrackle is a thrackle drawn in such a way that its edges are straight line segments. As Paul Erdős observed, every linear thrackle has at most as many edges as vertices. If a vertex v is connected to three or more edges vw, vx, and vy, at least one of those edges (say vw) lies on a line that separates two other edges. Then, w must have degree one, because no line segment ending at w, other than vw, can touch both vx and vy. Removing w and vw produces a smaller thrackle, without changing the difference between the numbers of edges and vertices. After removals like this lead to a thrackle in which every vertex has at most two neighbors, by the handshaking lemma the number of edges is at most the number of vertices.[2] Based on Erdős' proof, one can infer that every linear thrackle is a pseudoforest. Every cycle of odd length may be arranged to form a linear thrackle, but this is not possible for an even-length cycle, because if one edge of the cycle is chosen arbitrarily then the other cycle vertices must lie alternatingly on opposite sides of the line through this edge.
Micha Perles provided another simple proof that linear thrackles have at most n edges, based on the fact that in a linear thrackle every edge has an endpoint at which the edges span an angle of at most 180°, and for which it is the most clockwise edge within this span. For, if not, there would be two edges, incident to opposite endpoints of the edge and lying on opposite sides of the line through the edge, which could not cross each other. But each vertex can only have this property with respect to a single edge, so the number of edges is at most equal to the number of vertices.[3]
As Erdős also observed, the set of pairs of points realizing the diameter of a point set must form a linear thrackle: no two diameters can be disjoint from each other, because if they were then their four endpoints would have a pair at farther distance apart than the two disjoint edges. For this reason, every set of n points in the plane can have at most n diametral pairs, answering a question posed in 1934 by Heinz Hopf and Erika Pannwitz.[4] Andrew Vázsonyi conjectured bounds on the number of diameter pairs in higher dimensions, generalizing this problem.[2]
In computational geometry, the method of rotating calipers can be used to form a linear thrackle from any set of points in convex position, by connecting pairs of points that support parallel lines tangent to the convex hull of the points.[5] This graph contains as a subgraph the thrackle of diameter pairs.[6]
The diameters of the Reinhardt polygons form linear thrackles. An enumeration of linear thrackles may be used to solve the biggest little polygon problem, of finding an n-gon with maximum area relative to its diameter.[7]
Thrackle conjecture
Unsolved problem in mathematics:
Can a thrackle have more edges than vertices?
(more unsolved problems in mathematics)
John H. Conway conjectured that, in any thrackle, the number of edges is at most equal to the number of vertices. Conway himself used the terminology paths and spots (for edges and vertices respectively), so Conway's thrackle conjecture was originally stated in the form every thrackle has at least as many spots as paths. Conway offered a $1000 prize for proving or disproving this conjecture, as part of a set of prize problems also including Conway's 99-graph problem, the minimum spacing of Danzer sets, and the winner of Sylver coinage after the move 16.[8]
Equivalently, the thrackle conjecture may be stated as every thrackle is a pseudoforest. More specifically, if the thrackle conjecture is true, the thrackles may be exactly characterized by a result of Woodall: they are the pseudoforests in which there is no cycle of length four and at most one odd cycle.[1][9]
It has been proved that every cycle graph other than C4 has a thrackle embedding, which shows that the conjecture is sharp. That is, there are thrackles having the same number of spots as paths. At the other extreme, the worst-case scenario is that the number of spots is twice the number of paths; this is also attainable.
The thrackle conjecture is known to be true for thrackles drawn in such a way that every edge is an x-monotone curve, crossed at most once by every vertical line.[3]
Known bounds
Lovász, Pach & Szegedy (1997) proved that every bipartite thrackle is a planar graph, although not drawn in a planar way.[1] As a consequence, they show that every thrackleable graph with n vertices has at most 2n − 3 edges. Since then, this bound has been improved several times. First, it was improved to 3(n − 1)/2,[10] and another improvement led to a bound of roughly 1.428n.[11] Moreover, the method used to prove the latter result yields for any ε > 0 a finite algorithm that either improves the bound to (1 + ε)n or disproves the conjecture. The current record is due to Fulek & Pach (2017), who proved a bound of 1.3984n.[12]
If the conjecture is false, a minimal counterexample would have the form of two even cycles sharing a vertex.[9] Therefore, to prove the conjecture, it would suffice to prove that graphs of this type cannot be drawn as thrackles.
References
1. Lovász, L.; Pach, J.; Szegedy, M. (1997), "On Conway's thrackle conjecture", Discrete and Computational Geometry, 18 (4): 369–376, doi:10.1007/PL00009322, MR 1476318. A preliminary version of these results was reviewed in O'Rourke, J. (1995), "Computational geometry column 26", ACM SIGACT News, 26 (2): 15–17, arXiv:cs/9908007, doi:10.1145/202840.202842.
2. Erdős, P. (1946), "On sets of distances of n points" (PDF), American Mathematical Monthly, 53 (5): 248–250, doi:10.2307/2305092, JSTOR 2305092.
3. Pach, János; Sterling, Ethan (2011), "Conway's conjecture for monotone thrackles", American Mathematical Monthly, 118 (6): 544–548, doi:10.4169/amer.math.monthly.118.06.544, MR 2812285.
4. Hopf, H.; Pannwitz, E. (1934), "Aufgabe Nr. 167", Jahresbericht der Deutschen Mathematiker-Vereinigung, 43: 114.
5. Eppstein, David (May 1995), "The Rotating Caliper Graph", The Geometry Junkyard
6. For the fact that the rotating caliper graph contains all diameter pairs, see Shamos, Michael (1978), Computational Geometry (PDF), Doctoral dissertation, Yale University. For the fact that the diameter pairs form a thrackle, see, e.g., Pach & Sterling (2011).
7. Graham, R. L. (1975), "The largest small hexagon" (PDF), Journal of Combinatorial Theory, Series A, 18 (2): 165–170, doi:10.1016/0097-3165(75)90004-7.
8. Conway, John H., Five $1,000 Problems (Update 2017) (PDF), Online Encyclopedia of Integer Sequences, retrieved 2019-02-12
9. Woodall, D. R. (1969), "Thrackles and deadlock", in Welsh, D. J. A. (ed.), Combinatorial Mathematics and Its Applications, Academic Press, pp. 335–348, MR 0277421.
10. Cairns, G.; Nikolayevsky, Y. (2000), "Bounds for generalized thrackles", Discrete and Computational Geometry, 23 (2): 191–206, doi:10.1007/PL00009495, MR 1739605.
11. Fulek, R.; Pach, J. (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, arXiv:1002.3904, doi:10.1007/978-3-642-18469-7_21, MR 2785903.
12. Fulek, R.; Pach, J. (2017), "Thrackles: An improved upper bound", Graph Drawing and Network Visualization: 25th International Symposium, GD 2017, Boston, MA, USA, September 25-27, 2017, Revised Selected Papers, Lecture Notes in Computer Science, vol. 10692, pp. 160–166, arXiv:1708.08037, doi:10.1007/978-3-319-73915-1_14, ISBN 978-3-319-73914-4.
External links
• thrackle.org—website about the problem
| Wikipedia |
Cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here $E$), and is denoted by the symbol $\times $. Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b,[1] and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
If two vectors have the same direction or have the exact opposite direction from each other (that is, they are not linearly independent), or if either one has zero length, then their cross product is zero.[2] More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths.
The cross product is anticommutative (that is, a × b = − b × a) and is distributive over addition (that is, a × (b + c) = a × b + a × c).[1] The space $E$ together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it's why an oriented space is needed). In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space.
The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in n dimensions, take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[3] The cross-product in seven dimensions has undesirable properties, however (e.g. it fails to satisfy the Jacobi identity), so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time.[4] (See § Generalizations, below, for other dimensions.)
Definition
The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics and applied mathematics, the wedge notation a ∧ b is often used (in conjunction with the name vector product),[5][6][7] although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to n dimensions.
The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule[1] and a magnitude equal to the area of the parallelogram that the vectors span.[2]
The cross product is defined by the formula[8][9]
$\mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\ \mathbf {n} $
where:
• θ is the angle between a and b in the plane containing them (hence, it is between 0° and 180°)
• ‖a‖ and ‖b‖ are the magnitudes of vectors a and b
• and n is a unit vector perpendicular to the plane containing a and b, with direction such that the ordered set (a, b, n) is positively-oriented.
If the vectors a and b are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.
Direction
The direction of the vector n depends on the chosen orientation of the space. Conventionally, it is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is anti-commutative; that is, b × a = −(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.
As the cross product operator depends on the orientation of the space, in general the cross product of two vectors is not a "true" vector, but a pseudovector. See § Handedness for more detail.
Names and origin
In 1842, William Rowan Hamilton discovered the algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors.
In 1881, Josiah Willard Gibbs,[10] and independently Oliver Heaviside, introduced the notation for both the dot product and the cross product using a period (a ⋅ b) and an "×" (a × b), respectively, to denote them.[11]
In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations.[11] These alternative names are still widely used in the literature.
Both the cross notation (a × b) and the name cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a ⋅ b involves multiplications between corresponding components of a and b. As explained below, the cross product can be expressed in the form of a determinant of a special 3 × 3 matrix. According to Sarrus's rule, this involves multiplications between matrix elements identified by crossed diagonals.
Computing
Coordinate notation
If (i, j, k) is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities[1]
${\begin{alignedat}{2}\mathbf {\color {blue}{i}} &\times \mathbf {\color {red}{j}} &&=\mathbf {\color {green}{k}} \\\mathbf {\color {red}{j}} &\times \mathbf {\color {green}{k}} &&=\mathbf {\color {blue}{i}} \\\mathbf {\color {green}{k}} &\times \mathbf {\color {blue}{i}} &&=\mathbf {\color {red}{j}} \end{alignedat}}$
which imply, by the anticommutativity of the cross product, that
${\begin{alignedat}{2}\mathbf {\color {red}{j}} &\times \mathbf {\color {blue}{i}} &&=-\mathbf {\color {green}{k}} \\\mathbf {\color {green}{k}} &\times \mathbf {\color {red}{j}} &&=-\mathbf {\color {blue}{i}} \\\mathbf {\color {blue}{i}} &\times \mathbf {\color {green}{k}} &&=-\mathbf {\color {red}{j}} \end{alignedat}}$
The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that
$\mathbf {\color {blue}{i}} \times \mathbf {\color {blue}{i}} =\mathbf {\color {red}{j}} \times \mathbf {\color {red}{j}} =\mathbf {\color {green}{k}} \times \mathbf {\color {green}{k}} =\mathbf {0} $ (the zero vector).
These equalities, together with the distributivity and linearity of the cross product (though neither follows easily from the definition given above), are sufficient to determine the cross product of any two vectors a and b. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors:
${\begin{alignedat}{3}\mathbf {a} &=a_{1}\mathbf {\color {blue}{i}} &&+a_{2}\mathbf {\color {red}{j}} &&+a_{3}\mathbf {\color {green}{k}} \\\mathbf {b} &=b_{1}\mathbf {\color {blue}{i}} &&+b_{2}\mathbf {\color {red}{j}} &&+b_{3}\mathbf {\color {green}{k}} \end{alignedat}}$
Their cross product a × b can be expanded using distributivity:
${\begin{aligned}\mathbf {a} \times \mathbf {b} ={}&(a_{1}\mathbf {\color {blue}{i}} +a_{2}\mathbf {\color {red}{j}} +a_{3}\mathbf {\color {green}{k}} )\times (b_{1}\mathbf {\color {blue}{i}} +b_{2}\mathbf {\color {red}{j}} +b_{3}\mathbf {\color {green}{k}} )\\={}&a_{1}b_{1}(\mathbf {\color {blue}{i}} \times \mathbf {\color {blue}{i}} )+a_{1}b_{2}(\mathbf {\color {blue}{i}} \times \mathbf {\color {red}{j}} )+a_{1}b_{3}(\mathbf {\color {blue}{i}} \times \mathbf {\color {green}{k}} )+{}\\&a_{2}b_{1}(\mathbf {\color {red}{j}} \times \mathbf {\color {blue}{i}} )+a_{2}b_{2}(\mathbf {\color {red}{j}} \times \mathbf {\color {red}{j}} )+a_{2}b_{3}(\mathbf {\color {red}{j}} \times \mathbf {\color {green}{k}} )+{}\\&a_{3}b_{1}(\mathbf {\color {green}{k}} \times \mathbf {\color {blue}{i}} )+a_{3}b_{2}(\mathbf {\color {green}{k}} \times \mathbf {\color {red}{j}} )+a_{3}b_{3}(\mathbf {\color {green}{k}} \times \mathbf {\color {green}{k}} )\\\end{aligned}}$
This can be interpreted as the decomposition of a × b into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above-mentioned equalities and collecting similar terms, we obtain:
${\begin{aligned}\mathbf {a} \times \mathbf {b} ={}&\quad \ a_{1}b_{1}\mathbf {0} +a_{1}b_{2}\mathbf {\color {green}{k}} -a_{1}b_{3}\mathbf {\color {red}{j}} \\&-a_{2}b_{1}\mathbf {\color {green}{k}} +a_{2}b_{2}\mathbf {0} +a_{2}b_{3}\mathbf {\color {blue}{i}} \\&+a_{3}b_{1}\mathbf {\color {red}{j}} \ -a_{3}b_{2}\mathbf {\color {blue}{i}} \ +a_{3}b_{3}\mathbf {0} \\={}&(a_{2}b_{3}-a_{3}b_{2})\mathbf {\color {blue}{i}} +(a_{3}b_{1}-a_{1}b_{3})\mathbf {\color {red}{j}} +(a_{1}b_{2}-a_{2}b_{1})\mathbf {\color {green}{k}} \\\end{aligned}}$
meaning that the three scalar components of the resulting vector s = s1i + s2j + s3k = a × b are
${\begin{aligned}s_{1}&=a_{2}b_{3}-a_{3}b_{2}\\s_{2}&=a_{3}b_{1}-a_{1}b_{3}\\s_{3}&=a_{1}b_{2}-a_{2}b_{1}\end{aligned}}$
Using column vectors, we can represent the same result as follows:
${\begin{bmatrix}s_{1}\\s_{2}\\s_{3}\end{bmatrix}}={\begin{bmatrix}a_{2}b_{3}-a_{3}b_{2}\\a_{3}b_{1}-a_{1}b_{3}\\a_{1}b_{2}-a_{2}b_{1}\end{bmatrix}}$
Matrix notation
The cross product can also be expressed as the formal determinant:[note 1][1]
$\mathbf {a\times b} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\\end{vmatrix}}$
This determinant can be computed using Sarrus's rule or cofactor expansion. Using Sarrus's rule, it expands to
${\begin{aligned}\mathbf {a\times b} &=(a_{2}b_{3}\mathbf {i} +a_{3}b_{1}\mathbf {j} +a_{1}b_{2}\mathbf {k} )-(a_{3}b_{2}\mathbf {i} +a_{1}b_{3}\mathbf {j} +a_{2}b_{1}\mathbf {k} )\\&=(a_{2}b_{3}-a_{3}b_{2})\mathbf {i} +(a_{3}b_{1}-a_{1}b_{3})\mathbf {j} +(a_{1}b_{2}-a_{2}b_{1})\mathbf {k} .\end{aligned}}$
Using cofactor expansion along the first row instead, it expands to[12]
${\begin{aligned}\mathbf {a\times b} &={\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}\mathbf {i} -{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}\mathbf {j} +{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\mathbf {k} \\&=(a_{2}b_{3}-a_{3}b_{2})\mathbf {i} -(a_{1}b_{3}-a_{3}b_{1})\mathbf {j} +(a_{1}b_{2}-a_{2}b_{1})\mathbf {k} ,\end{aligned}}$
which gives the components of the resulting vector directly.
Using Levi-Civita tensors
• In any basis, the cross-product $a\times b$ is given by the tensorial formula $E_{ijk}a^{i}b^{j}$ where $E_{ijk}$ is the covariant Levi-Civita tensor (we note the position of the indices). That corresponds to the intrinsic formula given here.
• In an orthonormal basis having the same orientation as the space, $a\times b$ is given by the pseudo-tensorial formula $\varepsilon _{ijk}a^{i}b^{j}$ where $\varepsilon _{ijk}$ is the Levi-Civita symbol (which is a pseudo-tensor). That’s the formula used for everyday physics but it works only for this special choice of basis.
• In any orthonormal basis, $a\times b$ is given by the pseudo-tensorial formula $(-1)^{B}\varepsilon _{ijk}a^{i}b^{j}$ where $(-1)^{B}=\pm 1$ indicates whether the basis has the same orientation as the space or not.
The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.
Properties
Geometric meaning
See also: Triple product
The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides (see Figure 1):[1]
$\left\|\mathbf {a} \times \mathbf {b} \right\|=\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\left|\sin \theta \right|.$
Indeed, one can also compute the volume V of a parallelepiped having a, b and c as edges by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2):
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} ).$
Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value:
$V=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|.$
Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of perpendicularity in the same way that the dot product is a measure of parallelism. Given two unit vectors, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel.
Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive).
Algebraic properties
If the cross product of two vectors is the zero vector (that is, a × b = 0), then either one or both of the inputs is the zero vector, (a = 0 or b = 0) or else they are parallel or antiparallel (a ∥ b) so that the sine of the angle between them is zero (θ = 0° or θ = 180° and sin θ = 0).
The self cross product of a vector is the zero vector:
$\mathbf {a} \times \mathbf {a} =\mathbf {0} .$
The cross product is anticommutative,
$\mathbf {a} \times \mathbf {b} =-(\mathbf {b} \times \mathbf {a} ),$
distributive over addition,
$\mathbf {a} \times (\mathbf {b} +\mathbf {c} )=(\mathbf {a} \times \mathbf {b} )+(\mathbf {a} \times \mathbf {c} ),$
and compatible with scalar multiplication so that
$(r\,\mathbf {a} )\times \mathbf {b} =\mathbf {a} \times (r\,\mathbf {b} )=r\,(\mathbf {a} \times \mathbf {b} ).$
It is not associative, but satisfies the Jacobi identity:
$\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=\mathbf {0} .$
Distributivity, linearity and Jacobi identity show that the R3 vector space together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3). The cross product does not obey the cancellation law; that is, a × b = a × c with a ≠ 0 does not imply b = c, but only that:
${\begin{aligned}\mathbf {0} &=(\mathbf {a} \times \mathbf {b} )-(\mathbf {a} \times \mathbf {c} )\\&=\mathbf {a} \times (\mathbf {b} -\mathbf {c} ).\\\end{aligned}}$
This can be the case where b and c cancel, but additionally where a and b − c are parallel; that is, they are related by a scale factor t, leading to:
$\mathbf {c} =\mathbf {b} +t\,\mathbf {a} ,$
for some scalar t.
If, in addition to a × b = a × c and a ≠ 0 as above, it is the case that a ⋅ b = a ⋅ c then
${\begin{aligned}\mathbf {a} \times (\mathbf {b} -\mathbf {c} )&=\mathbf {0} \\\mathbf {a} \cdot (\mathbf {b} -\mathbf {c} )&=0,\end{aligned}}$
As b − c cannot be simultaneously parallel (for the cross product to be 0) and perpendicular (for the dot product to be 0) to a, it must be the case that b and c cancel: b = c.
From the geometrical definition, the cross product is invariant under proper rotations about the axis defined by a × b. In formulae:
$(R\mathbf {a} )\times (R\mathbf {b} )=R(\mathbf {a} \times \mathbf {b} )$, where $R$ is a rotation matrix with $\det(R)=1$.
More generally, the cross product obeys the following identity under matrix transformations:
$(M\mathbf {a} )\times (M\mathbf {b} )=(\det M)\left(M^{-1}\right)^{\mathrm {T} }(\mathbf {a} \times \mathbf {b} )=\operatorname {cof} M(\mathbf {a} \times \mathbf {b} )$
where $M$ is a 3-by-3 matrix and $\left(M^{-1}\right)^{\mathrm {T} }$ is the transpose of the inverse and $\operatorname {cof} $ is the cofactor matrix. It can be readily seen how this formula reduces to the former one if $M$ is a rotation matrix. If $M$ is a 3-by-3 symmetric matrix applied to a generic cross product $\mathbf {a} \times \mathbf {b} $, the following relation holds true:
$M(\mathbf {a} \times \mathbf {b} )=\operatorname {Tr} (M)(\mathbf {a} \times \mathbf {b} )-\mathbf {a} \times M\mathbf {b} +\mathbf {b} \times M\mathbf {a} $
The cross product of two vectors lies in the null space of the 2 × 3 matrix with the vectors as rows:
$\mathbf {a} \times \mathbf {b} \in NS\left({\begin{bmatrix}\mathbf {a} \\\mathbf {b} \end{bmatrix}}\right).$
For the sum of two cross products, the following identity holds:
$\mathbf {a} \times \mathbf {b} +\mathbf {c} \times \mathbf {d} =(\mathbf {a} -\mathbf {c} )\times (\mathbf {b} -\mathbf {d} )+\mathbf {a} \times \mathbf {d} +\mathbf {c} \times \mathbf {b} .$
Differentiation
Main article: Vector-valued function § Derivative and vector multiplication
The product rule of differential calculus applies to any bilinear operation, and therefore also to the cross product:
${\frac {d}{dt}}(\mathbf {a} \times \mathbf {b} )={\frac {d\mathbf {a} }{dt}}\times \mathbf {b} +\mathbf {a} \times {\frac {d\mathbf {b} }{dt}},$
where a and b are vectors that depend on the real variable t.
Triple product expansion
Main article: Triple product
The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ),$
It is the signed volume of the parallelepiped with edges a, b and c and as such the vectors can be used in any order that's an even permutation of the above ordering. The following therefore are equal:
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} ),$
The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula
${\begin{aligned}\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {c} \cdot \mathbf {a} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )\\(\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {b} (\mathbf {c} \cdot \mathbf {a} )-\mathbf {a} (\mathbf {b} \cdot \mathbf {c} )\end{aligned}}$
The mnemonic "BAC minus CAB" is used to remember the order of the vectors in the right hand member. This formula is used in physics to simplify vector calculations. A special case, regarding gradients and useful in vector calculus, is
${\begin{aligned}\nabla \times (\nabla \times \mathbf {f} )&=\nabla (\nabla \cdot \mathbf {f} )-(\nabla \cdot \nabla )\mathbf {f} \\&=\nabla (\nabla \cdot \mathbf {f} )-\nabla ^{2}\mathbf {f} ,\\\end{aligned}}$
where ∇2 is the vector Laplacian operator.
Other identities relate the cross product to the scalar triple product:
${\begin{aligned}(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )&=(\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\mathbf {a} \\(\mathbf {a} \times \mathbf {b} )\cdot (\mathbf {c} \times \mathbf {d} )&=\mathbf {b} ^{\mathrm {T} }\left(\left(\mathbf {c} ^{\mathrm {T} }\mathbf {a} \right)I-\mathbf {c} \mathbf {a} ^{\mathrm {T} }\right)\mathbf {d} \\&=(\mathbf {a} \cdot \mathbf {c} )(\mathbf {b} \cdot \mathbf {d} )-(\mathbf {a} \cdot \mathbf {d} )(\mathbf {b} \cdot \mathbf {c} )\end{aligned}}$
where I is the identity matrix.
Alternative formulation
The cross product and the dot product are related by:
$\left\|\mathbf {a} \times \mathbf {b} \right\|^{2}=\left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}.$
The right-hand side is the Gram determinant of a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle θ between the two vectors, as:
$\mathbf {a\cdot b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,$
the above given relationship can be rewritten as follows:
$\left\|\mathbf {a\times b} \right\|^{2}=\left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}\left(1-\cos ^{2}\theta \right).$
Invoking the Pythagorean trigonometric identity one obtains:
$\left\|\mathbf {a} \times \mathbf {b} \right\|=\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\left|\sin \theta \right|,$
which is the magnitude of the cross product expressed in terms of θ, equal to the area of the parallelogram defined by a and b (see definition above).
The combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product.[14]
Lagrange's identity
The relation:
$\left\|\mathbf {a} \times \mathbf {b} \right\|^{2}\equiv \det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {a} &\mathbf {a} \cdot \mathbf {b} \\\mathbf {a} \cdot \mathbf {b} &\mathbf {b} \cdot \mathbf {b} \\\end{bmatrix}}\equiv \left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}.$
can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as:[15]
$\sum _{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\equiv \left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a\cdot b} )^{2}\ ,$
where a and b may be n-dimensional vectors. This also shows that the Riemannian volume form for surfaces is exactly the surface element from vector calculus. In the case where n = 3, combining these two equations results in the expression for the magnitude of the cross product in terms of its components:[16]
${\begin{aligned}&\left\|\mathbf {a} \times \mathbf {b} \right\|^{2}\equiv \sum _{1\leq i<j\leq 3}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\\\equiv {}&\left(a_{1}b_{2}-b_{1}a_{2}\right)^{2}+\left(a_{2}b_{3}-a_{3}b_{2}\right)^{2}+\left(a_{3}b_{1}-a_{1}b_{3}\right)^{2}\ .\end{aligned}}$
The same result is found directly using the components of the cross product found from:
$\mathbf {a} \times \mathbf {b} \equiv \det {\begin{bmatrix}{\hat {\mathbf {i} }}&{\hat {\mathbf {j} }}&{\hat {\mathbf {k} }}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\\end{bmatrix}}.$
In R3, Lagrange's equation is a special case of the multiplicativity |vw| = |v||w| of the norm in the quaternion algebra.
It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet–Cauchy identity:[17][18]
$(\mathbf {a} \times \mathbf {b} )\cdot (\mathbf {c} \times \mathbf {d} )\equiv (\mathbf {a} \cdot \mathbf {c} )(\mathbf {b} \cdot \mathbf {d} )-(\mathbf {a} \cdot \mathbf {d} )(\mathbf {b} \cdot \mathbf {c} ).$
If a = c and b = d this simplifies to the formula above.
Infinitesimal generators of rotations
The cross product conveniently describes the infinitesimal generators of rotations in R3. Specifically, if n is a unit vector in R3 and R(φ, n) denotes a rotation about the axis through the origin specified by n, with angle φ (measured in radians, counterclockwise when viewed from the tip of n), then
$\left.{d \over d\phi }\right|_{\phi =0}R(\phi ,{\boldsymbol {n}}){\boldsymbol {x}}={\boldsymbol {n}}\times {\boldsymbol {x}}$
for every vector x in R3. The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the Lie algebra so(3) of the rotation group SO(3), and we obtain the result that the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3).
Alternative ways to compute
Conversion to matrix multiplication
The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:[17]
${\begin{aligned}\mathbf {a} \times \mathbf {b} =[\mathbf {a} ]_{\times }\mathbf {b} &={\begin{bmatrix}\,0&\!-a_{3}&\,\,a_{2}\\\,\,a_{3}&0&\!-a_{1}\\-a_{2}&\,\,a_{1}&\,0\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}\\\mathbf {a} \times \mathbf {b} ={[\mathbf {b} ]_{\times }}^{\mathrm {\!\!T} }\mathbf {a} &={\begin{bmatrix}\,0&\,\,b_{3}&\!-b_{2}\\-b_{3}&0&\,\,b_{1}\\\,\,b_{2}&\!-b_{1}&\,0\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}},\end{aligned}}$
where superscript T refers to the transpose operation, and [a]× is defined by:
$[\mathbf {a} ]_{\times }{\stackrel {\rm {def}}{=}}{\begin{bmatrix}\,\,0&\!-a_{3}&\,\,\,a_{2}\\\,\,\,a_{3}&0&\!-a_{1}\\\!-a_{2}&\,\,a_{1}&\,\,0\end{bmatrix}}.$
The columns [a]×,i of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit vectors. That is,
$[\mathbf {a} ]_{\times ,i}=\mathbf {a} \times \mathbf {{\hat {e}}_{i}} ,\;i\in \{1,2,3\}$
or
$[\mathbf {a} ]_{\times }=\sum _{i=1}^{3}\left(\mathbf {a} \times \mathbf {{\hat {e}}_{i}} \right)\otimes \mathbf {{\hat {e}}_{i}} ,$
where $\otimes $ is the outer product operator.
Also, if a is itself expressed as a cross product:
$\mathbf {a} =\mathbf {c} \times \mathbf {d} $
then
$[\mathbf {a} ]_{\times }=\mathbf {d} \mathbf {c} ^{\mathrm {T} }-\mathbf {c} \mathbf {d} ^{\mathrm {T} }.$
Proof by substitution
Evaluation of the cross product gives
$\mathbf {a} =\mathbf {c} \times \mathbf {d} ={\begin{pmatrix}c_{2}d_{3}-c_{3}d_{2}\\c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}\end{pmatrix}}$
Hence, the left hand side equals
$[\mathbf {a} ]_{\times }={\begin{bmatrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{bmatrix}}$
Now, for the right hand side,
$\mathbf {c} \mathbf {d} ^{\mathrm {T} }={\begin{bmatrix}c_{1}d_{1}&c_{1}d_{2}&c_{1}d_{3}\\c_{2}d_{1}&c_{2}d_{2}&c_{2}d_{3}\\c_{3}d_{1}&c_{3}d_{2}&c_{3}d_{3}\end{bmatrix}}$
And its transpose is
$\mathbf {d} \mathbf {c} ^{\mathrm {T} }={\begin{bmatrix}c_{1}d_{1}&c_{2}d_{1}&c_{3}d_{1}\\c_{1}d_{2}&c_{2}d_{2}&c_{3}d_{2}\\c_{1}d_{3}&c_{2}d_{3}&c_{3}d_{3}\end{bmatrix}}$
Evaluation of the right hand side gives
$\mathbf {d} \mathbf {c} ^{\mathrm {T} }-\mathbf {c} \mathbf {d} ^{\mathrm {T} }={\begin{bmatrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{bmatrix}}$
Comparison shows that the left hand side equals the right hand side.
This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector.[19] In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors.[19]
This notation is also often much easier to work with, for example, in epipolar geometry.
From the general properties of the cross product follows immediately that
$[\mathbf {a} ]_{\times }\,\mathbf {a} =\mathbf {0} $
and
$\mathbf {a} ^{\mathrm {T} }\,[\mathbf {a} ]_{\times }=\mathbf {0} $
and from fact that [a]× is skew-symmetric it follows that
$\mathbf {b} ^{\mathrm {T} }\,[\mathbf {a} ]_{\times }\,\mathbf {b} =0.$
The above-mentioned triple product expansion (bac–cab rule) can be easily proven using this notation.
As mentioned above, the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3), whose elements can be identified with the 3×3 skew-symmetric matrices. The map a → [a]× provides an isomorphism between R3 and so(3). Under this map, the cross product of 3-vectors corresponds to the commutator of 3x3 skew-symmetric matrices.
Matrix conversion for cross product with canonical base vectors
Denoting with $\mathbf {e} _{i}\in \mathbf {R} ^{3\times 1}$ the $i$-th canonical base vector, the cross product of a generic vector $\mathbf {v} \in \mathbf {R} ^{3\times 1}$ with $\mathbf {e} _{i}$ is given by: $\mathbf {v} \times \mathbf {e} _{i}=\mathbf {C} _{i}\mathbf {v} $, where
$\mathbf {C} _{1}={\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\end{bmatrix}},\quad \mathbf {C} _{2}={\begin{bmatrix}0&0&-1\\0&0&0\\1&0&0\end{bmatrix}},\quad \mathbf {C} _{3}={\begin{bmatrix}0&1&0\\-1&0&0\\0&0&0\end{bmatrix}}$
These matrices share the following properties:
• $\mathbf {C} _{i}^{\textrm {T}}=-\mathbf {C} _{i}$ (skew-symmetric);
• Both trace and determinant are zero;
• ${\text{rank}}(\mathbf {C} _{i})=2$;
• $\mathbf {C} _{i}\mathbf {C} _{i}^{\textrm {T}}=\mathbf {P} _{\mathbf {e} _{i}}^{^{\perp }}$ (see below);
The orthogonal projection matrix of a vector $\mathbf {v} \neq \mathbf {0} $ is given by $\mathbf {P} _{\mathbf {v} }=\mathbf {v} \left(\mathbf {v} ^{\textrm {T}}\mathbf {v} \right)^{-1}\mathbf {v} ^{T}$. The projection matrix onto the orthogonal complement is given by $\mathbf {P} _{\mathbf {v} }^{^{\perp }}=\mathbf {I} -\mathbf {P} _{\mathbf {v} }$, where $\mathbf {I} $ is the identity matrix. For the special case of $\mathbf {v} =\mathbf {e} _{i}$, it can be verified that
$\mathbf {P} _{\mathbf {e} _{1}}^{^{\perp }}={\begin{bmatrix}0&0&0\\0&1&0\\0&0&1\end{bmatrix}},\quad \mathbf {P} _{\mathbf {e} _{2}}^{^{\perp }}={\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\end{bmatrix}},\quad \mathbf {P} _{\mathbf {e} _{3}}^{^{\perp }}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}$
For other properties of orthogonal projection matrices, see projection (linear algebra).
Index notation for tensors
The cross product can alternatively be defined in terms of the Levi-Civita tensor Eijk and a dot product ηmi, which are useful in converting vector notation for tensor applications:
$\mathbf {c} =\mathbf {a\times b} \Leftrightarrow \ c^{m}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\eta ^{mi}E_{ijk}a^{j}b^{k}$
where the indices $i,j,k$ correspond to vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as
$\mathbf {c} =\mathbf {a\times b} \Leftrightarrow \ c^{m}=\eta ^{mi}E_{ijk}a^{j}b^{k}$
in which repeated indices are summed over the values 1 to 3.
In a positively-oriented orthonormal basis ηmi = δmi (the Kronecker delta) and $E_{ijk}=\varepsilon _{ijk}$ (the Levi-Civita symbol). In that case, this representation is another form of the skew-symmetric representation of the cross product:
$[\varepsilon _{ijk}a^{j}]=[\mathbf {a} ]_{\times }.$
In classical mechanics: representing the cross product by using the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are isotropic. (An example: consider a particle in a Hooke's Law potential in three-space, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the cross-product-represented angular momentum, which are made clear by the abovementioned Levi-Civita representation).
Mnemonic
The word "xyzzy" can be used to remember the definition of the cross product.
If
$\mathbf {a} =\mathbf {b} \times \mathbf {c} $
where:
$\mathbf {a} ={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}},\ \mathbf {b} ={\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}},\ \mathbf {c} ={\begin{bmatrix}c_{x}\\c_{y}\\c_{z}\end{bmatrix}}$
then:
$a_{x}=b_{y}c_{z}-b_{z}c_{y}$
$a_{y}=b_{z}c_{x}-b_{x}c_{z}$
$a_{z}=b_{x}c_{y}-b_{y}c_{x}.$
The second and third equations can be obtained from the first by simply vertically rotating the subscripts, x → y → z → x. The problem, of course, is how to remember the first equation, and two options are available for this purpose: either to remember the relevant two diagonals of Sarrus's scheme (those containing i), or to remember the xyzzy sequence.
Since the first diagonal in Sarrus's scheme is just the main diagonal of the above-mentioned 3×3 matrix, the first three letters of the word xyzzy can be very easily remembered.
Cross visualization
Similarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may be helpful for remembering the correct cross product formula.
If
$\mathbf {a} =\mathbf {b} \times \mathbf {c} $
then:
$\mathbf {a} ={\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}\times {\begin{bmatrix}c_{x}\\c_{y}\\c_{z}\end{bmatrix}}.$
If we want to obtain the formula for $a_{x}$ we simply drop the $b_{x}$ and $c_{x}$ from the formula, and take the next two components down:
$a_{x}={\begin{bmatrix}b_{y}\\b_{z}\end{bmatrix}}\times {\begin{bmatrix}c_{y}\\c_{z}\end{bmatrix}}.$
When doing this for $a_{y}$ the next two elements down should "wrap around" the matrix so that after the z component comes the x component. For clarity, when performing this operation for $a_{y}$, the next two components should be z and x (in that order). While for $a_{z}$ the next two components should be taken as x and y.
$a_{y}={\begin{bmatrix}b_{z}\\b_{x}\end{bmatrix}}\times {\begin{bmatrix}c_{z}\\c_{x}\end{bmatrix}},\ a_{z}={\begin{bmatrix}b_{x}\\b_{y}\end{bmatrix}}\times {\begin{bmatrix}c_{x}\\c_{y}\end{bmatrix}}$
For $a_{x}$ then, if we visualize the cross operator as pointing from an element on the left to an element on the right, we can take the first element on the left and simply multiply by the element that the cross points to in the right-hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as well. This results in our $a_{x}$ formula –
$a_{x}=b_{y}c_{z}-b_{z}c_{y}.$
We can do this in the same way for $a_{y}$ and $a_{z}$ to construct their associated formulas.
Applications
The cross product has applications in various contexts. For example, it is used in computational geometry, physics and engineering. A non-exhaustive list of examples follows.
Computational geometry
The cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space.
The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.
In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points $p_{1}=(x_{1},y_{1}),p_{2}=(x_{2},y_{2})$ and $p_{3}=(x_{3},y_{3})$. It corresponds to the direction (upward or downward) of the cross product of the two coplanar vectors defined by the two pairs of points $(p_{1},p_{2})$ and $(p_{1},p_{3})$. The sign of the acute angle is the sign of the expression
$P=(x_{2}-x_{1})(y_{3}-y_{1})-(y_{2}-y_{1})(x_{3}-x_{1}),$
which is the signed length of the cross product of the two vectors.
In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around $p_{1}$ from $p_{2}$ to $p_{3}$, otherwise a negative angle. From another point of view, the sign of $P$ tells whether $p_{3}$ lies to the left or to the right of line $p_{1},p_{2}.$
The cross product is used in calculating the volume of a polyhedron such as a tetrahedron or parallelepiped.
Angular momentum and torque
The angular momentum L of a particle about a given origin is defined as:
$\mathbf {L} =\mathbf {r} \times \mathbf {p} ,$
where r is the position vector of the particle relative to the origin, p is the linear momentum of the particle.
In the same way, the moment M of a force FB applied at point B around point A is given as:
$\mathbf {M} _{\mathrm {A} }=\mathbf {r} _{\mathrm {AB} }\times \mathbf {F} _{\mathrm {B} }\,$
In mechanics the moment of a force is also called torque and written as $\mathbf {\tau } $
Since position r, linear momentum p and force F are all true vectors, both the angular momentum L and the moment of a force M are pseudovectors or axial vectors.
Rigid body
The cross product frequently appears in the description of rigid motions. Two points P and Q on a rigid body can be related by:
$\mathbf {v} _{P}-\mathbf {v} _{Q}={\boldsymbol {\omega }}\times \left(\mathbf {r} _{P}-\mathbf {r} _{Q}\right)\,$
where $\mathbf {r} $ is the point's position, $\mathbf {v} $ is its velocity and ${\boldsymbol {\omega }}$ is the body's angular velocity.
Since position $\mathbf {r} $ and velocity $\mathbf {v} $ are true vectors, the angular velocity ${\boldsymbol {\omega }}$ is a pseudovector or axial vector.
Lorentz force
The cross product is used to describe the Lorentz force experienced by a moving electric charge qe:
$\mathbf {F} =q_{e}\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$
Since velocity v, force F and electric field E are all true vectors, the magnetic field B is a pseudovector.
Other
In vector calculus, the cross product is used to define the formula for the vector operator curl.
The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints.
As an external product
The cross product can be defined in terms of the exterior product. It can be generalized to an external product in other than three dimensions.[20] This view allows a natural geometric interpretation of the cross product. In exterior algebra the exterior product of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector a ∧ b as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge star of the bivector a ∧ b, mapping 2-vectors to vectors:
$a\times b=\star (a\wedge b).$
This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented one-dimensional element – a vector – whereas, for example, in four dimensions the Hodge dual of a bivector is two-dimensional – a bivector. So, only in three dimensions can a vector cross product of a and b be defined as the vector dual to the bivector a ∧ b: it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as a ∧ b has relative to the unit bivector; precisely the properties described above.
Handedness
Consistency
When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. One should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two polar vectors, one must take into account that the result is an axial vector. Therefore, for consistency, the other side must also be an axial vector. More generally, the result of a cross product may be either a polar vector or an axial vector, depending on the type of its operands (polar vectors or axial vectors). Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product:
• polar vector × polar vector = axial vector
• axial vector × axial vector = axial vector
• polar vector × axial vector = polar vector
• axial vector × polar vector = polar vector
or symbolically
• polar × polar = axial
• axial × axial = axial
• polar × axial = polar
• axial × polar = polar
Because the cross product may also be a polar vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a vector triple product involving three polar vectors is a polar vector.
A handedness-free approach is possible using exterior algebra.
The paradox of the orthonormal basis
Let (i, j,k) be an orthonormal basis. The vectors i, j and k don't depend on the orientation of the space. They can even be defined in the absence of any orientation. They can not therefore be axial vectors. But if i and j are polar vectors then k is an axial vector for i × j = k or j × i = k. This is a paradox.
"Axial" and "polar" are physical qualifiers for physical vectors; that is, vectors which represent physical quantities such as the velocity or the magnetic field. The vectors i, j and k are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. There is no contradiction.
Generalizations
There are several ways to generalize the cross product to higher dimensions.
Lie algebra
Main article: Lie algebra
The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity. Many Lie algebras exist, and their study is a major field of mathematics, called Lie theory.
For example, the Heisenberg algebra gives another Lie algebra structure on $\mathbf {R} ^{3}.$ In the basis $\{x,y,z\},$ the product is $[x,y]=z,[x,z]=[y,z]=0.$
Quaternions
Further information: quaternions and spatial rotation
The cross product can also be described in terms of quaternions. In general, if a vector [a1, a2, a3] is represented as the quaternion a1i + a2j + a3k, the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.
Octonions
See also: Seven-dimensional cross product and Octonion
A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.
Exterior product
Main articles: Exterior algebra and Comparison of vector algebra and geometric algebra § Cross and exterior products
In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the exterior product, which has similar properties, except that the exterior product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the exterior product in three dimensions by using the Hodge star operator to map 2-vectors to vectors. The Hodge dual of the exterior product yields an (n − 2)-vector, which is a natural generalization of the cross product in any number of dimensions.
The exterior product and dot product can be combined (through summation) to form the geometric product in geometric algebra.
External product
As mentioned above, the cross product can be interpreted in three dimensions as the Hodge dual of the exterior product. In any finite n dimensions, the Hodge dual of the exterior product of n − 1 vectors is a vector. So, instead of a binary operation, in arbitrary finite dimensions, the cross product is generalized as the Hodge dual of the exterior product of some given n − 1 vectors. This generalization is called external product.[21]
Commutator product
Main articles: Geometric algebra § Extensions of the inner and exterior products, Cross product § Cross product and handedness, and Cross product § Lie algebra
Interpreting the three-dimensional vector space of the algebra as the 2-vector (not the 1-vector) subalgebra of the three-dimensional geometric algebra, where $\mathbf {i} =\mathbf {e_{2}} \mathbf {e_{3}} $, $\mathbf {j} =\mathbf {e_{1}} \mathbf {e_{3}} $, and $\mathbf {k} =\mathbf {e_{1}} \mathbf {e_{2}} $, the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol $\times $. The commutator product is defined for 2-vectors $A$ and $B$ in geometric algebra as:
$A\times B={\tfrac {1}{2}}(AB-BA)$
where $AB$ is the geometric product.[22]
The commutator product could be generalised to arbitrary multivectors in three dimensions, which results in a multivector consisting of only elements of grades 1 (1-vectors/true vectors) and 2 (2-vectors/pseudovectors). While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator of a 1-vector and a 2-vector yields a true vector, corresponding instead to the left and right contractions in geometric algebra. The commutator product of two 2-vectors has no corresponding equivalent product, which is why the commutator product is defined in the first place for 2-vectors. Furthermore, the commutator triple product of three 2-vectors is the same as the vector triple product of the same three pseudovectors in vector algebra. However, the commutator triple product of three 1-vectors in geometric algebra is instead the negative of the vector triple product of the same three true vectors in vector algebra.
Generalizations to higher dimensions is provided by the same commutator product of 2-vectors in higher-dimensional geometric algebras, but the 2-vectors are no longer pseudovectors. Just as the commutator product/cross product of 2-vectors in three dimensions correspond to the simplest Lie algebra, the 2-vector subalgebras of higher dimensional geometric algebra equipped with the commutator product also correspond to the Lie algebras.[23] Also as in three dimensions, the commutator product could be further generalised to arbitrary multivectors.
Multilinear algebra
In the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form,[note 2] a (0,3)-tensor, by raising an index.
In detail, the 3-dimensional volume form defines a product $V\times V\times V\to \mathbf {R} ,$ by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function $V\times V\to V^{*},$ (fixing any two inputs gives a function $V\to \mathbf {R} $ by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism $V\to V^{*},$ and thus this yields a map $V\times V\to V,$ which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".
Translating the above algebra into geometry, the function "volume of the parallelepiped defined by $(a,b,-)$" (where the first two vectors are fixed and the last is an input), which defines a function $V\to \mathbf {R} $, can be represented uniquely as the dot product with a vector: this vector is the cross product $a\times b.$ From this perspective, the cross product is defined by the scalar triple product, $\mathrm {Vol} (a,b,c)=(a\times b)\cdot c.$
In the same way, in higher dimensions one may define generalized cross products by raising indices of the n-dimensional volume form, which is a $(0,n)$-tensor. The most direct generalizations of the cross product are to define either:
• a $(1,n-1)$-tensor, which takes as input $n-1$ vectors, and gives as output 1 vector – an $(n-1)$-ary vector-valued product, or
• a $(n-2,2)$-tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank n − 2 – a binary product with rank n − 2 tensor values. One can also define $(k,n-k)$-tensors for other k.
These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity.
The $(n-1)$-ary product can be described as follows: given $n-1$ vectors $v_{1},\dots ,v_{n-1}$ in $\mathbf {R} ^{n},$ define their generalized cross product $v_{n}=v_{1}\times \cdots \times v_{n-1}$ as:
• perpendicular to the hyperplane defined by the $v_{i},$
• magnitude is the volume of the parallelotope defined by the $v_{i},$ which can be computed as the Gram determinant of the $v_{i},$
• oriented so that $v_{1},\dots ,v_{n}$ is positively oriented.
This is the unique multilinear, alternating product which evaluates to $e_{1}\times \cdots \times e_{n-1}=e_{n}$, $e_{2}\times \cdots \times e_{n}=e_{1},$ and so forth for cyclic permutations of indices.
In coordinates, one can give a formula for this $(n-1)$-ary analogue of the cross product in Rn by:
$\bigwedge _{i=0}^{n-1}\mathbf {v} _{i}={\begin{vmatrix}v_{1}{}^{1}&\cdots &v_{1}{}^{n}\\\vdots &\ddots &\vdots \\v_{n-1}{}^{1}&\cdots &v_{n-1}{}^{n}\\\mathbf {e} _{1}&\cdots &\mathbf {e} _{n}\end{vmatrix}}.$
This formula is identical in structure to the determinant formula for the normal cross product in R3 except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v1, ..., vn−1, Λn–1
i=0
vi) have a positive orientation with respect to (e1, ..., en). If n is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is even, however, the distinction must be kept. This $(n-1)$-ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments. Moreover, the product $[v_{1},\ldots ,v_{n}]:=\bigwedge _{i=0}^{n}v_{i}$ satisfies the Filippov identity,
$[[x_{1},\ldots ,x_{n}],y_{2},\ldots ,y_{n}]]=\sum _{i=1}^{n}[x_{1},\ldots ,x_{i-1},[x_{i},y_{2},\ldots ,y_{n}],x_{i+1},\ldots ,x_{n}],$
and so it endows Rn+1 with a structure of n-Lie algebra (see Proposition 1 of [24]).
History
In 1773, Joseph-Louis Lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions.[25][note 3]
In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms vector and scalar. Given two quaternions [0, u] and [0, v], where u and v are vectors in R3, their quaternion product can be summarized as [−u ⋅ v, u × v]. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.
In 1844, Hermann Grassmann published a geometric algebra not tied to dimension two or three. Grassmann developed several products, including a cross product represented then by [uv].[26] (See also: exterior algebra.)
In 1853, Augustin-Louis Cauchy, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product.[27][28]
In 1878, William Kingdon Clifford published Elements of Dynamic, in which the term vector product is attested. In the book, this product of two vectors is defined to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.[29] (See also: Clifford algebra.)
In 1881 lecture notes, Gibbs represented the cross product by $u\times v$ and called it the skew product.[30][31] In 1901, Gibb's student Edwin Bidwell Wilson edited and extended these lecture notes into the textbook Vector Analysis. Wilson kept the term skew product, but observed that the alternative terms cross product[note 4] and vector product were more frequent.[32]
In 1908, Cesare Burali-Forti and Roberto Marcolongo introduced the vector product notation u ∧ v.[26] This is used in France and other areas until this day, as the symbol $\times $ is already used to denote multiplication and the cartesian product.
See also
• Cartesian product – a product of two sets
• Geometric algebra: Rotating systems
• Multiple cross products – products involving more than three vectors
• Multiplication of vectors
• Quadruple product
• × (the symbol)
Notes
1. Here, "formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product.
2. By a volume form one means a function that takes in n vectors and gives out a scalar, the volume of the parallelotope defined by the vectors: $V\times \cdots \times V\to \mathbf {R} .$ This is an n-ary multilinear skew-symmetric form. In the presence of a basis, such as on $\mathbf {R} ^{n},$ this is given by the determinant, but in an abstract vector space, this is added structure. In terms of G-structures, a volume form is an $SL$-structure.
3. In modern notation, Lagrange defines $\mathbf {\xi } =\mathbf {y} \times \mathbf {z} $, ${\boldsymbol {\eta }}=\mathbf {z} \times \mathbf {x} $, and ${\boldsymbol {\zeta }}=\mathbf {x} \times {\boldsymbol {y}}$. Thereby, the modern $\mathbf {x} $ corresponds to the three variables $(x,x',x'')$ in Lagrange's notation.
4. since A × B is read as "A cross B"
References
1. Weisstein, Eric W. "Cross Product". mathworld.wolfram.com. Retrieved 2020-09-06.
2. "Cross Product". www.mathsisfun.com. Retrieved 2020-09-06.
3. Massey, William S. (December 1983). "Cross products of vectors in higher dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.
4. Arfken, George B. Mathematical Methods for Physicists (4th ed.). Elsevier.
5. Jeffreys, H; Jeffreys, BS (1999). Methods of mathematical physics. Cambridge University Press. OCLC 41158050.
6. Acheson, DJ (1990). Elementary Fluid Dynamics. Oxford University Press. ISBN 0198596790.
7. Howison, Sam (2005). Practical Applied Mathematics. Cambridge University Press. ISBN 0521842743.
8. Wilson 1901, p. 60–61
9. Dennis G. Zill; Michael R. Cullen (2006). "Definition 7.4: Cross product of two vectors". Advanced engineering mathematics (3rd ed.). Jones & Bartlett Learning. p. 324. ISBN 0-7637-4591-X.
10. Edwin Bidwell Wilson (1913). "Chapter II. Direct and Skew Products of Vectors". Vector Analysis. Founded upon the lectures of J. William Gibbs. New Haven: Yale University Press. The dot product is called "direct product", and cross product is called "skew product".
11. A History of Vector Analysis by Michael J. Crowe, Math. UC Davis.
12. Dennis G. Zill; Michael R. Cullen (2006). "Equation 7: a × b as sum of determinants". cited work. Jones & Bartlett Learning. p. 321. ISBN 0-7637-4591-X.
13. M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum's outlines. McGraw Hill. p. 29. ISBN 978-0-07-161545-7.
14. WS Massey (Dec 1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. The American Mathematical Monthly, Vol. 90, No. 10. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537.
15. Vladimir A. Boichenko; Gennadiĭ Alekseevich Leonov; Volker Reitmann (2005). Dimension theory for ordinary differential equations. Vieweg+Teubner Verlag. p. 26. ISBN 3-519-00437-2.
16. Pertti Lounesto (2001). Clifford algebras and spinors (2nd ed.). Cambridge University Press. p. 94. ISBN 0-521-00551-5.
17. Shuangzhe Liu; Gõtz Trenkler (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products". Int J Information and Systems Sciences. Institute for scientific computing and education. 4 (1): 160–177.
18. by Eric W. Weisstein (2003). "Binet-Cauchy identity". CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 228. ISBN 1-58488-347-2.
19. Lounesto, Pertti (2001). Clifford algebras and spinors. Cambridge: Cambridge University Press. pp. 193. ISBN 978-0-521-00551-7.
20. Greub, W. (1978). Multilinear Algebra.
21. Hogben, L, ed. (2007). Handbook of Linear Algebra.
22. Arthur, John W. (2011). Understanding Geometric Algebra for Electromagnetic Theory. IEEE Press. p. 49. ISBN 978-0470941638.
23. Doran, Chris; Lasenby, Anthony (2003). Geometric Algebra for Physicists. Cambridge University Press. pp. 401–408. ISBN 978-0521715959.
24. Filippov, V.T. (1985). "n-Lie algebras". Sibirsk. Mat. Zh. 26 (6): 879–891. doi:10.1007/BF00969110. S2CID 125051596.
25. Lagrange, Joseph-Louis (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires". Oeuvres. Vol. 3. p. 661.
26. Cajori (1929), p. 134.
27. Crowe (1994), p. 83.
28. Cauchy, Augustin-Louis (1900). Ouvres. Vol. 12. p. 16.
29. William Kingdon Clifford (1878) Elements of Dynamic, Part I, page 95, London: MacMillan & Co
30. Gibbs, Josiah Willard (1884). Elements of vector analysis : arranged for the use of students in physics. New Haven : Printed by Tuttle, Morehouse & Taylor.
31. Crowe (1994), p. 154.
32. Wilson (1901), p. 61.
Bibliography
• Cajori, Florian (1929). A History Of Mathematical Notations Volume II. Open Court Publishing. p. 134. ISBN 978-0-486-67766-8.
• Crowe, Michael J. (1994). A History of Vector Analysis. Dover. ISBN 0-486-67910-1.
• E. A. Milne (1948) Vectorial Mechanics, Chapter 2: Vector Product, pp 11 –31, London: Methuen Publishing.
• Wilson, Edwin Bidwell (1901). Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs. Yale University Press.
• T. Levi-Civita; U. Amaldi (1949). Lezioni di meccanica razionale (in Italian). Bologna: Zanichelli editore.
External links
• "Cross product", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• A quick geometrical derivation and interpretation of cross products
• An interactive tutorial created at Syracuse University – (requires java)
• W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).
• The vector product, Mathcentre (UK), 2009
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| Wikipedia |
Solid geometry
Solid geometry or stereometry is the geometry of three-dimensional Euclidean space.[1]
A solid figure is the region bounded by a two-dimensional surface; for example, a solid ball consists of a sphere and its interior.
Solid geometry deals with the measurements of volumes of various solids, including pyramids, prisms (and other polyhedrons), cubes, cylinders, cones (and truncated cones).[2]
History
The Pythagoreans dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius.[3]
Topics
Basic topics in solid geometry and stereometry include:
• incidence of planes and lines
• dihedral angle and solid angle
• the cube, cuboid, parallelepiped
• the tetrahedron and other pyramids
• prisms
• octahedron, dodecahedron, icosahedron
• cones and cylinders
• the sphere
• other quadrics: spheroid, ellipsoid, paraboloid and hyperboloids.
Advanced topics include:
• projective geometry of three dimensions (leading to a proof of Desargues' theorem by using an extra dimension)
• further polyhedra
• descriptive geometry.
List of solid figures
For a more complete list and organization, see List of mathematical shapes.
Whereas a sphere is the surface of a ball, for other solid figures it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder.
Major types of shapes that either constitute or define a volume.
FigureDefinitionsImages
Parallelepiped
• A polyhedron with six faces (hexahedron), each of which is a parallelogram
• A hexahedron with three pairs of parallel faces
• A prism of which the base is a parallelogram
Rhombohedron
• A parallelepiped where all edges are the same length
• A cube, except that its faces are not squares but rhombi
Cuboid
• A convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube[4]
• Some sources also require that each of the faces is a rectangle (so each pair of adjacent faces meets in a right angle). This more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped.[5]
Polyhedron Flat polygonal faces, straight edges and sharp corners or vertices
• Small stellated dodecahedron
• Toroidal polyhedron
Uniform polyhedron Regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)
• Tetrahedron
• Snub dodecahedron
PrismA polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases
Cone Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex
Cylinder Straight parallel sides and a circular or oval cross section
• A solid elliptic cylinder
• A right and an oblique circular cylinder
Ellipsoid A surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation
Lemon A lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)[6]
Hyperboloid A surface that is generated by rotating a hyperbola around one of its principal axes
Techniques
Various techniques and tools are used in solid geometry. Among them, analytic geometry and vector techniques have a major impact by allowing the systematic use of linear equations and matrix algebra, which are important for higher dimensions.
Applications
A major application of solid geometry and stereometry is in 3D computer graphics.
See also
• Ball regions
• Euclidean geometry
• Dimension
• Point
• Planimetry
• Shape
• Lists of shapes
• Solid modeling
• Surface
• Surface area
• Archimedes
Notes
1. The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68.
2. Kiselev 2008.
3. Paraphrased and taken in part from the 1911 Encyclopædia Britannica.
4. Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.
5. Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
6. Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.
References
• Kiselev, A. P. (2008). Geometry. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.
Authority control: National
• France
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| Wikipedia |
The geometry and topology of three-manifolds
The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes for a graduate course taught at Princeton University by William Thurston from 1978 to 1980 describing his work on 3-manifolds. The notes introduced several new ideas into geometric topology, including orbifolds, pleated manifolds, and train tracks.
Distribution
Copies of the original 1980 notes were circulated by Princeton University. Later the Geometry Center at the University of Minnesota sold a loosely bound copy of the notes. In 2002, Sheila Newbery typed the notes in TeX and made a PDF file of the notes available, which can be downloaded from MSRI using the links below. The book (Thurston 1997) is an expanded version of the first three chapters of the notes.
Contents
Chapters 1 to 3 mostly describe basic background material on hyperbolic geometry.
Chapter 4 cover Dehn surgery on hyperbolic manifolds
Chapter 5 covers results related to Mostow's theorem on rigidity
Chapter 6 describes Gromov's invariant and his proof of Mostow's theorem.
Chapter 7 (by Milnor) describes the Lobachevsky function and its applications to computing volumes of hyperbolic 3-manifolds.
Chapter 8 on Kleinian groups introduces Thurston's work on train track and pleated manifolds
Chapter 9 covers convergence of Kleinian groups and hyperbolic manifolds.
Chapter 10 does not exist.
Chapter 11 covers deformations of Kleinian groups.
Chapter 12 does not exist.
Chapter 13 introduces orbifolds.
References
• Canary, R. D.; Epstein, D. B. A.; Green, P. (2006) [1987], "Notes on notes of Thurston", in Canary, Richard D.; Epstein, David; Marden, Albert (eds.), Fundamentals of hyperbolic geometry: selected expositions, London Mathematical Society Lecture Note Series, vol. 328, Cambridge University Press, doi:10.1017/CBO9781139106986, ISBN 978-0-521-61558-7, MR 0903850
• Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes (original notes)
• Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes (TeX version)
• Thurston, William P. (1997), Levy, Silvio (ed.), Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, doi:10.1515/9781400865321, ISBN 978-0-691-08304-9, MR 1435975
| Wikipedia |
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
Introduction
Definition
A topological space $M$ is a 3-manifold if it is a second-countable Hausdorff space and if every point in $M$ has a neighbourhood that is homeomorphic to Euclidean 3-space.
Mathematical theory of 3-manifolds
The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.
Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology.
A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case.
Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
The fundamental groups of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group theory and topological methods.
Invariants describing 3-manifolds
3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let $M$ be a 3-manifold and $\pi =\pi _{1}(M)$ be its fundamental group, then a lot of information can be derived from them. For example, using Poincare duality and the Hurewicz theorem, we have the following homology groups:
${\begin{aligned}H_{0}(M)&=H^{3}(M)=&\mathbb {Z} \\H_{1}(M)&=H^{2}(M)=&\pi /[\pi ,\pi ]\\H_{2}(M)&=H^{1}(M)=&{\text{Hom}}(\pi ,\mathbb {Z} )\\H_{3}(M)&=H^{0}(M)=&\mathbb {Z} \end{aligned}}$
where the last two groups are isomorphic to the group homology and cohomology of $\pi $, respectively; that is,
${\begin{aligned}H_{1}(\pi ;\mathbb {Z} )&\cong \pi /[\pi ,\pi ]\\H^{1}(\pi ;\mathbb {Z} )&\cong {\text{Hom}}(\pi ,\mathbb {Z} )\end{aligned}}$ ;\mathbb {Z} )&\cong \pi /[\pi ,\pi ]\\H^{1}(\pi ;\mathbb {Z} )&\cong {\text{Hom}}(\pi ,\mathbb {Z} )\end{aligned}}}
From this information a basic homotopy theoretic classification of 3-manifolds[1] can be found. Note from the Postnikov tower there is a canonical map
$q:M\to B\pi $
If we take the pushforward of the fundamental class $[M]\in H_{3}(M)$ into $H_{3}(B\pi )$ we get an element $\zeta _{M}=q_{*}([M])$. It turns out the group $\pi $ together with the group homology class $\zeta _{M}\in H_{3}(\pi ,\mathbb {Z} )$ gives a complete algebraic description of the homotopy type of $M$.
Connected sums
One important topological operation is the connected sum of two 3-manifolds $M_{1}\#M_{2}$. In fact, from general theorems in topology, we find for a three manifold with a connected sum decomposition $M=M_{1}\#\cdots \#M_{n}$ the invariants above for $M$ can be computed from the $M_{i}$. In particular
${\begin{aligned}H_{1}(M)&=H_{1}(M_{1})\oplus \cdots \oplus H_{1}(M_{n})\\H_{2}(M)&=H_{2}(M_{1})\oplus \cdots \oplus H_{2}(M_{n})\\\pi _{1}(M)&=\pi _{1}(M_{1})*\cdots *\pi _{1}(M_{n})\end{aligned}}$
Moreover, a 3-manifold $M$ which cannot be described as a connected sum of two 3-manifolds is called prime.
Second homotopy groups
For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it turns out there is a nice description of the second fundamental group as a $\mathbb {Z} [\pi ]$-module.[2] For the special case of having each $\pi _{1}(M_{i})$ is infinite but not cyclic, if we take based embeddings of a 2-sphere
$\sigma _{i}:S^{2}\to M$ where $\sigma _{i}(S^{2})\subset M_{i}-\{B^{3}\}\subset M$
then the second fundamental group has the presentation
$\pi _{2}(M)={\frac {\mathbb {Z} [\pi ]\{\sigma _{1},\ldots ,\sigma _{n}\}}{(\sigma _{1}+\cdots +\sigma _{n})}}$
giving a straightforward computation of this group.
Important examples of 3-manifolds
Euclidean 3-space
Main article: Euclidean 3-space
Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it. This is just the standard 3-dimensional vector space over the real numbers.
3-sphere
Main article: 3-sphere
A 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Many examples of 3-manifolds can be constructed by taking quotients of the 3-sphere by a finite group $\pi $ acting freely on $S^{3}$ via a map $\pi \to {\text{SO}}(4)$, so $M=S^{3}/\pi $.[3]
Real projective 3-space
Main article: Real projective space
Real projective 3-space, or RP3, is the topological space of lines passing through the origin 0 in R4. It is a compact, smooth manifold of dimension 3, and is a special case Gr(1, R4) of a Grassmannian space.
RP3 is (diffeomorphic to) SO(3), hence admits a group structure; the covering map S3 → RP3 is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).
3-torus
Main article: Torus § n-dimensional torus
The 3-dimensional torus is the product of 3 circles. That is:
$\mathbf {T} ^{3}=S^{1}\times S^{1}\times S^{1}.$
The 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer lattice Z3 (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.
A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
Hyperbolic 3-space
Main article: hyperbolic 3-space
Hyperbolic space is a homogeneous space that can be characterized by a constant negative curvature. It is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and models of elliptic geometry (like the 3-sphere) that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.
Poincaré dodecahedral space
Main article: Homology sphere § Poincaré homology sphere
The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone.
In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere.[4][5] In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.[6] However, there is no strong support for the correctness of the model, as yet.
Seifert–Weber space
Main article: Seifert–Weber space
In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.
With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
Gieseking manifold
Main article: Gieseking manifold
In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by Hugo Gieseking (1912).
The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner.[7] Moreover, the angle made by the faces is $\pi /3$. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
Some important classes of 3-manifolds
• Graph manifold
• Haken manifold
• Homology spheres
• Hyperbolic 3-manifold
• I-bundles
• Knot and link complements
• Lens space
• Seifert fiber spaces, Circle bundles
• Spherical 3-manifold
• Surface bundles over the circle
• Torus bundle
Hyperbolic link complements
A hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.
The following examples are particularly well-known and studied.
• Figure eight knot
• Whitehead link
• Borromean rings
The classes are not necessarily mutually exclusive.
Some important structures on 3-manifolds
Contact geometry
Main article: Contact geometry
Contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation on the manifold ('complete integrability').
Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the odd-dimensional extended phase space that includes the time variable.
Haken manifold
A Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
Haken manifolds were introduced by Wolfgang Haken. Haken proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. Jaco and Oertel gave an algorithm to determine if a 3-manifold was Haken.
Essential lamination
An essential lamination is a lamination where every leaf is incompressible and end incompressible, if the complementary regions of the lamination are irreducible, and if there are no spherical leaves.
Essential laminations generalize the incompressible surfaces found in Haken manifolds.
Heegaard splitting
Main article: Heegaard splitting
A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.
Taut foliation
Main article: Taut foliation
A taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface.
Taut foliations were brought to prominence by the work of William Thurston and David Gabai.
Foundational results
Some results are named as conjectures as a result of historical artifacts.
We begin with the purely topological:
Moise's theorem
Main article: Moise's theorem
In geometric topology, Moise's theorem, proved by Edwin E. Moise in, states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure.
As corollary, every compact 3-manifold has a Heegaard splitting.
Prime decomposition theorem
The prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds.
A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.
Kneser–Haken finiteness
Kneser-Haken finiteness says that for each 3-manifold, there is a constant C such that any collection of surfaces of cardinality greater than C must contain parallel elements.
Loop and Sphere theorems
The loop theorem is a generalization of Dehn's lemma and should more properly be called the "disk theorem". It was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.
A simple and useful version of the loop theorem states that if there is a map
$f\colon (D^{2},\partial D^{2})\to (M,\partial M)\,$
with $f|\partial D^{2}$ not nullhomotopic in $\partial M$, then there is an embedding with the same property.
The sphere theorem of Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
One example is the following:
Let $M$ be an orientable 3-manifold such that $\pi _{2}(M)$ is not the trivial group. Then there exists a non-zero element of $\pi _{2}(M)$ having a representative that is an embedding $S^{2}\to M$.
Annulus and Torus theorems
The annulus theorem states that if a pair of disjoint simple closed curves on the boundary of a three manifold are freely homotopic then they cobound a properly embedded annulus. This should not be confused with the high dimensional theorem of the same name.
The torus theorem is as follows: Let M be a compact, irreducible 3-manifold with nonempty boundary. If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus[8]
JSJ decomposition
Main article: JSJ decomposition
The JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.
The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently.[9][10]
Scott core theorem
Main article: Scott core theorem
The Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott.[11] The precise statement is as follows:
Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable.
A simplified proof is given in,[12] and a stronger uniqueness statement is proven in.[13]
Lickorish–Wallace theorem
Main article: Lickorish–Wallace theorem
The Lickorish–Wallace theorem states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with $\pm 1$ surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
Waldhausen's theorems on topological rigidity
Friedhelm Waldhausen's theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.
Waldhausen conjecture on Heegaard splittings
Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.
Smith conjecture
Main article: Smith conjecture
The Smith conjecture (now proven) states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.
Cyclic surgery theorem
Main article: Cyclic surgery theorem
The cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s (the minimal number of times that two simple closed curves in T representing r and s must intersect) is at most 1. Consequently, there are at most three Dehn fillings of M with cyclic fundamental group.
Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem
Thurston's hyperbolic Dehn surgery theorem states: $M(u_{1},u_{2},\dots ,u_{n})$ is hyperbolic as long as a finite set of exceptional slopes $E_{i}$ is avoided for the i-th cusp for each i. In addition, $M(u_{1},u_{2},\dots ,u_{n})$ converges to M in H as all $p_{i}^{2}+q_{i}^{2}\rightarrow \infty $ for all $p_{i}/q_{i}$ corresponding to non-empty Dehn fillings $u_{i}$.
This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.
Another important result by Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the Gromov norm.
Jørgensen also showed that the volume function on this space is a continuous, proper function. Thus by the previous results, nontrivial limits in H are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type $\omega ^{\omega }$. This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov.
Also, Gabai, Meyerhoff & Milley showed that the Weeks manifold has the smallest volume of any closed orientable hyperbolic 3-manifold.
Thurston's hyperbolization theorem for Haken manifolds
Main article: Hyperbolization theorem
One form of Thurston's geometrization theorem states: If M is an compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.
The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
The conditions that the manifold M should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurston geometrization conjecture.
Tameness conjecture, also called the Marden conjecture or tame ends conjecture
Main article: Tameness conjecture
The tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.
The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure conjecture.
Ending lamination conjecture
The ending lamination theorem, originally conjectured by William Thurston and later proven by Jeffrey Brock, Richard Canary, and Yair Minsky, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
Poincaré conjecture
Main article: Poincaré conjecture
The 3-sphere is an especially important 3-manifold because of the now-proven Poincaré conjecture. Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.
After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard S. Hamilton to use the Ricci flow to attack the problem. Perelman introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.
Thurston's geometrization conjecture
Main article: Thurston's geometrization conjecture
Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery. There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
Virtually fibered conjecture and Virtually Haken conjecture
Main articles: Virtually fibered conjecture and Virtually Haken conjecture
The virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.
The virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.
In a posting on the ArXiv on 25 Aug 2009,[14] Daniel Wise implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the Virtually fibered conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.[15] Several more preprints[16] have followed, including the aforementioned longer manuscript by Wise.[17] In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds.[18] The proof built on results of Kahn and Markovic[19][20] in their proof of the Surface subgroup conjecture and results of Wise in proving the Malnormal Special Quotient Theorem[17] and results of Bergeron and Wise for the cubulation of groups.[14] Taken together with Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.
Simple loop conjecture
If $f\colon S\rightarrow T$ is a map of closed connected surfaces such that $f_{\star }\colon \pi _{1}(S)\rightarrow \pi _{1}(T)$ is not injective, then there exists a non-contractible simple closed curve $\alpha \subset S$ such that $f|_{a}$ is homotopically trivial. This conjecture was proven by David Gabai.
Surface subgroup conjecture
Main article: Surface subgroup conjecture
The surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.[21]
Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds. A proof of this case was announced in the Summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared on the arxiv in October 2009.[22] Their paper was published in the Annals of Mathematics in 2012.[23] In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.[24]
Important conjectures
Cabling conjecture
The cabling conjecture states that if Dehn surgery on a knot in the 3-sphere yields a reducible 3-manifold, then that knot is a $(p,q)$-cable on some other knot, and the surgery must have been performed using the slope $pq$.
Lubotzky–Sarnak conjecture
The fundamental group of any finite volume hyperbolic n-manifold does not have Property τ.
References
1. Swarup, G. Ananda (1974). "On a Theorem of C. B. Thomas". Journal of the London Mathematical Society. s2-8 (1): 13–21. doi:10.1112/jlms/s2-8.1.13. ISSN 1469-7750.
2. Swarup, G. Ananda (1973-06-01). "On embedded spheres in 3-manifolds". Mathematische Annalen. 203 (2): 89–102. doi:10.1007/BF01431437. ISSN 1432-1807. S2CID 120672504.
3. Zimmermann, Bruno. On the Classification of Finite Groups Acting on Homology 3-Spheres. CiteSeerX 10.1.1.218.102.
4. "Is the universe a dodecahedron?", article at PhysicsWorld.
5. Luminet, Jean-Pierre; Weeks, Jeffrey; Riazuelo, Alain; Lehoucq, Roland; Uzan, Jean-Phillipe (2003-10-09). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background". Nature. 425 (6958): 593–595. arXiv:astro-ph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579. S2CID 4380713.
6. Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy and Astrophysics. 482 (3): 747–753. arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777. S2CID 1616362.
7. Epstein, David B.A.; Penner, Robert C. (1988). "Euclidean decompositions of noncompact hyperbolic manifolds". Journal of Differential Geometry. 27 (1): 67–80. doi:10.4310/jdg/1214441650. MR 0918457.
8. Feustel, Charles D (1976). "On the torus theorem and its applications". Transactions of the American Mathematical Society. 217: 1–43. doi:10.1090/s0002-9947-1976-0394666-3.
9. Jaco, William; Shalen, Peter B. A new decomposition theorem for irreducible sufficiently-large 3-manifolds. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 71–84, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
10. Johannson, Klaus, Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. ISBN 3-540-09714-7
11. Scott, G. Peter (1973), "Compact submanifolds of 3-manifolds", Journal of the London Mathematical Society, Second Series, 7 (2): 246–250, doi:10.1112/jlms/s2-7.2.246, MR 0326737
12. Rubinstein, J. Hyam; Swarup, Gadde A. (1990), "On Scott's core theorem", Bulletin of the London Mathematical Society, 22 (5): 495–498, doi:10.1112/blms/22.5.495, MR 1082023
13. Harris, Luke; Scott, G. Peter (1996), "The uniqueness of compact cores for 3-manifolds", Pacific Journal of Mathematics, 172 (1): 139–150, doi:10.2140/pjm.1996.172.139, MR 1379290
14. Bergeron, Nicolas; Wise, Daniel T. (2009). "A boundary criterion for cubulation". arXiv:0908.3609 [math.GT].
15. Wise, Daniel T. (2009-10-29), "Research announcement: The structure of groups with a quasiconvex hierarchy", Electronic Research Announcements in Mathematical Sciences, 16: 44–55, doi:10.3934/era.2009.16.44, MR 2558631
16. Haglund and Wise, A combination theorem for special cube complexes,
Hruska and Wise, Finiteness properties of cubulated groups,
Hsu and Wise, Cubulating malnormal amalgams,
http://comet.lehman.cuny.edu/behrstock/cbms/program.html
17. Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
18. Agol, Ian; Groves, Daniel; Manning, Jason (2012). "The virtual Haken conjecture". arXiv:1204.2810 [math.GT].
19. Kahn, Jeremy; Markovic, Vladimir (2009). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". arXiv:0910.5501 [math.GT].
20. Kahn, Jeremy; Markovic, Vladimir (2010). "Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold". arXiv:1012.2828 [math.GT].
21. Robion Kirby, Problems in low-dimensional topology
22. Kahn, Jeremy; Markovic, Vladimir (2009). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". arXiv:0910.5501 [math.GT].
23. Kahn, Jeremy; Markovic, Vladimir (2012), "Immersing almost geodesic surfaces in a closed hyperbolic three manifold", Annals of Mathematics, 175 (3): 1127–1190, arXiv:0910.5501, doi:10.4007/annals.2012.175.3.4, S2CID 32593851
24. "2012 Clay Research Conference". Archived from the original on June 4, 2012. Retrieved Apr 30, 2020.
Further reading
• Hempel, John (2004), 3-manifolds, Providence, RI: American Mathematical Society, doi:10.1090/chel/349, ISBN 0-8218-3695-1, MR 2098385
• Jaco, William H. (1980), Lectures on three-manifold topology, Providence, RI: American Mathematical Society, ISBN 0-8218-1693-4, MR 0565450
• Rolfsen, Dale (1976), Knots and Links, Providence, RI: American Mathematical Society, ISBN 0-914098-16-0, MR 1277811
• Thurston, William P. (1997), Three-dimensional geometry and topology, Princeton, NJ: Princeton University Press, ISBN 0-691-08304-5, MR 1435975
• Adams, Colin Conrad (2004), The Knot Book. An elementary introduction to the mathematical theory of knots. Revised reprint of the 1994 original., Providence, RI: American Mathematical Society, pp. xiv+307, ISBN 0-8050-7380-9, MR 2079925
• Bing, R. H. (1983), The Geometric Topology of 3-Manifolds, Colloquium Publications, vol. 40, Providence, RI: American Mathematical Society, pp. x+238, ISBN 0-8218-1040-5, MR 0928227
• Thurston, William P. (1982). "Three dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. 6 (3): 357–382. doi:10.1090/s0273-0979-1982-15003-0. ISSN 0273-0979.
• Papakyriakopoulos, Christos D. (1957-01-15). "On Dehn's Lemma and the Asphericity of Knots". Proceedings of the National Academy of Sciences. 43 (1): 169–172. Bibcode:1957PNAS...43..169P. doi:10.1073/pnas.43.1.169. ISSN 0027-8424. PMC 528404. PMID 16589993.
• "Topologische Fragen der Differentialgeometrie 43. Gewebe und Gruppen [31–32h]", Gesammelte Abhandlungen / Collected Papers, DE GRUYTER, 2005, doi:10.1515/9783110894516.239, ISBN 978-3-11-089451-6
External links
Wikimedia Commons has media related to 3-manifolds.
• Hatcher, Allen, Notes on basic 3-manifold topology, Cornell University
• Strickland, Neil, A Bestiary of Topological Objects
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| Wikipedia |
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, $\mathbb {T} ^{3}=S^{1}\times S^{1}\times S^{1}.$ In contrast, the usual torus is the Cartesian product of only two circles.
This article is about the three-dimensional space. For the two-dimensional surface with three holes, see triple torus.
The 3-torus is a three-dimensional compact manifold with no boundary. It can be obtained by "gluing" the three pairs of opposite faces of a cube, where being "glued" can be intuitively understood to mean that when a particle moving in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face, producing periodic boundary conditions. Gluing only one pair of opposite faces produces a solid torus while gluing two of these pairs produces the solid space between two nested tori.
In 1984, Alexei Starobinsky and Yakov Borisovich Zel'dovich at the Landau Institute in Moscow proposed a cosmological model where the shape of the universe is a 3-torus.[1]
References
1. Overbeye, Dennis. New York Times 11 March 2003: Web. 16 January 2011. “Universe as Doughnut: New Data, New Debate”
Sources
• Thurston, William P. (1997), Three-dimensional Geometry and Topology, Volume 1, Princeton University Press, p. 31, ISBN 9780691083049.
• Weeks, Jeffrey R. (2001), The Shape of Space (2nd ed.), CRC Press, p. 13, ISBN 9780824748371.
Manifolds (Glossary)
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| Wikipedia |
Three Prisoners problem
The Three Prisoners problem appeared in Martin Gardner's "Mathematical Games" column in Scientific American in 1959.[1][2] It is mathematically equivalent to the Monty Hall problem with car and goat replaced respectively with freedom and execution.
Not to be confused with Prisoner's dilemma, Hangman paradox, or Prisoners and hats puzzle.
Problem
Three prisoners, A, B, and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the two who are going to be executed. "If B is to be pardoned, give me C's name. If C is to be pardoned, give me B's name. And if I'm to be pardoned, secretly flip a coin to decide whether to name B or C."
The warden tells A that B is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and C. Prisoner A secretly tells C the news, who reasons that A's chance of being pardoned is unchanged at 1/3, but he is pleased because his own chance has gone up to 2/3. Which prisoner is correct?
Solution
The answer is that prisoner A did not gain any information about his own fate, since he already knew that the warden would give him the name of someone else. Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as 1/3, the same as both B and C. As the warden says B will be executed, it is either because C will be pardoned (1/3 chance), or A will be pardoned (1/3 chance) and the coin to decide whether to name B or C the warden flipped came up B (1/2 chance; for an overall 1/2 × 1/3 = 1/6 chance B was named because A will be pardoned). Hence, after hearing that B will be executed, the estimate of A's chance of being pardoned is half that of C. This means his chances of being pardoned, now knowing B is not, again are 1/3, but C has a 2/3 chance of being pardoned.
Table
The explanation above may be summarised in the following table. As the warden is asked by A, he can only answer B or C to be executed (or "not pardoned").
Being pardonedWarden: "not B"Warden: "not C"Sum
A1/61/61/3
B01/31/3
C1/301/3
As the warden has answered that B will not be pardoned, the solution comes from the second column "not B". It appears that the odds for A vs. C to be pardoned are 1:2.
Mathematical formulation
Call $A$, $B$ and $C$ the events that the corresponding prisoner will be pardoned, and $b$ the event that the warden tells A that prisoner B is to be executed, then, using Bayes' theorem, the posterior probability of A being pardoned, is:[3]
${\begin{aligned}P(A|b)&={\frac {P(b|A)P(A)}{P(b|A)P(A)+P(b|B)P(B)+P(b|C)P(C)}}\\&={\frac {{\tfrac {1}{2}}\times {\tfrac {1}{3}}}{{\tfrac {1}{2}}\times {\tfrac {1}{3}}+0\times {\tfrac {1}{3}}+1\times {\tfrac {1}{3}}}}={\frac {1}{3}}.\end{aligned}}$
The probability of C being pardoned, on the other hand, is:
${\begin{aligned}P(C|b)&={\frac {P(b|C)P(C)}{P(b|A)P(A)+P(b|B)P(B)+P(b|C)P(C)}}\\&={\frac {1\times {\tfrac {1}{3}}}{{\tfrac {1}{2}}\times {\tfrac {1}{3}}+0\times {\tfrac {1}{3}}+1\times {\tfrac {1}{3}}}}={\frac {2}{3}}.\end{aligned}}$
The crucial difference making A and C unequal is that $P(b|A)={\tfrac {1}{2}}$ but $P(b|C)=1$. If A will be pardoned, the warden can tell A that either B or C is to be executed, and hence $P(b|A)={\tfrac {1}{2}}$; whereas if C will be pardoned, the warden can only tell A that B is executed, so $P(b|C)=1$.
An intuitive explanation
Prisoner A only has a 1/3 chance of pardon. Knowing whether B or C will be executed does not change his chance. After he hears B will be executed, Prisoner A realizes that if he will not get the pardon himself it must only be going to C. That means there is a 2/3 chance for C to get a pardon. This is comparable to the Monty Hall problem.
Enumeration of possible cases
The following scenarios may arise:
1. A is pardoned and the warden mentions B to be executed: 1/3 × 1/2 = 1/6 of the cases
2. A is pardoned and the warden mentions C to be executed: 1/3 × 1/2 = 1/6 of the cases
3. B is pardoned and the warden mentions C to be executed: 1/3 of the cases
4. C is pardoned and the warden mentions B to be executed: 1/3 of the cases
With the stipulation that the warden will choose randomly, in the 1/3 of the time that A is to be pardoned, there is a 1/2 chance he will say B and 1/2 chance he will say C. This means that taken overall, 1/6 of the time (1/3 [that A is pardoned] × 1/2 [that warden says B]), the warden will say B because A will be pardoned, and 1/6 of the time (1/3 [that A is pardoned] × 1/2 [that warden says C]) he will say C because A is being pardoned. This adds up to the total of 1/3 of the time (1/6 + 1/6) A is being pardoned, which is accurate.
It is now clear that if the warden answers B to A (1/2 of all cases), then 1/3 of the time C is pardoned and A will still be executed (case 4), and only 1/6 of the time A is pardoned (case 1). Hence C's chances are (1/3)/(1/2) = 2/3 and A's are (1/6)/(1/2) = 1/3.
The key to this problem is that the warden may not reveal the name of a prisoner who will be pardoned. If we eliminate this requirement, it can demonstrate the original problem in another way. The only change in this example is that prisoner A asks the warden to reveal the fate of one of the other prisoners (not specifying one that will be executed). In this case, the warden flips a coin and chooses one of B and C to reveal the fate of. The cases are as follows:
1. A pardoned, warden says: B executed (1/6)
2. A pardoned, warden says: C executed (1/6)
3. B pardoned, warden says: B pardoned (1/6)
4. B pardoned, warden says: C executed (1/6)
5. C pardoned, warden says: B executed (1/6)
6. C pardoned, warden says: C pardoned (1/6)
Each scenario has a 1/6 probability. The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a 1/6 probability of occurring. However, the warden in the original case cannot reveal the fate of a pardoned prisoner. Therefore, in case 3 for example, since saying "B is pardoned" is not an option, the warden says "C is executed" instead (making it the same as case 4). That leaves cases 4 and 5 each with a 1/3 probability of occurring and leaves us with the same probability as before.
Why the paradox?
The tendency of people to provide the answer 1/2 is likely due to a tendency to ignore context that may seem unimpactful. For example, how the question is posed to the warden can affect the answer. This can be shown by considering a modified case, where $P(A)={\frac {1}{4}},P(B)={\frac {1}{4}},P(C)={\frac {1}{2}}$ and everything else about the problem remains the same.[3] Using Bayes' Theorem once again:
${\begin{aligned}P(A|b)&={\frac {{\tfrac {1}{2}}\times {\tfrac {1}{4}}}{{\tfrac {1}{2}}\times {\tfrac {1}{4}}+0\times {\tfrac {1}{4}}+1\times {\tfrac {1}{2}}}}={\frac {1}{5}}.\end{aligned}}$
However, if A simply asks if B will be executed, and the warden responds with "yes", the probability that A is pardoned becomes:
${\begin{aligned}P(A|b)&={\frac {1\times {\tfrac {1}{4}}}{1\times {\tfrac {1}{4}}+0\times {\tfrac {1}{4}}+1\times {\tfrac {1}{2}}}}={\frac {1}{3}}.\end{aligned}}$ [3]
A similar assumption is that A plans beforehand to ask the warden for this information. A similar case to the above arises if A does not plan to ask the warden anything and the warden simply informs him that he will be executing B.[4]
Another likely overlooked assumption is that the warden has a probabilistic choice. Let us define $p$ as the conditional probability that the warden will name B given that C will be executed. The conditional probability $P(A|b)$ can be then expressed as:[5]
${\begin{aligned}P(A|b)&={\frac {p}{p+1}}\end{aligned}}$
If we assume that $p=1$, that is, that we do not take into account that the warden is making a probabilistic choice, then $P(A|b)={\frac {1}{2}}$. However, the reality of the problem is that the warden is flipping a coin ($p={\frac {1}{2}}$), so $P(A|b)={\frac {1}{3}}$.[4]
Judea Pearl (1988) used a variant of this example to demonstrate that belief updates must depend not merely on the facts observed but also on the experiment (i.e., query) that led to those facts.[6]
Related problems and applications
• Monty Hall problem
• Boy or Girl paradox
• Principle of restricted choice, an application in the card game bridge
• Prisoner's dilemma, a game theory problem
• Sleeping Beauty problem
• Two envelopes problem
References
1. Gardner, Martin (October 1959). "Mathematical Games: Problems involving questions of probability and ambiguity". Scientific American. 201 (4): 174–182. doi:10.1038/scientificamerican1059-174.
2. Gardner, Martin (1959). "Mathematical Games: How three modern mathematicians disproved a celebrated conjecture of Leonhard Euler". Scientific American. 201 (5): 188. doi:10.1038/scientificamerican1159-181.
3. Shimojo, Shinsuke; Ichikawa, Shin'Ichi (August 1990). "Intuitive reasoning about probability: Theoretical and experimental analyses of the "problem of three prisoners"". Cognition. 36 (2): 205. doi:10.1016/0010-0277(89)90012-7. PMID 2752704. S2CID 45658299.
4. Wechsler, Sergio; Esteves, L. G.; Simonis, A.; Peixoto, C. (February 2005). "Indifference, Neutrality and Informativeness: Generalizing the Three Prisoners Paradox". Synthese. 143 (3): 255–272. doi:10.1007/s11229-005-7016-1. JSTOR 20118537. S2CID 16773272. Retrieved 15 December 2021.
5. Billingsley, Patrick (1995). Probability and measure. Wiley Series in Probability and Mathematical Statistics (Third edition of 1979 original ed.). New York: John Wiley & Sons, Inc. Exercise 33.3, pp. 441 and 576. ISBN 0-471-00710-2. MR 1324786.
6. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (First ed.). San Mateo, CA: Morgan Kaufmann.
Further reading
• Frederick Mosteller, Fifty Challenging Problems in Probability, p. 28, at Google Books.
• Richard Isaac, Pleasures of Probability, p. 24, at Google Books.
| Wikipedia |
Three-gap theorem
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two.[1] Unless θ is a rational multiple of π, there will also be at least two distinct distances.
This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi, and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square.
Statement
The three-gap theorem can be stated geometrically in terms of points on a circle. In this form, it states that if one places $n$ points on a circle, at angles of $\theta ,2\theta ,\dots ,n\theta $ from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. An equivalent and more algebraic form involves the fractional parts of multiples of a real number. It states that, for any positive real number $\alpha $ and integer $n$, the fractional parts of the numbers $\alpha ,2\alpha ,\dots ,n\alpha $ divide the unit interval into subintervals with at most three different lengths. The two problems are equivalent under a linear correspondence between the unit interval and the circumference of the circle, and a correspondence between the real number $\alpha $ and the angle $\theta =2\pi \alpha $.[2][3][4]
Applications
Plant growth
In the study of phyllotaxis, the arrangements of leaves on plant stems, it has been observed that each successive leaf on the stems of many plants is turned from the previous leaf by the golden angle, approximately 137.5°. It has been suggested that this angle maximizes the sun-collecting power of the plant's leaves.[5] If one looks end-on at a plant stem that has grown in this way, there will be at most three distinct angles between two leaves that are consecutive in the cyclic order given by this end-on view.[6]
For example, in the figure, the largest of these three angles occurs three times, between the leaves numbered 3 and 6, between leaves 4 and 7, and between leaves 5 and 8. The second-largest angle occurs five times, between leaves 6 and 1, 9 and 4, 7 and 2, 10 and 5, and 8 and 3. And the smallest angle occurs only twice, between leaves 1 and 9 and between leaves 2 and 10. The phenomenon of having three types of distinct gaps depends only on fact that the growth pattern uses a constant rotation angle, and not on the relation of this angle to the golden ratio; the same phenomenon would happen for any other rotation angle, and not just for the golden angle. However, other properties of this growth pattern do depend on the golden ratio. For instance, the fact that golden ratio is a badly approximable number implies that points spaced at this angle along the Fermat spiral (as they are in some models of plant growth) form a Delone set; intuitively, this means that they are uniformly spaced.[7]
Music theory
In music theory, a musical interval describes the ratio in frequency between two musical tones. Intervals are commonly considered consonant or harmonious when they are the ratio of two small integers; for instance, the octave corresponds to the ratio 2:1, while the perfect fifth corresponds to the ratio 3:2.[8] Two tones are commonly considered to be equivalent when they differ by a whole number of octaves; this equivalence can be represented geometrically by the chromatic circle, the points of which represent classes of equivalent tones. Mathematically, this circle can be described as the unit circle in the complex plane, and the point on this circle that represents a given tone can be obtained by the mapping the frequency $\nu $ to the complex number $ \exp(2\pi i\log _{2}\nu )$. An interval with ratio $\rho $ corresponds to the angle $2\pi \log _{2}\rho $ between points on this circle, meaning that two musical tones differ by the given interval when their two points on the circle differ by this angle. For instance, this formula gives $2\pi $ (a whole circle) as the angle corresponding to an octave. Because 3/2 is not a rational power of two, the angle on the chromatic circle that represents a perfect fifth is not a rational multiple of $2\pi $, and similarly other common musical intervals other than the octave do not correspond to rational angles.[9]
A tuning system is a collection of tones used to compose and play music. For instance, the equal temperament commonly used for the piano is a tuning system, consisting of 12 tones equally spaced around the chromatic circle. Some other tuning systems do not space their tones equally, but instead generate them by some number of consecutive multiples of a given interval. An example is the Pythagorean tuning, which is constructed in this way from twelve tones, generated as the consecutive multiples of a perfect fifth in the circle of fifths. The irrational angle formed on the chromatic circle by a perfect fifth is close to 7/12 of a circle, and therefore the twelve tones of the Pythagorean tuning are close to, but not the same as, the twelve tones of equal temperament, which could be generated in the same way using an angle of exactly 7/12 of a circle.[10] Instead of being spaced at angles of exactly 1/12 of a circle, as the tones of equal temperament would be, the tones of the Pythagorean tuning are separated by intervals of two different angles, close to but not exactly 1/12 of a circle, representing two different types of semitones.[11] If the Pythagorean tuning system were extended by one more perfect fifth, to a set of 13 tones, then the sequence of intervals between its tones would include a third, much shorter interval, the Pythagorean comma.[12]
In this context, the three-gap theorem can be used to describe any tuning system that is generated in this way by consecutive multiples of a single interval. Some of these tuning systems (like equal temperament) may have only one interval separating the closest pairs of tones, and some (like the Pythagorean tuning) may have only two different intervals separating the tones, but the three-gap theorem implies that there are always at most three different intervals separating the tones.[13][14]
Mirrored reflection
A Sturmian word is infinite sequences of two symbols (for instance, "H" and "V") describing the sequence of horizontal and vertical reflections of a light ray within a mirrored square, starting along a line of irrational slope. Equivalently, the same sequence describes the sequence of horizontal and vertical lines of the integer grid that are crossed by the starting line. One property that all such sequences have is that, for any positive integer n, the sequence has exactly n + 1 distinct consecutive subsequences of length n. Each subsequence occurs infinitely often with a certain frequency, and the three-gap theorem implies that these n + 1 subsequences occur with at most three distinct frequencies. If there are three frequencies, then the largest frequency must equal the sum of the other two. One proof of this result involves partitioning the y-intercepts of the starting lines (modulo 1) into n + 1 subintervals within which the initial n elements of the sequence are the same, and applying the three-gap theorem to this partition.[15][16]
History and proof
The three-gap theorem was conjectured by Hugo Steinhaus, and its first[17] proofs were found in the late 1950s by Vera T. Sós,[18] János Surányi,[19] and Stanisław Świerczkowski.[20] Later researchers published additional proofs,[21] generalizing this result to higher dimensions[22][23][24][25], and connecting it to topics including continued fractions,[4][26] symmetries and geodesics of Riemannian manifolds,[27] ergodic theory,[28] and the space of planar lattices.[3] Mayero (2000) formalizes a proof using the Coq interactive theorem prover.[2]
The following simple proof is due to Frank Liang. Let θ be the rotation angle generating a set of points as some number of consecutive multiples of θ on a circle. Define a gap to be an arc A of the circle that extends between two adjacent points of the given set, and define a gap to be rigid if its endpoints occur later in the sequence of multiples of θ than any other gap of the same length. From this definition, it follows that every gap has the same length as a rigid gap. If A is a rigid gap, then A + θ is not a gap, because it has the same length and would be one step later. The only ways for this to happen are for one of the endpoints of A to be the last point in the sequence of multiples of θ (so that the corresponding endpoint of A + θ is missing) or for one of the given points to land within A + θ, preventing it from being a gap. A point can only land within A + θ if it is the first point in the sequence of multiples of θ, because otherwise its predecessor in the sequence would land within A, contradicting the assumption that A is a gap. So there can be at most three rigid gaps, the two on either side of the last point and the one in which the predecessor of the first point (if it were part of the sequence) would land. Because there are at most three rigid gaps, there are at most three lengths of gaps.[29][30]
Related results
Liang's proof additionally shows that, when there are exactly three gap lengths, the longest gap length is the sum of the other two. For, in this case, the rotated copy A + θ that has the first point in it is partitioned by that point into two smaller gaps, which must be the other two gaps.[29][30] Liang also proves a more general result, the "$3d$ distance theorem", according to which the union of $d$ different arithmetic progressions on a circle has at most $3d$ different gap lengths.[29] In the three-gap theorem, there is a constant bound on the ratios between the three gaps, if and only if θ/2π is a badly approximable number.[7]
A closely related but earlier theorem, also called the three-gap theorem, is that if A is any arc of the circle, then the integer sequence of multiples of θ that land in A has at most three lengths of gaps between sequence values. Again, if there are three gap lengths then one is the sum of the other two.[31][32]
See also
• Equidistribution theorem
• Lonely runner conjecture
References
1. Allouche, Jean-Paul; Shallit, Jeffrey (2003), "2.6 The Three-Distance Theorem", Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, pp. 53–55, ISBN 9780521823326
2. Mayero, Micaela (2000), "The three gap theorem (Steinhaus conjecture)", Types for Proofs and Programs: International Workshop, TYPES'99, Lökeberg, Sweden, June 12–16, 1999, Selected Papers, Lecture Notes in Computer Science, vol. 1956, Springer, pp. 162–173, arXiv:cs/0609124, doi:10.1007/3-540-44557-9_10, ISBN 978-3-540-41517-6, S2CID 3228597
3. Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663
4. van Ravenstein, Tony (1988), "The three-gap theorem (Steinhaus conjecture)", Journal of the Australian Mathematical Society, Series A, 45 (3): 360–370, doi:10.1017/S1446788700031062, MR 0957201
5. Adam, John A. (2011), A Mathematical Nature Walk, Princeton University Press, pp. 35–41, ISBN 9781400832903
6. van Ravenstein, Tony (1987), "Number sequences and phyllotaxis", Bulletin of the Australian Mathematical Society, 36 (2): 333, doi:10.1017/s0004972700026605
7. Akiyama, Shigeki (March 2020), "Spiral Delone sets and three distance theorem", Nonlinearity, 33 (5): 2533–2540, arXiv:1904.10815, Bibcode:2020Nonli..33.2533A, doi:10.1088/1361-6544/ab74ad, S2CID 129945118
8. Haack, Joel K. (1999), "The mathematics of the just intonation used in the music of Terry Riley", in Sarhangi, Reza (ed.), Bridges: Mathematical Connections in Art, Music, and Science, Southwestern College, Winfield, Kansas: Bridges Conference, pp. 101–110, ISBN 0-9665201-1-4
9. Baroin, Gilles; Calvet, André (2019), "Visualizing temperaments: squaring the circle?", in Montiel, Mariana; Gomez-Martin, Francisco; Agustín-Aquino, Octavio A. (eds.), Mathematics and Computation in Music: 7th International Conference, MCM 2019, Madrid, Spain, June 18–21, 2019, Proceedings, Springer International Publishing, pp. 333–337, doi:10.1007/978-3-030-21392-3_27, S2CID 184482714
10. Carey, Norman; Clampitt, David (October 1989), "Aspects of well-formed scales", Music Theory Spectrum, 11 (2): 187–206, doi:10.2307/745935, JSTOR 745935
11. Strohm, Reinhard; Blackburn, Bonnie J., eds. (2001), Music as Concept and Practice in the Late Middle Ages, Volume 3, Part 1, New Oxford history of music, Oxford University Press, p. 252, ISBN 9780198162056
12. Benson, Donald C. (2003), A Smoother Pebble: Mathematical Explorations, Oxford University Press, p. 51, ISBN 9780198032977
13. Carey, Norman (2007), "Coherence and sameness in well-formed and pairwise well-formed scales", Journal of Mathematics and Music, 1 (2): 79–98, doi:10.1080/17459730701376743, S2CID 120586231
14. Narushima, Terumi (2017), Microtonality and the Tuning Systems of Erv Wilson: Mapping the Harmonic Spectrum, Routledge Studies in Music Theory, Routledge, pp. 90–91, ISBN 9781317513421
15. Lothaire, M. (2002), "Sturmian Words", Algebraic Combinatorics on Words, Cambridge: Cambridge University Press, pp. 40–97, ISBN 978-0-521-81220-7, Zbl 1001.68093. Lothaire uses the property of having $d+1$ words of length $d$ as a definition of Sturmian words, rather than as a consequence of the definition. For the equivalence of this property with the definition stated here, see Theorem 2.1.13, p. 51. For the three frequencies of these words see Theorem 2.2.37, p. 73.
16. Alessandri, Pascal; Berthé, Valérie (1998), "Three distance theorems and combinatorics on words", L'Enseignement mathématique, 44 (1–2): 103–132, MR 1643286; see in particular Section 2.1, "Complexity and frequencies of codings of rotations"
17. Haynes, Alan; Marklof, Jens (2020), "Higher dimensional Steinhaus and Slater problems via homogeneous dynamics", Annales Scientifiques de l'École Normale Supérieure, 53 (2): 537–557, arXiv:1707.04094, doi:10.24033/asens.2427, MR 4094564, S2CID 67851217, The first proofs of this remarkable fact were published in 1957 by Sós, in 1958 by Surányi, and in 1959 by Świerczkowski
18. Sós, V. T. (1958), "On the distribution mod 1 of the sequence $n\alpha $", Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1: 127–134
19. Surányi, J. (1958), "Über die Anordnung der Vielfachen einer reelen Zahl mod 1", Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1: 107–111
20. Świerczkowski, S. (1959), "On successive settings of an arc on the circumference of a circle", Fundamenta Mathematicae, 46 (2): 187–189, doi:10.4064/fm-46-2-187-189, MR 0104651
21. These proofs are briefly surveyed and classified by Marklof & Strömbergsson (2017), from which the following classification of these proofs and many of their references are taken.
22. Halton, John H. (1965), "The distribution of the sequence $\{n\xi \}\,(n=0,\,1,\,2,\,\ldots )$", Mathematical Proceedings of the Cambridge Philosophical Society, 61 (3): 665–670, doi:10.1017/S0305004100039013, MR 0202668, S2CID 123400321
23. Chevallier, Nicolas (2007), "Cyclic groups and the three distance theorem", Canadian Journal of Mathematics, 59 (3): 503–552, doi:10.4153/CJM-2007-022-3, MR 2319157, S2CID 123011205
24. Vijay, Sujith (2008), "Eleven Euclidean distances are enough", Journal of Number Theory, 128 (6): 1655–1661, doi:10.1016/j.jnt.2007.08.016, MR 2419185, S2CID 119655772
25. Bleher, Pavel M.; Homma, Youkow; Ji, Lyndon L.; Roeder, Roland K. W.; Shen, Jeffrey D. (2012), "Nearest neighbor distances on a circle: multidimensional case", Journal of Statistical Physics, 146 (2): 446–465, arXiv:1107.4134, Bibcode:2012JSP...146..446B, doi:10.1007/s10955-011-0367-8, MR 2873022, S2CID 99723
26. Slater, Noel B. (1967), "Gaps and steps for the sequence $n\theta {\bmod {1}}$", Mathematical Proceedings of the Cambridge Philosophical Society, 63 (4): 1115–1123, doi:10.1017/S0305004100042195, MR 0217019, S2CID 121496726
27. Biringer, Ian; Schmidt, Benjamin (2008), "The three gap theorem and Riemannian geometry", Geometriae Dedicata, 136: 175–190, arXiv:0803.1250, doi:10.1007/s10711-008-9283-8, MR 2443351, S2CID 6389675
28. Haynes, Alan; Koivusalo, Henna; Walton, James; Sadun, Lorenzo (2016), "Gaps problems and frequencies of patches in cut and project sets" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (1): 65–85, Bibcode:2016MPCPS.161...65H, doi:10.1017/S0305004116000128, MR 3505670, S2CID 55686324
29. Liang, Frank M. (1979), "A short proof of the $3d$ distance theorem", Discrete Mathematics, 28 (3): 325–326, doi:10.1016/0012-365X(79)90140-7, MR 0548632
30. Shiu, Peter (2018), "A footnote to the three gaps theorem", The American Mathematical Monthly, 125 (3): 264–266, doi:10.1080/00029890.2018.1412210, MR 3768035, S2CID 125810745
31. Slater, N. B. (1950), "The distribution of the integers $N$ for which $\theta N<\phi $", Mathematical Proceedings of the Cambridge Philosophical Society, 46 (4): 525–534, doi:10.1017/S0305004100026086, MR 0041891, S2CID 120454265
32. Florek, K. (1951), "Une remarque sur la répartition des nombres $n\xi \,(\operatorname {mod} 1)$", Colloquium Mathematicum, 2: 323–324
| Wikipedia |
Compound of three tetrahedra
In geometry, a compound of three tetrahedra can be constructed by three tetrahedra rotated by 60 degree turns along an axis of the middle of an edge. It has dihedral symmetry, D3d, order 12. It is a uniform prismatic compound of antiprisms, UC23.
Compound of 3 digonal antiprisms
TypeUniform
compound
Uniform indexUC23 (n=3, p=2, q=1)
Polyhedra3 digonal antiprisms
(tetrahedra)
Faces12 triangles
Edges24
Vertices12
Symmetry groupD6d, order 12
Subgroup restricting
to one constituent
D2d, order 4
It is similar to the compound of two tetrahedra with 90 degree turns. It has the same vertex arrangement as the convex hexagonal antiprism.
Related polytopes
A subset of edges of this compound polyhedron can generate a compound regular skew polygon, with 3 skew squares. Each tetrahedron contains one skew square. This regular compound polygon containing the same symmetry as the uniform polyhedral compound.
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.
External links
• Weisstein, Eric W. "Compound of three tetrahedra". MathWorld.
| Wikipedia |
Three spheres inequality
In mathematics, the three spheres inequality bounds the $L^{2}$ norm of a harmonic function on a given sphere in terms of the $L^{2}$ norm of this function on two spheres, one with bigger radius and one with smaller radius.
Statement of the three spheres inequality
Let $u$ be an harmonic function on $\mathbb {R} ^{n}$. Then for all $0<r_{1}<r<r_{2}$ one has
$\|u\|_{L^{2}(S_{r})}\leq \|u\|_{L^{2}(S_{r_{1}})}^{\alpha }\|u\|_{L^{2}(S_{r_{2}})}^{1-\alpha }$
where $S_{\rho }:=\{x\in \mathbb {R} ^{n}\colon \vert x\vert =\rho \}$ for $\rho >0$ is the sphere of radius $\rho $ centred at the origin and where
$\alpha :={\frac {\log(r_{2}/r)}{\log(r_{2}/r_{1})}}.$ :={\frac {\log(r_{2}/r)}{\log(r_{2}/r_{1})}}.}
Here we use the following normalisation for the $L^{2}$ norm:
$\|u\|_{L^{2}(S_{\rho })}^{2}:=\rho ^{1-n}\int _{\mathbb {S} ^{n-1}}\vert u(\rho {\hat {x}})\vert ^{2}\,d\sigma ({\hat {x}}).$
References
• Korevaar, J.; Meyers, J. L. H. (1994), "Logarithmic convexity for supremum norms of harmonic functions", Bull. London Math. Soc., 26 (4): 353–362, doi:10.1112/blms/26.4.353, MR 1302068
| Wikipedia |
Three-twist knot
In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot[1] in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.
Three-twist knot
Common nameFigure-of-nine knot
Arf invariant0
Braid length6
Braid no.3
Bridge no.2
Crosscap no.2
Crossing no.5
Genus1
Hyperbolic volume2.82812
Stick no.8
Unknotting no.1
Conway notation[32]
A–B notation52
Dowker notation4, 8, 10, 2, 6
Last /Next51 / 61
Other
alternating, hyperbolic, prime, reversible, twist
Properties
The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is
$\Delta (t)=2t-3+2t^{-1},\,$
its Conway polynomial is
$\nabla (z)=2z^{2}+1,\,$
and its Jones polynomial is
$V(q)=q^{-1}-q^{-2}+2q^{-3}-q^{-4}+q^{-5}-q^{-6}.\,$[2]
Because the Alexander polynomial is not monic, the three-twist knot is not fibered.
The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.
If the fibre of the knot in the initial image of this page were cut at the bottom right of the image, and the ends were pulled apart, it would result in a single-stranded figure-of-nine knot (not the figure-of-nine loop).
Example
References
1. Pinsky, Tali (1 September 2017). "On the topology of the Lorenz system". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. The Royal Society. 473 (2205): 20170374. doi:10.1098/rspa.2017.0374. PMC 5627380. PMID 28989313. Retrieved 26 August 2018. (b) the knot with three half-twists, called the 52 knot.
2. "5_2", The Knot Atlas.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
Threefish
Threefish is a symmetric-key tweakable block cipher designed as part of the Skein hash function, an entry in the NIST hash function competition. Threefish uses no S-boxes or other table lookups in order to avoid cache timing attacks;[1] its nonlinearity comes from alternating additions with exclusive ORs. In that respect, it is similar to Salsa20, TEA, and the SHA-3 candidates CubeHash and BLAKE.
Threefish
General
DesignersBruce Schneier, Niels Ferguson, Stefan Lucks, Doug Whiting, Mihir Bellare, Tadayoshi Kohno, Jon Callas, Jesse Walker
First published2008
Related toBlowfish, Twofish
Cipher detail
Key sizes256, 512 or 1024 bits
(key size is equal to block size)
Block sizes256, 512 or 1024 bits
Rounds72 (80 for 1024-bit block size)
Speed6.1 cpb on Core 2.[1]
Best public cryptanalysis
In October 2010, an attack that combines rotational cryptanalysis with the rebound attack was published. The attack mounts a known-key distinguisher against 53 of 72 rounds in Threefish-256, and 57 of 72 rounds in Threefish-512. It also affects the Skein hash function.[2]
Threefish and the Skein hash function were designed by Bruce Schneier, Niels Ferguson, Stefan Lucks, Doug Whiting, Mihir Bellare, Tadayoshi Kohno, Jon Callas, and Jesse Walker.
Description of the cipher
Threefish works on words of 64 bits (unsigned Little endian integers). $w\in \{4,8,16\}$ is the number of plaintext words and also of key words. The tweak consists of two words. All additions and subtractions are defined modulo $2^{64}$.
Key schedule
Threefish encrypts in $r$ rounds and uses ${\frac {r}{4}}+1$ different round keys. After every four rounds, and before the first, $w$ round key words are added to the $w$ data words. To calculate the round keys an additional key word $k_{w}$ is appended to the original key words $k_{0},k_{1},\dots ,k_{w-1}$. Also, an additional tweak word $t_{2}$ is appended to the tweak words $t_{0},t_{1}$.
$k_{w}=C\oplus k_{0}\oplus k_{1}\oplus \dots \oplus k_{w-1};\quad C={\text{0x1BD11BDAA9FC1A22}}$
$t_{2}=t_{0}\oplus t_{1}$
The purpose of the seemingly arbitrary constant $C$ is to frustrate some attacks that take advantage of the relationship between $k_{w}$ and the other keywords.
The round key words $k_{s,i}$ are now defined like this:
$k_{s,i}={\begin{cases}k_{(s+i){\bmod {(}}w+1)}&i=0,\dots ,w-4\\k_{(s+i){\bmod {(}}w+1)}+t_{s{\bmod {3}}}&i=w-3\\k_{(s+i){\bmod {(}}w+1)}+t_{(s+1){\bmod {3}}}&i=w-2\\k_{(s+i){\bmod {(}}w+1)}+s&i=w-1\end{cases}}$
Here $s=0,1,\dots ,r/4$, where $4s$ is the number of the round in which the round key word $k_{s,i}$ is used.
Mix function
The mix function takes a tuple of words $(x_{0},x_{1})$ and returns another tuple of words $(y_{0},y_{1})$. The function is defined like this:
$y_{0}=(x_{0}+x_{1}){\bmod {2}}^{64}$
$y_{1}=(x_{1}\lll R_{(d{\bmod {8}}),j})\oplus y_{0}$
$R_{d,j}$ is a fixed set of rotation constants chosen to achieve quick diffusion.
Permute
The permutation step swaps the positions of the words according to a constant pattern. Bit-level permutation is not achieved in this step, but this is not necessary since the MIX functions provides bit-level permutations in the form of bitwise rotations. The Permute step and rotation constants in the MIX functions are chosen in such a way that the overall effect is complete diffusion of all the bits in a data block.
Because this permutation is fixed and independent of the key, the time needed to compute it does not provide information about the key or plaintext. This is important because on most modern microprocessors performance optimisations can make the time taken to compute an array operation dependent on where the data is stored in memory. In ciphers where array lookup depends on either the key or plaintext (as is the case for the substitution step in AES), it can make the cipher vulnerable to timing attacks by examining the time required for encryption. The permutation is therefore deliberately designed to ensure that it should execute in the same fashion independent of the key being used or the data encrypted.
A full Threefish round
• if $d\;{\bmod {\;}}4=0$ the round key $k_{d/4,i}$ is added to word $i$
• the mix function is applied to pairs of words, the rotation widths $R_{d{\bmod {8}},j}$ depend on round number $d$ and word pair $j\in \{0,\cdots ,w/2-1\}$
• the words are permutated using a permutation independent from the round number
Threefish256 and Threefish512 apply this round $r=72$ times ($d=0,1,\dots ,71$). Threefish1024 applies it 80 times ($d=0,1,\dots ,79$).
Final operations
After all rounds are applied, the last round key words $k_{r/4,i}$ are added to the words and the words are converted back to a string of bytes.
Security
In October 2010, an attack that combines rotational cryptanalysis with the rebound attack was published. The attack mounts a known-key distinguisher against 53 of 72 rounds in Threefish-256, and 57 of 72 rounds in Threefish-512. It also affects the Skein hash function.[2] This is a follow-up to the earlier attack published in February, which breaks 39 and 42 rounds respectively.[3] In response to this attack, the Skein team tweaked the rotation constants used in Threefish and thereby the key schedule constants for round 3 of the NIST hash function competition.[1]
In 2009, a related key boomerang attack against a reduced round Threefish version was published. For the 32-round version, the time complexity is $2^{226}$ and the memory complexity is $2^{12}$; for the 33-round version, the time complexity is $2^{352.17}$ with a negligible memory usage. The attacks also work against the tweaked version of Threefish: for the 32-round version, the time complexity is $2^{222}$ and the memory complexity is $2^{12}$; for the 33-round version, the time complexity is $2^{355.5}$ with a negligible memory usage.[4]
See also
• Twofish
• Blowfish (cipher)
References
1. Ferguson; et al. (2010-10-01). "The Skein Hash Function Family" (PDF). {{cite journal}}: Cite journal requires |journal= (help) The paper in which Threefish was introduced.
2. Dmitry Khovratovich; Ivica Nikolic; Christian Rechberger (2010-10-20). "Rotational Rebound Attacks on Reduced Skein". Cryptology ePrint Archive.
3. Dmitry Khovratovich & Ivica Nikolić (2010). "Rotational Cryptanalysis of ARX" (PDF). University of Luxembourg. {{cite journal}}: Cite journal requires |journal= (help)
4. Jiazhe Chen; Keting Jia (2009-11-01). "Improved Related-key Boomerang Attacks on Round-Reduced Threefish-512". Cryptology ePrint Archive.
External links
• "The Skein Hash Function Family" Homepage of the Skein Hash Function Family.
Block ciphers (security summary)
Common
algorithms
• AES
• Blowfish
• DES (internal mechanics, Triple DES)
• Serpent
• Twofish
Less common
algorithms
• ARIA
• Camellia
• CAST-128
• GOST
• IDEA
• LEA
• RC2
• RC5
• RC6
• SEED
• Skipjack
• TEA
• XTEA
Other
algorithms
• 3-Way
• Akelarre
• Anubis
• BaseKing
• BassOmatic
• BATON
• BEAR and LION
• CAST-256
• Chiasmus
• CIKS-1
• CIPHERUNICORN-A
• CIPHERUNICORN-E
• CLEFIA
• CMEA
• Cobra
• COCONUT98
• Crab
• Cryptomeria/C2
• CRYPTON
• CS-Cipher
• DEAL
• DES-X
• DFC
• E2
• FEAL
• FEA-M
• FROG
• G-DES
• Grand Cru
• Hasty Pudding cipher
• Hierocrypt
• ICE
• IDEA NXT
• Intel Cascade Cipher
• Iraqi
• Kalyna
• KASUMI
• KeeLoq
• KHAZAD
• Khufu and Khafre
• KN-Cipher
• Kuznyechik
• Ladder-DES
• LOKI (97, 89/91)
• Lucifer
• M6
• M8
• MacGuffin
• Madryga
• MAGENTA
• MARS
• Mercy
• MESH
• MISTY1
• MMB
• MULTI2
• MultiSwap
• New Data Seal
• NewDES
• Nimbus
• NOEKEON
• NUSH
• PRESENT
• Prince
• Q
• REDOC
• Red Pike
• S-1
• SAFER
• SAVILLE
• SC2000
• SHACAL
• SHARK
• Simon
• SM4
• Speck
• Spectr-H64
• Square
• SXAL/MBAL
• Threefish
• Treyfer
• UES
• xmx
• XXTEA
• Zodiac
Design
• Feistel network
• Key schedule
• Lai–Massey scheme
• Product cipher
• S-box
• P-box
• SPN
• Confusion and diffusion
• Round
• Avalanche effect
• Block size
• Key size
• Key whitening (Whitening transformation)
Attack
(cryptanalysis)
• Brute-force (EFF DES cracker)
• MITM
• Biclique attack
• 3-subset MITM attack
• Linear (Piling-up lemma)
• Differential
• Impossible
• Truncated
• Higher-order
• Differential-linear
• Distinguishing (Known-key)
• Integral/Square
• Boomerang
• Mod n
• Related-key
• Slide
• Rotational
• Side-channel
• Timing
• Power-monitoring
• Electromagnetic
• Acoustic
• Differential-fault
• XSL
• Interpolation
• Partitioning
• Rubber-hose
• Black-bag
• Davies
• Rebound
• Weak key
• Tau
• Chi-square
• Time/memory/data tradeoff
Standardization
• AES process
• CRYPTREC
• NESSIE
Utilization
• Initialization vector
• Mode of operation
• Padding
Cryptography
General
• History of cryptography
• Outline of cryptography
• Cryptographic protocol
• Authentication protocol
• Cryptographic primitive
• Cryptanalysis
• Cryptocurrency
• Cryptosystem
• Cryptographic nonce
• Cryptovirology
• Hash function
• Cryptographic hash function
• Key derivation function
• Digital signature
• Kleptography
• Key (cryptography)
• Key exchange
• Key generator
• Key schedule
• Key stretching
• Keygen
• Cryptojacking malware
• Ransomware
• Random number generation
• Cryptographically secure pseudorandom number generator (CSPRNG)
• Pseudorandom noise (PRN)
• Secure channel
• Insecure channel
• Subliminal channel
• Encryption
• Decryption
• End-to-end encryption
• Harvest now, decrypt later
• Information-theoretic security
• Plaintext
• Codetext
• Ciphertext
• Shared secret
• Trapdoor function
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| Wikipedia |
Threshold graph
In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations:
1. Addition of a single isolated vertex to the graph.
2. Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices.
For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered.
Threshold graphs were first introduced by Chvátal & Hammer (1977). A chapter on threshold graphs appears in Golumbic (1980), and the book Mahadev & Peled (1995) is devoted to them.
Alternative definitions
An equivalent definition is the following: a graph is a threshold graph if there are a real number $S$ and for each vertex $v$ a real vertex weight $w(v)$ such that for any two vertices $v,u$, $uv$ is an edge if and only if $w(u)+w(v)>S$.
Another equivalent definition is this: a graph is a threshold graph if there are a real number $T$ and for each vertex $v$ a real vertex weight $a(v)$ such that for any vertex set $X\subseteq V$, $X$ is independent if and only if $\sum _{v\in X}a(v)\leq T.$
The name "threshold graph" comes from these definitions: S is the "threshold" for the property of being an edge, or equivalently T is the threshold for being independent.
Threshold graphs also have a forbidden graph characterization: A graph is a threshold graph if and only if it no four of its vertices form an induced subgraph that is a three-edge path graph, a four-edge cycle graph, or a two-edge matching.
Decomposition
From the definition which uses repeated addition of vertices, one can derive an alternative way of uniquely describing a threshold graph, by means of a string of symbols. $\epsilon $ is always the first character of the string, and represents the first vertex of the graph. Every subsequent character is either $u$, which denotes the addition of an isolated vertex (or union vertex), or $j$, which denotes the addition of a dominating vertex (or join vertex). For example, the string $\epsilon uuj$ represents a star graph with three leaves, while $\epsilon uj$ represents a path on three vertices. The graph of the figure can be represented as $\epsilon uuujuuj$
Related classes of graphs and recognition
Threshold graphs are a special case of cographs, split graphs, and trivially perfect graphs. A graph is a threshold graph if and only if it is both a cograph and a split graph. Every graph that is both a trivially perfect graph and the complementary graph of a trivially perfect graph is a threshold graph. Threshold graphs are also a special case of interval graphs. All these relations can be explained in terms of their characterisation by forbidden induced subgraphs. A cograph is a graph with no induced path on four vertices, P4, and a threshold graph is a graph with no induced P4, C4 nor 2K2. C4 is a cycle of four vertices and 2K2 is its complement, that is, two disjoint edges. This also explains why threshold graphs are closed under taking complements; the P4 is self-complementary, hence if a graph is P4-, C4- and 2K2-free, its complement is as well.
Heggernes & Kratsch (2007) showed that threshold graphs can be recognized in linear time; if a graph is not threshold, an obstruction (one of P4, C4, or 2K2) will be output.
See also
• Indifference graph
• Series–parallel graph
• Threshold hypergraphs[1]
References
1. Reiterman, Jan; Rödl, Vojtěch; Šiňajová, Edita; Tůma, Miroslav (1985-04-01). "Threshold hypergraphs". Discrete Mathematics. 54 (2): 193–200. doi:10.1016/0012-365X(85)90080-9. ISSN 0012-365X.
• Chvátal, Václav; Hammer, Peter L. (1977), "Aggregation of inequalities in integer programming", in Hammer, P. L.; Johnson, E. L.; Korte, B. H.; et al. (eds.), Studies in Integer Programming (Proc. Worksh. Bonn 1975), Annals of Discrete Mathematics, vol. 1, Amsterdam: North-Holland, pp. 145–162.
• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, New York: Academic Press. 2nd edition, Annals of Discrete Mathematics, 57, Elsevier, 2004.
• Heggernes, Pinar; Kratsch, Dieter (2007), "Linear-time certifying recognition algorithms and forbidden induced subgraphs" (PDF), Nordic Journal of Computing, 14 (1–2): 87–108 (2008), MR 2460558.
• Mahadev, N. V. R.; Peled, Uri N. (1995), Threshold Graphs and Related Topics, Elsevier.
External links
• Threshold graphs, Information System on Graph Classes and their Inclusions.
| Wikipedia |
Threshold theorem
In quantum computing, the threshold theorem (or quantum fault-tolerance theorem) states that a quantum computer with a physical error rate below a certain threshold can, through application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels. This shows that quantum computers can be made fault-tolerant, as an analogue to von Neumann's threshold theorem for classical computation.[1] This result was proven (for various error models) by the groups of Dorit Aharanov and Michael Ben-Or;[2] Emanuel Knill, Raymond Laflamme, and Wojciech Zurek;[3] and Alexei Kitaev[4] independently.[3] These results built off a paper of Peter Shor,[5] which proved a weaker version of the threshold theorem.
Explanation
The key question that the threshold theorem resolves is whether quantum computers in practice could perform long computations without succumbing to noise. Since a quantum computer will not be able to perform gate operations perfectly, some small constant error is inevitable; hypothetically, this could mean that quantum computers with imperfect gates can only apply a constant number of gates before the computation is destroyed by noise.
Surprisingly, the quantum threshold theorem shows that if the error to perform each gate is a small enough constant, one can perform arbitrarily long quantum computations to arbitrarily good precision, with only some small added overhead in the number of gates. The formal statement of the threshold theorem depends on the types of error correction codes and error model being considered. Quantum Computation and Quantum Information, by Michael Nielsen and Isaac Chuang, gives the general framework for such a theorem:
Threshold theorem for quantum computation[6]: 481 : A quantum circuit on n qubits and containing p(n) gates may be simulated with probability of error at most ε using
$O(\log ^{c}(p(n)/\varepsilon )p(n))$
gates (for some constant c) on hardware whose components fail with probability at most p, provided p is below some constant threshold, $p<p_{\rm {th}}$, and given reasonable assumptions about the noise in the underlying hardware.
Threshold theorems for classical computation have the same form as above, except for classical circuits instead of quantum. The proof strategy for quantum computation is similar to that of classical computation: for any particular error model (such as having each gate fail with independent probability p), use error correcting codes to build better gates out of existing gates. Though these "better gates" are larger, and so are more prone to errors within them, their error-correction properties mean that they have a lower chance of failing than the original gate (provided p is a small-enough constant). Then, one can use these better gates to recursively create even better gates, until one has gates with the desired failure probability, which can be used for the desired quantum circuit. According to quantum information theorist Scott Aaronson:
"The entire content of the Threshold Theorem is that you're correcting errors faster than they're created. That's the whole point, and the whole non-trivial thing that the theorem shows. That's the problem it solves."[7]
Threshold value in practice
Current estimates put the threshold for the surface code on the order of 1%,[8] though estimates range widely and are difficult to calculate due to the exponential difficulty of simulating large quantum systems.[lower-alpha 1] At a 0.1% probability of a depolarizing error, the surface code would require approximately 1,000-10,000 physical qubits per logical data qubit,[9] though more pathological error types could change this figure drastically.
See also
• Quantum error correction schemes
• Physical and logical qubits
• Fault tolerance
Notes
1. It is widely believed that it is exponentially difficult for classical computers to simulate quantum systems. This problem is known as the quantum many body problem. However, quantum computers can simulate many (though not all) Hamiltonians in polynomial time with bounded errors, which is one of the main appeals of quantum computing. This is applicable to chemical simulations, drug discovery, energy production, climate modeling and fertilizer production (e.g. FeMoco) as well. Because of this, quantum computers may be better than classical computers at aiding design of further quantum computers.
References
1. Neumann, J. von (1956-12-31), "Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Components", Automata Studies. (AM-34), Princeton: Princeton University Press, pp. 43–98, doi:10.1515/9781400882618-003, ISBN 978-1-4008-8261-8, retrieved 2020-10-10
2. Aharonov, Dorit; Ben-Or, Michael (2008-01-01). "Fault-Tolerant Quantum Computation with Constant Error Rate". SIAM Journal on Computing. 38 (4): 1207–1282. arXiv:quant-ph/9906129. doi:10.1137/S0097539799359385. ISSN 0097-5397. S2CID 8969800.
3. Knill, E. (1998-01-16). "Resilient Quantum Computation". Science. 279 (5349): 342–345. arXiv:quant-ph/9702058. Bibcode:1998Sci...279..342K. doi:10.1126/science.279.5349.342.
4. Kitaev, A. Yu. (2003-01-01). "Fault-tolerant quantum computation by anyons". Annals of Physics. 303 (1): 2–30. arXiv:quant-ph/9707021. Bibcode:2003AnPhy.303....2K. doi:10.1016/S0003-4916(02)00018-0. ISSN 0003-4916. S2CID 119087885.
5. Shor, P.W. (1996). "Fault-tolerant quantum computation". Proceedings of 37th Conference on Foundations of Computer Science. Burlington, VT, USA: IEEE Comput. Soc. Press. pp. 56–65. doi:10.1109/SFCS.1996.548464. ISBN 978-0-8186-7594-2. S2CID 7508572.
6. Nielsen, Michael A.; Chuang, Isaac L. (June 2012). Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge: Cambridge University Press. ISBN 9780511992773. OCLC 700706156.
7. Aaronson, Scott; Granade, Chris (Fall 2006). "Lecture 14: Skepticism of Quantum Computing". PHYS771: Quantum Computing Since Democritus. Shtetl Optimized. Retrieved 2018-12-27.
8. Fowler, Austin G.; Stephens, Ashley M.; Groszkowski, Peter (2009-11-11). "High-threshold universal quantum computation on the surface code". Physical Review A. 80 (5): 052312. arXiv:0803.0272. Bibcode:2009PhRvA..80e2312F. doi:10.1103/physreva.80.052312. ISSN 1050-2947. S2CID 119228385.
9. Campbell, Earl T.; Terhal, Barbara M.; Vuillot, Christophe (2017-09-13). "Roads towards fault-tolerant universal quantum computation". Nature. 549 (7671): 172–179. arXiv:1612.07330. Bibcode:2017Natur.549..172C. doi:10.1038/nature23460. ISSN 0028-0836. PMID 28905902. S2CID 4446310.
External links
• Gil Kalai. "Perpetual Motion of The 21st Century?".
• Scott Aaronson. "PHYS771 Lecture 14: Skepticism of Quantum Computing": «The entire content of the Threshold Theorem is that you're correcting errors faster than they're created. That's the whole point, and the whole non-trivial thing that the theorem shows. That's the problem it solves.»
Quantum information science
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Quantum
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| Wikipedia |
Critical value
Critical value may refer to:
• In differential topology, a critical value of a differentiable function ƒ : M → N between differentiable manifolds is the image (value of) ƒ(x) in N of a critical point x in M.[1]
• In statistical hypothesis testing, the critical values of a statistical test are the boundaries of the acceptance region of the test.[2] The acceptance region is the set of values of the test statistic for which the null hypothesis is not rejected. Depending on the shape of the acceptance region, there can be one or more than one critical value.
• In complex dynamics, a critical value is the image of a critical point.
• In medicine, a critical value or panic value is a value of a laboratory test that indicates a serious risk to the patient.[3] Laboratory staff may be required to directly notify a physician or clinical staff of these values.
References
1. do Carmo, Manfredo Perdigão (1976). Differential Geometry of Curves and Surfaces. Prentice Hall. ISBN 0-13-212589-7.
2. Hughes, Ann J.; Grawoig, Dennis E. (1971). Statistics: A Foundation for Analysis. Reading, Mass.: Addison-Wesley. p. 191. ISBN 0-201-03021-7.
3. "Laboratory Critical/Panic Value List". Stanford Health Care.
| Wikipedia |
Thue's lemma
In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus.
This article is about modular arithmetic. For Thue's theorem on Diophantine approximations, see Roth's theorem § Discussion.
More precisely, for every pair of integers (a, m) with m > 1, given two positive integers X and Y such that X ≤ m < XY, there are two integers x and y such that
$ay\equiv x{\pmod {m}}$
and
$|x|<X,\quad 0<y<Y.$
Usually, one takes X and Y equal to the smallest integer greater than the square root of m, but the general form is sometimes useful, and makes the uniqueness theorem (below) easier to state.[1]
The first known proof is attributed to Axel Thue (1902)[2] who used a pigeonhole argument.[3][4] It can be used to prove Fermat's theorem on sums of two squares by taking m to be a prime p that is congruent to 1 modulo 4 and taking a to satisfy a2 + 1 = 0 mod p. (Such an "a" is guaranteed for "p" by Wilson's theorem.[5])
Uniqueness
In general, the solution whose existence is asserted by Thue's lemma is not unique. For example, when a = 1 there are usually several solutions (x, y) = (1, 1), (2, 2), (3, 3), ..., provided that X and Y are not too small. Therefore, one may only hope for uniqueness for the rational number x/y, to which a is congruent modulo m if y and m are coprime. Nevertheless, this rational number need not be unique; for example, if m = 5, a = 2 and X = Y = 3, one has the two solutions
$2a+1\equiv -a+2\equiv 0{\pmod {5}}$.
However, for X and Y small enough, if a solution exists, it is unique. More precisely, with above notation, if
$2XY<m,$
and
$ay_{1}-x_{1}\equiv ay_{2}-x_{2}\equiv 0{\pmod {m}}$,
with
$\left|x_{1}\right|<X,\quad \left|y_{1}\right|<Y,$
and
$\left|x_{2}\right|<X,\quad \left|y_{2}\right|<Y,$
then
${\frac {x_{1}}{y_{1}}}={\frac {x_{2}}{y_{2}}}.$
This result is the basis for rational reconstruction, which allows using modular arithmetic for computing rational numbers for which one knows bounds for numerators and denominators.[6]
The proof is rather easy: by multiplying each congruence by the other yi and subtracting, one gets
$y_{2}x_{1}-y_{1}x_{2}\equiv 0{\pmod {m}}.$
The hypotheses imply that each term has an absolute value lower than XY < m/2, and thus that the absolute value of their difference is lower than m. This implies that $y_{2}x_{1}-y_{1}x_{2}=0$, hence the result.
Computing solutions
The original proof of Thue's lemma is not efficient, in the sense that it does not provide any fast method for computing the solution. The extended Euclidean algorithm, allows us to provide a proof that leads to an efficient algorithm that has the same computational complexity of the Euclidean algorithm.[7]
More precisely, given the two integers m and a appearing in Thue's lemma, the extended Euclidean algorithm computes three sequences of integers (ti), (xi) and (yi) such that
$t_{i}m+y_{i}a=x_{i}\quad {\text{for }}i=0,1,...,$
where the xi are non-negative and strictly decreasing. The desired solution is, up to the sign, the first pair (xi, yi) such that xi < X.
See also
• Padé approximant, a similar theory, for approximating Taylor series by rational functions
References
• Shoup, Victor (2005). A Computational Introduction to Number Theory and Algebra (PDF). Cambridge University Press. Retrieved 26 February 2016.
1. Shoup, theorem 2.33
2. Thue, A. (1902), "Et par antydninger til en taltheoretisk metode", Kra. Vidensk. Selsk. Forh., 7: 57–75
3. Clark, Pete L. "Thue's Lemma and Binary Forms". CiteSeerX 10.1.1.151.636. {{cite journal}}: Cite journal requires |journal= (help)
4. Löndahl, Carl (2011-03-22). "Lecture on sums of squares" (PDF). Retrieved 26 February 2016. {{cite journal}}: Cite journal requires |journal= (help)
5. Ore, Oystein (1948), Number Theory and its History, pp. 262–263
6. Shoup, section 4.6
7. Shoup, section 4.5
| Wikipedia |
Thue–Morse sequence
In mathematics, the Thue–Morse sequence or Prouhet–Thue–Morse sequence or parity sequence[1] is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins:
01101001100101101001011001101001.... [1]
The sequence is named after Axel Thue and Marston Morse.
Definition
There are several equivalent ways of defining the Thue–Morse sequence.
Direct definition
To compute the nth element tn, write the number n in binary. If the number of ones in this binary expansion is odd then tn = 1, if even then tn = 0.[2] That is, tn is the even parity bit for n. John H. Conway et al. called numbers n satisfying tn = 1 odious (for odd) numbers and numbers for which tn = 0 evil (for even) numbers. In other words, tn = 0 if n is an evil number and tn = 1 if n is an odious number.
Fast sequence generation
This method leads to a fast method for computing the Thue–Morse sequence: start with t0 = 0, and then, for each n, find the highest-order bit in the binary representation of n that is different from the same bit in the representation of n − 1. If this bit is at an even index, tn differs from tn − 1, and otherwise it is the same as tn − 1.
In pseudo-code form:
def generate_sequence(seq_length: int):
"""Thue–Morse sequence."""
value = 0
for n = 0 to seq_length-1 by 1:
# Note: assumes an even number of bits in the word size, and two's complement arithmetic so that when n == 0, x is odd (e.g. 31 or 63)
x = index_of_highest_one_bit(n ^ (n - 1))
if ((x & 1) == 0):
# bit index is even, so toggle value
value = 1 - value
yield value
The resulting algorithm takes constant time to generate each sequence element, using only a logarithmic number of bits (constant number of words) of memory.[3]
Recurrence relation
The Thue–Morse sequence is the sequence tn satisfying the recurrence relation
${\begin{aligned}t_{0}&=0,\\t_{2n}&=t_{n},\\t_{2n+1}&=1-t_{n},\end{aligned}}$
for all non-negative integers n.[2]
L-system
The Thue–Morse sequence is a morphic word:[4] it is the output of the following Lindenmayer system:
Variables 0, 1
Constants None
Start 0
Rules (0 → 01), (1 → 10)
Characterization using bitwise negation
The Thue–Morse sequence in the form given above, as a sequence of bits, can be defined recursively using the operation of bitwise negation. So, the first element is 0. Then once the first 2n elements have been specified, forming a string s, then the next 2n elements must form the bitwise negation of s. Now we have defined the first 2n+1 elements, and we recurse.
Spelling out the first few steps in detail:
• We start with 0.
• The bitwise negation of 0 is 1.
• Combining these, the first 2 elements are 01.
• The bitwise negation of 01 is 10.
• Combining these, the first 4 elements are 0110.
• The bitwise negation of 0110 is 1001.
• Combining these, the first 8 elements are 01101001.
• And so on.
So
• T0 = 0.
• T1 = 01.
• T2 = 0110.
• T3 = 01101001.
• T4 = 0110100110010110.
• T5 = 01101001100101101001011001101001.
• T6 = 0110100110010110100101100110100110010110011010010110100110010110.
• And so on.
Infinite product
The sequence can also be defined by:
$\prod _{i=0}^{\infty }\left(1-x^{2^{i}}\right)=\sum _{j=0}^{\infty }(-1)^{t_{j}}x^{j},$
where tj is the jth element if we start at j = 0.
Properties
The Thue–Morse sequence contains many squares: instances of the string $XX$, where $X$ denotes the string $A$, ${\overline {A}}$, $A{\overline {A}}A$, or ${\overline {A}}A{\overline {A}}$, where $A=T_{k}$ for some $k\geq 0$ and ${\overline {A}}$ is the bitwise negation of $A$.[5] For instance, if $k=0$, then $A=T_{0}=0$. The square $A{\overline {A}}AA{\overline {A}}A=010010$ appears in $T$ starting at the 16th bit. Since all squares in $T$ are obtained by repeating one of these 4 strings, they all have length $2^{n}$ or $3\cdot 2^{n}$ for some $n\geq 0$. $T$ contains no cubes: instances of $XXX$. There are also no overlapping squares: instances of $0X0X0$ or $1X1X1$.[6][7] The critical exponent of $T$ is 2.[8]
The Thue–Morse sequence is a uniformly recurrent word: given any finite string X in the sequence, there is some length nX (often much longer than the length of X) such that X appears in every block of length nX.[9][10] Notably, the Thue-Morse sequence is uniformly recurrent without being either periodic or eventually periodic (i.e., periodic after some initial nonperiodic segment).[11]
The sequence T2n is a palindrome for any n. Furthermore, let qn be a word obtained by counting the ones between consecutive zeros in T2n . For instance, q1 = 2 and q2 = 2102012. Since Tn does not contain overlapping squares, the words qn are palindromic squarefree words.
The Thue–Morse morphism μ is defined on alphabet {0,1} by the substitution map μ(0) = 01, μ(1) = 10: every 0 in a sequence is replaced with 01 and every 1 with 10.[12] If T is the Thue–Morse sequence, then μ(T) is also T. Thus, T is a fixed point of μ. The morphism μ is a prolongable morphism on the free monoid {0,1}∗ with T as fixed point: T is essentially the only fixed point of μ; the only other fixed point is the bitwise negation of T, which is simply the Thue–Morse sequence on (1,0) instead of on (0,1). This property may be generalized to the concept of an automatic sequence.
The generating series of T over the binary field is the formal power series
$t(Z)=\sum _{n=0}^{\infty }T(n)Z^{n}\ .$
This power series is algebraic over the field of rational functions, satisfying the equation[13]
$Z+(1+Z)^{2}t+(1+Z)^{3}t^{2}=0$
In combinatorial game theory
The set of evil numbers (numbers $n$ with $t_{n}=0$) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim-values.
The Prouhet–Tarry–Escott problem
The Prouhet–Tarry–Escott problem can be defined as: given a positive integer N and a non-negative integer k, partition the set S = { 0, 1, ..., N-1 } into two disjoint subsets S0 and S1 that have equal sums of powers up to k, that is:
$\sum _{x\in S_{0}}x^{i}=\sum _{x\in S_{1}}x^{i}$ for all integers i from 1 to k.
This has a solution if N is a multiple of 2k+1, given by:
• S0 consists of the integers n in S for which tn = 0,
• S1 consists of the integers n in S for which tn = 1.
For example, for N = 8 and k = 2,
0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,
02 + 32 + 52 + 62 = 12 + 22 + 42 + 72.
The condition requiring that N be a multiple of 2k+1 is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of kth powers of any set of N numbers in arithmetic progression can be partitioned into two sets with equal sums. This follows directly from the expansion given by the binomial theorem applied to the binomial representing the nth element of an arithmetic progression.
For generalizations of the Thue–Morse sequence and the Prouhet–Tarry–Escott problem to partitions into more than two parts, see Bolker, Offner, Richman and Zara, "The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences".[14]
Fractals and turtle graphics
Using turtle graphics, a curve can be generated if an automaton is programmed with a sequence. When Thue–Morse sequence members are used in order to select program states:
• If t(n) = 0, move ahead by one unit,
• If t(n) = 1, rotate by an angle of π/3 radians (60°)
The resulting curve converges to the Koch curve, a fractal curve of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence.[15]
It is also possible to draw the curve precisely using the following instructions:[16]
• If t(n) = 0, rotate by an angle of π radians (180°),
• If t(n) = 1, move ahead by one unit, then rotate by an angle of π/3 radians.
Equitable sequencing
In their book on the problem of fair division, Steven Brams and Alan Taylor invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called balanced alternation, or taking turns taking turns taking turns . . . , as a way to circumvent the favoritism inherent when one party chooses before the other. An example showed how a divorcing couple might reach a fair settlement in the distribution of jointly-owned items. The parties would take turns to be the first chooser at different points in the selection process: Ann chooses one item, then Ben does, then Ben chooses one item, then Ann does.[17]
Lionel Levine and Katherine E. Stange, in their discussion of how to fairly apportion a shared meal such as an Ethiopian dinner, proposed the Thue–Morse sequence as a way to reduce the advantage of moving first. They suggested that “it would be interesting to quantify the intuition that the Thue–Morse order tends to produce a fair outcome.”[18]
Robert Richman addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication.[19] He presented the sequences Tn as step functions on the interval [0,1] and described their relationship to the Walsh and Rademacher functions. He showed that the nth derivative can be expressed in terms of Tn. As a consequence, the step function arising from Tn is orthogonal to polynomials of order n − 1. A consequence of this result is that a resource whose value is expressed as a monotonically decreasing continuous function is most fairly allocated using a sequence that converges to Thue–Morse as the function becomes flatter. An example showed how to pour cups of coffee of equal strength from a carafe with a nonlinear concentration gradient, prompting a whimsical article in the popular press.[20]
Joshua Cooper and Aaron Dutle showed why the Thue–Morse order provides a fair outcome for discrete events.[21] They considered the fairest way to stage a Galois duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other's a priori probability of winning exceeded their own. They proved that, as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue–Morse sequence. In so doing, they demonstrated that the Thue–Morse order produces a fair outcome not only for sequences Tn of length 2n, but for sequences of any length.
Thus the mathematics supports using the Thue–Morse sequence instead of alternating turns when the goal is fairness but earlier turns differ monotonically from later turns in some meaningful quality, whether that quality varies continuously[19] or discretely.[21]
Sports competitions form an important class of equitable sequencing problems, because strict alternation often gives an unfair advantage to one team. Ignacio Palacios-Huerta proposed changing the sequential order to Thue–Morse to improve the ex post fairness of various tournament competitions, such as the kicking sequence of a penalty shoot-out in soccer.[22] He did a set of field experiments with pro players and found that the team kicking first won 60% of games using ABAB (or T1), 54% using ABBA (or T2), and 51% using full Thue–Morse (or Tn). As a result, ABBA is undergoing extensive trials in FIFA (European and World Championships) and English Federation professional soccer (EFL Cup).[23] An ABBA serving pattern has also been found to improve the fairness of tennis tie-breaks.[24] In competitive rowing, T2 is the only arrangement of port- and starboard-rowing crew members that eliminates transverse forces (and hence sideways wiggle) on a four-membered coxless racing boat, while T3 is one of only four rigs to avoid wiggle on an eight-membered boat.[25]
Fairness is especially important in player drafts. Many professional sports leagues attempt to achieve competitive parity by giving earlier selections in each round to weaker teams. By contrast, fantasy football leagues have no pre-existing imbalance to correct, so they often use a “snake” draft (forward, backward, etc.; or T1).[26] Ian Allan argued that a “third-round reversal” (forward, backward, backward, forward, etc.; or T2) would be even more fair.[27] Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a pick-up game of basketball mirrors T3: captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices.[19]
Hash collisions
The initial 2k bits of the Thue–Morse sequence are mapped to 0 by a wide class of polynomial hash functions modulo a power of two, which can lead to hash collisions.[28]
History
The Thue–Morse sequence was first studied by Eugène Prouhet in 1851,[29] who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent the threefold repetition rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw. At the time, consecutive identical board states were required to trigger the rule; the rule was later amended to the same board position reoccurring three times at any point, as the sequence shows that the consecutive criterion can be evaded forever.
See also
• Dejean's theorem
• Fabius function
• Gray code[30][31][32]
• Komornik–Loreti constant
• Prouhet–Thue–Morse constant
Notes
1. Sloane, N. J. A. (ed.). "Sequence A010060 (Thue-Morse sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Allouche & Shallit (2003, p. 15)
3. Arndt (2011).
4. Lothaire (2011, p. 11)
5. Brlek (1989).
6. Lothaire (2011, p. 113)
7. Pytheas Fogg (2002, p. 103)
8. Krieger (2006).
9. Lothaire (2011, p. 30)
10. Berthé & Rigo (2010).
11. Lothaire (2011, p. 31)
12. Berstel et al. (2009, p. 70)
13. Berstel et al. (2009, p. 80)
14. Bolker et al. (2016).
15. Ma & Holdener (2005).
16. Abel, Zachary (January 23, 2012). "Thue-Morse Navigating Turtles". Three-Cornered Things.
17. Brams & Taylor (1999).
18. Levine & Stange (2012).
19. Richman (2001)
20. Abrahams (2010).
21. Cooper & Dutle (2013)
22. Palacios-Huerta (2012).
23. Palacios-Huerta (2014).
24. Cohen-Zada, Krumer & Shapir (2018).
25. Barrow (2010).
26. "Fantasy Draft Types". NFL.com. Archived from the original on October 12, 2018.
27. Allan, Ian (July 16, 2014). "Third-Round Reversal Drafts". Fantasy Index. Retrieved September 1, 2020.
28. Pachocki, Jakub; Radoszewski, Jakub (2013). "Where to Use and How not to Use Polynomial String Hashing" (PDF). Olympiads in Informatics. 7: 90–100.
29. The ubiquitous Prouhet-Thue-Morse sequence by Jean-Paul Allouche and Jeffrey Shallit
30. Fredricksen, Harold (1992). "Gray codes and the Thue-Morse-Hedlund sequence". Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC). Naval Postgraduate School, Department of Mathematics, Monterey, California, USA. 11: 3–11.
31. Erickson, John (2018-10-30). "On the Asymptotic Relative Change for Sequences of Permutations". Retrieved 2021-01-31.
32. Plousos, George (2020-06-21). "What is the relationship between the Gray code and the Thue–Morse sequence". Quora. Archived from the original on 2020-12-17. Retrieved 2021-01-31.
References
• Abrahams, Marc (12 July 2010). "How to pour the perfect cup of coffee". The Guardian.
• Arndt, Jörg (2011). "1.16.4 The Thue–Morse sequence" (PDF). Matters Computational: Ideas, Algorithms, Source Code. Springer. p. 44.
• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Barrow, John D. (2010). "Rowing and the Same-Sum Problem Have Their Moments". American Journal of Physics. 78 (7): 728–732. arXiv:0911.3551. Bibcode:2010AmJPh..78..728B. doi:10.1119/1.3318808. S2CID 119207447.
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 27. Providence, RI, USA: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
• Berthé, Valérie; Rigo, Michel, eds. (2010). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. p. 7. ISBN 978-0-521-51597-9. Zbl 1197.68006.
• Bolker, Ethan; Offner, Carl; Richman, Robert; Zara, Catalin (2016). "The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences". Journal of Combinatorics. 7 (1): 117–133. arXiv:1304.6756. doi:10.4310/JOC.2016.v7.n1.a5. S2CID 118040795.}
• Brams, Steven J.; Taylor, Alan D. (1999). The Win-Win Solution: Guaranteeing Fair Shares to Everybody. W. W. Norton & Co., Inc. pp. 36–44. ISBN 978-0-393-04729-5.
• Brlek, Srećko (1989). "Enumeration of Factors in the Thue-Morse Word". Discrete Applied Mathematics. 24 (1–3): 83–96. doi:10.1016/0166-218x(92)90274-e.
• Cohen-Zada, Danny; Krumer, Alex; Shapir, Offer Moshe (2018). "Testing the effect of serve order in tennis tiebreak". Journal of Economic Behavior and Organization. 146: 106–115. doi:10.1016/j.jebo.2017.12.012. S2CID 89610106.
• Cooper, Joshua; Dutle, Aaron (2013). "Greedy Galois Games" (PDF). American Mathematical Monthly. 120 (5): 441–451. arXiv:1110.1137. doi:10.4169/amer.math.monthly.120.05.441. S2CID 1291901.
• Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe (eds.). Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, California, USA, June 26-29, 2006. Lecture Notes in Computer Science. Vol. 4036. Springer-Verlag. pp. 280–291. ISBN 978-3-540-35428-4. Zbl 1227.68074.
• Levine, Lionel; Stange, Katherine E. (2012). "How to Make the Most of a Shared Meal: Plan the Last Bite First" (PDF). American Mathematical Monthly. 119 (7): 550–565. arXiv:1104.0961. doi:10.4169/amer.math.monthly.119.07.550. S2CID 14537479.
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
• Ma, Jun; Holdener, Judy (2005). "When Thue-Morse meets Koch" (PDF). Fractals. 13 (3): 191–206. doi:10.1142/S0218348X05002908. MR 2166279.
• Palacios-Huerta, Ignacio (2012). "Tournaments, fairness and the Prouhet–Thue–Morse sequence" (PDF). Economic Inquiry. 50 (3): 848–849. doi:10.1111/j.1465-7295.2011.00435.x. S2CID 54036493.
• Palacios-Huerta, Ignacio (2014). Beautiful Game Theory. Princeton University Press. ISBN 978-0691144023.
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin, Germany: Springer-Verlag. ISBN 978-3-540-44141-0. Zbl 1014.11015.
• Richman, Robert (2001). "Recursive Binary Sequences of Differences" (PDF). Complex Systems. 13 (4): 381–392.
Further reading
• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl 1260.11001.
• Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 978-0-521-84802-2. Zbl 1133.68067.
External links
Wikimedia Commons has media related to Thue-Morse sequence.
• "Thue-Morse sequence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Thue-Morse Sequence". MathWorld.
• Allouche, J.-P.; Shallit, J. O. The Ubiquitous Prouhet-Thue-Morse Sequence. (contains many applications and some history)
• Thue–Morse Sequence over (1,2) (sequence A001285 in the OEIS)
• OEIS sequence A000069 (Odious numbers: numbers with an odd number of 1's in their binary expansion)
• OEIS sequence A001969 (Evil numbers: numbers with an even number of 1's in their binary expansion)
• Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence. A technical application of the Thue–Morse Sequence
• MusiNum - The Music in the Numbers. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences.
• Parker, Matt. "The Fairest Sharing Sequence Ever" (video). standupmaths. Retrieved 20 January 2016.
| Wikipedia |
Roth's theorem
In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).
For Roth's theorem on arithmetic progressions, see Roth's theorem on arithmetic progressions.
Statement
Roth's theorem states that every irrational algebraic number $\alpha $ has approximation exponent equal to 2. This means that, for every $\varepsilon >0$, the inequality
$\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2+\varepsilon }}}$
can have only finitely many solutions in coprime integers $p$ and $q$. Roth's proof of this fact resolved a conjecture by Siegel. It follows that every irrational algebraic number α satisfies
$\left|\alpha -{\frac {p}{q}}\right|>{\frac {C(\alpha ,\varepsilon )}{q^{2+\varepsilon }}}$
with $C(\alpha ,\varepsilon )$ a positive number depending only on $\varepsilon >0$ and $\alpha $.
Discussion
The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of d for an algebraic number α of degree d ≥ 2. This is already enough to demonstrate the existence of transcendental numbers. Thue realised that an exponent less than d would have applications to the solution of Diophantine equations and in Thue's theorem from 1909 established an exponent $d/2+1+\varepsilon $ which he applied to prove the finiteness of the solutions of Thue equation. Siegel's theorem improves this to an exponent about 2√d, and Dyson's theorem of 1947 has exponent about √2d.
Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting $\varepsilon =0$: by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case. However, there is a stronger conjecture of Serge Lang that
$\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}\log(q)^{1+\varepsilon }}}$
can have only finitely many solutions in integers p and q. If one lets α run over the whole of the set of real numbers, not just the algebraic reals, then both Roth's conclusion and Lang's hold for almost all $\alpha $. So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero.[1]
The theorem is not currently effective: that is, there is no bound known on the possible values of p,q given $\alpha $.[2] Davenport & Roth (1955) showed that Roth's techniques could be used to give an effective bound for the number of p/q satisfying the inequality, using a "gap" principle.[2] The fact that we do not actually know C(ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach.
Proof technique
The proof technique involves constructing an auxiliary multivariate polynomial in an arbitrarily large number of variables depending upon $\varepsilon $, leading to a contradiction in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). By its nature, it was ineffective (see effective results in number theory); this is of particular interest since a major application of this type of result is to bound the number of solutions of some diophantine equations.
Generalizations
There is a higher-dimensional version, Schmidt's subspace theorem, of the basic result. There are also numerous extensions, for example using the p-adic metric,[3] based on the Roth method.
William J. LeVeque generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field. Define the height H(ξ) of an algebraic number ξ to be the maximum of the absolute values of the coefficients of its minimal polynomial. Fix κ>2. For a given algebraic number α and algebraic number field K, the equation
$|\alpha -\xi |<{\frac {1}{H(\xi )^{\kappa }}}$
has only finitely many solutions in elements ξ of K.[4]
See also
• Davenport–Schmidt theorem
• Granville–Langevin conjecture
• Størmer's theorem
• Diophantine geometry
Notes
1. It is also closely related to the Manin–Mumford conjecture.
2. Hindry, Marc; Silverman, Joseph H. (2000), Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, vol. 201, pp. 344–345, ISBN 0-387-98981-1
3. Ridout, D. (1958), "The p-adic generalization of the Thue–Siegel–Roth theorem", Mathematika, 5: 40–48, doi:10.1112/s0025579300001339, Zbl 0085.03501
4. LeVeque, William J. (2002) [1956], Topics in Number Theory, Volumes I and II, New York: Dover Publications, pp. II:148–152, ISBN 978-0-486-42539-9, Zbl 1009.11001
References
• Davenport, H.; Roth, Klaus Friedrich (1955), "Rational approximations to algebraic numbers", Mathematika, 2 (2): 160–167, doi:10.1112/S0025579300000814, ISSN 0025-5793, MR 0077577, Zbl 0066.29302
• Dyson, Freeman J. (1947), "The approximation to algebraic numbers by rationals", Acta Mathematica, 79: 225–240, doi:10.1007/BF02404697, ISSN 0001-5962, MR 0023854, Zbl 0030.02101
• Roth, Klaus Friedrich (1955), "Rational approximations to algebraic numbers", Mathematika, 2: 1–20, 168, doi:10.1112/S0025579300000644, ISSN 0025-5793, MR 0072182, Zbl 0064.28501
• Wolfgang M. Schmidt (1996) [1980], Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, doi:10.1007/978-3-540-38645-2, ISBN 978-3-540-09762-4
• Wolfgang M. Schmidt (1991), Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, doi:10.1007/BFb0098246, ISBN 978-3-540-54058-8, S2CID 118143570
• Siegel, Carl Ludwig (1921), "Approximation algebraischer Zahlen", Mathematische Zeitschrift, 10 (3): 173–213, doi:10.1007/BF01211608, ISSN 0025-5874, MR 1544471, S2CID 119577458
• Thue, A. (1909), "Über Annäherungswerte algebraischer Zahlen", Journal für die reine und angewandte Mathematik, 1909 (135): 284–305, doi:10.1515/crll.1909.135.284, ISSN 0075-4102, S2CID 125903243
Further reading
• Baker, Alan (1975), Transcendental Number Theory, Cambridge University Press, ISBN 0-521-20461-5, Zbl 0297.10013
• Baker, Alan; Wüstholz, Gisbert (2007), Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, vol. 9, Cambridge University Press, ISBN 978-0-521-88268-2, Zbl 1145.11004
• Bombieri, Enrico; Gubler, Walter (2006), Heights in Diophantine Geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, ISBN 978-0-521-71229-3, Zbl 1130.11034
• Vojta, Paul (1987), Diophantine Approximations and Value Distribution Theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, ISBN 3-540-17551-2, Zbl 0609.14011
| Wikipedia |
Prouhet–Thue–Morse constant
In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,
$\tau =\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}=0.412454033640\ldots $
where tn is the nth element of the Prouhet–Thue–Morse sequence.
Other representations
The Prouhet–Thue–Morse constant can also be expressed, without using tn , as an infinite product,[1]
$\tau ={\frac {1}{4}}\left[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]$
This formula is obtained by substituting x = 1/2 into generating series for tn
$F(x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})$
The continued fraction expansion of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] (sequence A014572 in the OEIS)
Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.[2]
Transcendence
The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.[3]
He also showed that the number
$\sum _{i=0}^{\infty }t_{n}\,\alpha ^{n}$
is also transcendental for any algebraic number α, where 0 < |α| < 1.
Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2.[4]
Appearances
The Prouhet–Thue–Morse constant appears in probability. If a language L over {0, 1} is chosen at random, by flipping a fair coin to decide whether each word w is in L, the probability that it contains at least one word for each possible length is [5]
$p=\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)=\sum _{n=0}^{\infty }{\frac {(-1)^{t_{n}}}{2^{n+1}}}=2-4\tau =0.35018386544\ldots $
See also
• Euler-Mascheroni constant
• Fibonacci word
• Golay–Rudin–Shapiro sequence
• Komornik–Loreti constant
Notes
1. Weisstein, Eric W. "Thue-Morse Constant". MathWorld.
2. Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences. 16 (13.2.3).
3. Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929.
4. Bugaeud, Yann (2011). "On the rational approximation to the Thue–Morse–Mahler numbers". Annales de l'Institut Fourier. 61 (5): 2065–2076. doi:10.5802/aif.2666.
5. Allouche, Jean-Paul; Shallit, Jeffrey (1999). "The Ubiquitous Prouhet-Thue-Morse Sequence". Discrete Mathematics and Theoretical Computer Science: 11.
References
• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015..
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
External links
• OEIS sequence A010060 (Thue-Morse sequence)
• The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
• PlanetMath entry
| Wikipedia |
Thue equation
In mathematics, a Thue equation is a Diophantine equation of the form
ƒ(x,y) = r,
where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers x and y, a result known as Thue's theorem, [1]
The Thue equation is solvable effectively: there is an explicit bound on the solutions x, y of the form $(C_{1}r)^{C_{2}}$ where constants C1 and C2 depend only on the form ƒ. A stronger result holds: if K is the field generated by the roots of ƒ, then the equation has only finitely many solutions with x and y integers of K, and again these may be effectively determined.[2]
Finiteness of solutions and diophantine approximation
Thue's original proof that the equation named in his honour has finitely many solutions is through the proof of what is now known as Thue's theorem: it asserts that for any algebraic number $\alpha $ having degree $d\geq 3$ and for any $\varepsilon >0$ there exists only finitely many co-prime integers $p,q$ with $q>0$ such that $|\alpha -p/q|<q^{-(d+1+\varepsilon )/2}$. Applying this theorem allows one to almost immediately deduce the finiteness of solutions. However, Thue's proof, as well as subsequent improvements by Siegel, Dyson, and Roth were all ineffective.
Solution algorithm
Finding all solutions to a Thue equation can be achieved by a practical algorithm,[3] which has been implemented in the following computer algebra systems:
• in PARI/GP as functions thueinit() and thue().
• in Magma computer algebra system as functions ThueObject() and ThueSolve().
• in Mathematica through Reduce
Bounding the number of solutions
While there are several effective methods to solve Thue equations (including using Baker's method and Skolem's $p$-adic method), these are not able to give the best theoretical bounds on the number of solutions. One may qualify an effective bound $C(f,r)$ of the Thue equation $f(x,y)=r$ by the parameters it depends on, and how "good" the dependence is.
The best result known today, essentially building on pioneering work of Bombieri and Schmidt,[4] gives a bound of the shape $C(f,r)=C\cdot (\deg f)^{1+\omega (r)}$, where $C$ is an absolute constant (that is, independent of both $f$ and $r$) and $\omega (\cdot )$ is the number of distinct prime divisors of $r$. The most significant qualitative improvement to the theorem of Bombieri and Schmidt is due to Stewart,[5] who obtained a bound of the form $C(f,r)=C\cdot (\deg f)^{1+\omega (g)}$ where $g$ is a divisor of $r$ exceeding $|r|^{3/4}$ in absolute value. It is conjectured that one may take the bound $C(f,r)=C(\deg f)$; that is, depending only on the degree of $f$ but not its coefficients, and completely independent of the integer $r$ on the right hand side of the equation.
This is a weaker form of a conjecture of Stewart, and is a special case of the uniform boundedness conjecture for rational points. This conjecture has been proven for "small" integers $r$, where smallness is measured in terms of the discriminant of the form $f$, by various authors, including Evertse, Stewart, and Akhtari. Stewart and Xiao demonstrated a strong form of this conjecture, asserting that the number of solutions is absolutely bounded, holds on average (as $r$ ranges over the interval $|r|\leq Z$ with $Z\rightarrow \infty $) [6]
See also
• Roth's theorem
• Faltings's Theorem
References
1. A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen". Journal für die reine und angewandte Mathematik. 1909 (135): 284–305. doi:10.1515/crll.1909.135.284. S2CID 125903243.
2. Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 38. ISBN 0-521-20461-5.
3. N. Tzanakis and B. M. M. de Weger (1989). "On the practical solution of the Thue equation". Journal of Number Theory. 31 (2): 99–132. doi:10.1016/0022-314X(89)90014-0.
4. E. Bombieri and W. M. Schmidt (1987). "On Thue's equation". Inventiones Mathematicae. 88 (2): 69–81. Bibcode:1987InMat..88...69B. doi:10.1007/BF01405092.
5. C.L. Stewart (1991). "On the number of solutions to polynomial congruences and Thue equations". Journal of the American Mathematical Society. 4 (4): 793–835. doi:10.2307/2939289. JSTOR 2939289.
6. C.L. Stewart and Stanley Yao Xiao (2019). "On the representation of integers by binary forms". Mathematische Annalen. 375 (4): 133–163. arXiv:1605.03427. doi:10.1007/s00208-019-01855-y.
Further reading
• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge University Press. ISBN 978-0-521-88268-2.
| Wikipedia |
Thue number
In the mathematical area of graph theory, the Thue number of a graph is a variation of the chromatic index, defined by Alon et al. (2002) and named after mathematician Axel Thue, who studied the squarefree words used to define this number.
Alon et al. define a nonrepetitive coloring of a graph to be an assignment of colors to the edges of the graph, such that there does not exist any even-length simple path in the graph in which the colors of the edges in the first half of the path form the same sequence as the colors of the edges in the second half of the path. The Thue number of a graph is the minimum number of colors needed in any nonrepetitive coloring.[1]
Variations on this concept involving vertex colorings or more general walks on a graph have been studied by several authors including Barát and Varjú, Barát and Wood (2005), Brešar and Klavžar (2004), and Kündgen and Pelsmajer.
Example
Consider a pentagon, that is, a cycle of five vertices. If we color the edges with two colors, some two adjacent edges will have the same color x; the path formed by those two edges will have the repetitive color sequence xx. If we color the edges with three colors, one of the three colors will be used only once; the path of four edges formed by the other two colors will either have two consecutive edges or will form the repetitive color sequence xyxy. However, with four colors it is not difficult to avoid all repetitions. Therefore, the Thue number of C5 is four.[1]
Results
Alon et al. use the Lovász local lemma to prove that the Thue number of any graph is at most quadratic in its maximum degree; they provide an example showing that for some graphs this quadratic dependence is necessary. In addition they show that the Thue number of a path of four or more vertices is exactly three, that the Thue number of any cycle is at most four, and that the Thue number of the Petersen graph is exactly five.[1]
The known cycles with Thue number four are C5, C7, C9, C10, C14, and C17. Alon et al. conjecture that the Thue number of any larger cycle is three; they verified computationally that the cycles listed above are the only ones of length ≤ 2001 with Thue number four. Currie resolved this in a 2002 paper, showing that all cycles with 18 or more vertices have Thue number 3.
Computational complexity
Testing whether a coloring has a repetitive path is in NP, so testing whether a coloring is nonrepetitive is in co-NP, and Manin showed that it is co-NP-complete. The problem of finding such a coloring belongs to $\Sigma _{2}^{P}$ in the polynomial hierarchy, and again Manin showed that it is complete for this level.Manin (2007)
References
• Alon, Noga; Grytczuk, Jaroslaw; Hałuszczak, Mariusz; Riordan, Oliver (2002). "Nonrepetitive colorings of graphs" (PDF). Random Structures & Algorithms. 21 (3–4): 336–346. doi:10.1002/rsa.10057. MR 1945373. S2CID 5724512.
• Barát, János; Varjú, P. P. (2008). "On square-free edge colorings of graphs". Ars Combinatoria. 87: 377–383. MR 2414029.
• Barát, János; Wood, David (2005). "Notes on nonrepetitive graph colouring". Electronic Journal of Combinatorics. 15 (1). R99. arXiv:math.CO/0509608. Bibcode:2005math......9608B. MR 2426162.
• Brešar, Boštjan; Klavžar, Sandi (2004). "Square-free coloring of graphs". Ars Combin. 70: 3–13. MR 2023057.
• Currie, James D. (2002). "There are ternary circular square-free words of length n for n ≥ 18". Electronic Journal of Combinatorics. 9 (1). N10. doi:10.37236/1671. MR 1936865.
• Grytczuk, Jarosław (2007). "Nonrepetitive colorings of graphs—a survey". International Journal of Mathematics and Mathematical Sciences. Art. ID 74639. doi:10.1155/2007/74639. MR 2272338.
• Kündgen, André; Pelsmajer, Michael J. (2008). "Nonrepetitive colorings of graphs of bounded tree-width". Discrete Mathematics. 308 (19): 4473–4478. doi:10.1016/j.disc.2007.08.043. MR 2433774.
• Manin, Fedor (2007). "The complexity of nonrepetitive edge coloring of graphs". arXiv:0709.4497. Bibcode:2007arXiv0709.4497M. {{cite journal}}: Cite journal requires |journal= (help)
• Schaefer, Marcus; Umans, Christopher (2005). "Completeness in the polynomial-time hierarchy: a compendium". {{cite journal}}: Cite journal requires |journal= (help)
External links
• Media related to Thue number at Wikimedia Commons
1. Alon et al. (2002).
| Wikipedia |
Circle packing
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.
This article is about the packing of circles on surfaces. For circle packing with a prescribed intersection graph, see Circle packing theorem.
The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.
Densest packing
In the two-dimensional Euclidean plane, Joseph Louis Lagrange proved in 1773 that the highest-density lattice packing of circles is the hexagonal packing arrangement,[1] in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, like a honeycomb), and each circle is surrounded by six other circles. For circles of diameter D and hexagons of side length D, the hexagon area and the circle area are, respectively:
${\begin{aligned}A_{\mathrm {H} }&={\frac {3{\sqrt {3}}}{2}}D^{2}\\[4pt]A_{\mathrm {C} }&={\frac {\pi }{4}}D^{2}\end{aligned}}$
The area covered within each hexagon by circles is:
$A_{\mathrm {HC} }=3A_{\mathrm {C} }={\frac {3\pi }{4}}D^{2}$
Finally, the packing density is:
${\begin{aligned}\eta ={\frac {A_{\mathrm {HC} }}{A_{\mathrm {H} }}}&={\frac {{\frac {3\pi }{4}}D^{2}}{{\frac {3{\sqrt {3}}}{2}}D^{2}}}\\[4pt]&={\frac {\pi }{2{\sqrt {3}}}}\approx 0.9069\end{aligned}}$
In 1890, Axel Thue published a proof that this same density is optimal among all packings, not just lattice packings, but his proof was considered by some to be incomplete. The first rigorous proof is attributed to László Fejes Tóth in 1942.[1][2]
While the circle has a relatively low maximum packing density, it does not have the lowest possible, even among centrally-symmetric convex shapes: the smoothed octagon has a packing density of about 0.902414, the smallest known for centrally-symmetric convex shapes and conjectured to be the smallest possible.[3] (Packing densities of concave shapes such as star polygons can be arbitrarily small.)
Other packings
At the other extreme, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist.[4][5]
There are eleven circle packings based on the eleven uniform tilings of the plane.[6] In these packings, every circle can be mapped to every other circle by reflections and rotations. The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with seven circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing. The snub hexagonal tiling has two mirror-image forms.
On the sphere
A related problem is to determine the lowest-energy arrangement of identically interacting points that are constrained to lie within a given surface. The Thomson problem deals with the lowest energy distribution of identical electric charges on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere. This is analogous to distributing non-point charges on a sphere.
In bounded areas
Packing circles in simple bounded shapes is a common type of problem in recreational mathematics. The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles. Specific problems of this type that have been studied include:
• Circle packing in a circle
• Circle packing in a square
• Circle packing in a rectangle
• Circle packing in an equilateral triangle
• Circle packing in an isosceles right triangle
See the linked articles for details.
Unequal circles
See also: Unequal sphere packing
There are also a range of problems which permit the sizes of the circles to be non-uniform. One such extension is to find the maximum possible density of a system with two specific sizes of circle (a binary system). Only nine particular radius ratios permit compact packing, which is when every pair of circles in contact is in mutual contact with two other circles (when line segments are drawn from contacting circle-center to circle-center, they triangulate the surface).[7] For all these radius ratios a compact packing is known that achieves the maximum possible packing fraction (above that of uniformly-sized discs) for mixtures of discs with that radius ratio.[9] All nine have ratio-specific packings denser than the uniform hexagonal packing, as do some radius ratios without compact packings.[10]
It is also known that if the radius ratio is above 0.742, a binary mixture cannot pack better than uniformly-sized discs.[8] Upper bounds for the density that can be obtained in such binary packings at smaller ratios have also been obtained.[11]
Applications
Quadrature amplitude modulation is based on packing circles into circles within a phase-amplitude space. A modem transmits data as a series of points in a two-dimensional phase-amplitude plane. The spacing between the points determines the noise tolerance of the transmission, while the circumscribing circle diameter determines the transmitter power required. Performance is maximized when the constellation of code points are at the centres of an efficient circle packing. In practice, suboptimal rectangular packings are often used to simplify decoding.
Circle packing has become an essential tool in origami design, as each appendage on an origami figure requires a circle of paper.[12] Robert J. Lang has used the mathematics of circle packing to develop computer programs that aid in the design of complex origami figures.
See also
• Apollonian gasket
• Circle packing in a rectangle
• Circle packing in a square
• Circle packing in a circle
• Inversive distance
• Kepler conjecture
• Malfatti circles
• Packing problem
References
1. Chang, Hai-Chau; Wang, Lih-Chung (2010). "A Simple Proof of Thue's Theorem on Circle Packing". arXiv:1009.4322 [math.MG].
2. Tóth, László Fejes (1942). "Über die dichteste Kugellagerung". Math. Z. 48: 676–684. doi:10.1007/BF01180035. S2CID 123697077.
3. Weisstein, Eric W. "Smoothed Octagon". MathWorld.
4. Böröczky, K. (1964). "Über stabile Kreis- und Kugelsysteme". Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica. 7: 79–82.
5. Kahle, Matthew (2012). "Sparse locally-jammed disk packings". Annals of Combinatorics. 16 (4): 773–780. doi:10.1007/s00026-012-0159-0. S2CID 1559383.
6. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 35-39. ISBN 0-486-23729-X.
7. Tom Kennedy (2006). "Compact packings of the plane with two sizes of discs". Discrete and Computational Geometry. 35 (2): 255–267. arXiv:math/0407145. doi:10.1007/s00454-005-1172-4. S2CID 11688453.
8. Heppes, Aladár (1 August 2003). "Some Densest Two-Size Disc Packings in the Plane". Discrete and Computational Geometry. 30 (2): 241–262. doi:10.1007/s00454-003-0007-6.
9. Bédaride, Nicolas; Fernique, Thomas (17 February 2020). "Density of Binary Compact Disc Packings". arXiv:2002.07168. {{cite journal}}: Cite journal requires |journal= (help)
10. Kennedy, Tom (2004-07-21). "Circle Packings". Retrieved 2018-10-11.
11. de Laat, David; de Oliveira Filho, Fernando Mario; Vallentin, Frank (12 June 2012). "Upper bounds for packings of spheres of several radii". Forum of Mathematics, Sigma. 2. arXiv:1206.2608. doi:10.1017/fms.2014.24. S2CID 11082628.
12. TED.com lecture on modern origami "Robert Lang on TED Archived 2011-10-15 at the Wayback Machine."
Bibliography
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 30–31, 167. ISBN 0-14-011813-6.
• Stephenson, Kenneth (December 2003). "Circle Packing: A Mathematical Tale" (PDF). Notices of the American Mathematical Society. 50 (11).
Packing problems
Abstract packing
• Bin
• Set
Circle packing
• In a circle / equilateral triangle / isosceles right triangle / square
• Apollonian gasket
• Circle packing theorem
• Tammes problem (on sphere)
Sphere packing
• Apollonian
• Finite
• In a sphere
• In a cube
• In a cylinder
• Close-packing
• Kissing number
• Sphere-packing (Hamming) bound
Other 2-D packing
• Square packing
Other 3-D packing
• Tetrahedron
• Ellipsoid
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| Wikipedia |
Double limit theorem
In hyperbolic geometry, Thurston's double limit theorem gives condition for a sequence of quasi-Fuchsian groups to have a convergent subsequence. It was introduced in Thurston (1998, theorem 4.1) and is a major step in Thurston's proof of the hyperbolization theorem for the case of manifolds that fiber over the circle.
Statement
By Bers's theorem, quasi-Fuchsian groups (of some fixed genus) are parameterized by points in T×T, where T is Teichmüller space of the same genus. Suppose that there is a sequence of quasi-Fuchsian groups corresponding to points (gi, hi) in T×T. Also suppose that the sequences gi, hi converge to points μ,μ′ in the Thurston boundary of Teichmüller space of projective measured laminations. If the points μ,μ′ have the property that any nonzero measured lamination has positive intersection number with at least one of them, then the sequence of quasi-Fuchsian groups has a subsequence that converges algebraically.
References
• Holt, John (2001), The double limit theorem, archived from the original on 2011-09-27, retrieved 2011-03-20
• Kapovich, Michael (2009) [2001], Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8, MR 1792613
• Otal, Jean-Pierre (1996), "Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3", Astérisque (235), ISSN 0303-1179, MR 1402300 Translated into English as Otal, Jean-Pierre (2001) [1996], Kay, Leslie D. (ed.), The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, vol. 7, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2153-4, MR 1855976
• Thurston, William P. (1998) [1986], Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, arXiv:math/9801045
| Wikipedia |
Thurston–Bennequin number
In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps. It is named after William Thurston and Daniel Bennequin.
The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological knot invariant.
References
• "Thurston–Bennequin number", The Knot Atlas.
• Lee Rudolph (1997). "The slice genus and the Thurston–Bennequin invariant of a knot". Proceedings of the American Mathematical Society. 125: 3049–3050. doi:10.1090/S0002-9939-97-04258-5. MR 1443854.
| Wikipedia |
Thurston elliptization conjecture
William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.
Thurston elliptization conjecture
FieldGeometric topology
Conjectured byWilliam Thurston
Conjectured in1980
First proof byGrigori Perelman
First proof in2006
Implied byGeometrization conjecture
Equivalent toPoincaré conjecture
Spherical space form conjecture
Relation to other conjectures
A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere. If the original 3-manifold had in fact a trivial fundamental group, then it is homeomorphic to the 3-sphere (via the covering map). Thus, proving the elliptization conjecture would prove the Poincaré conjecture as a corollary. In fact, the elliptization conjecture is logically equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture.
The elliptization conjecture is a special case of Thurston's geometrization conjecture, which was proved in 2003 by G. Perelman.
References
For the proof of the conjectures, see the references in the articles on geometrization conjecture or Poincaré conjecture.
• William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5.
• William Thurston. The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture near the beginning of section 3.
| Wikipedia |
Geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Geometrization theorem
FieldGeometric topology
Conjectured byWilliam Thurston
Conjectured in1982
First proof byGrigori Perelman
First proof in2006
ConsequencesPoincaré conjecture
Thurston elliptization conjecture
Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. Verification was essentially complete in time for Perelman to be awarded the 2006 Fields Medal for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined to accept either award.
The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
The conjecture
A 3-manifold is called closed if it is compact and has no boundary.
Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime 3-manifolds (this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds). This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.
Here is a statement of Thurston's conjecture:
Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
There are 8 possible geometric structures in 3 dimensions, described in the next section. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an Anosov map of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.)
For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover. It is also possible to work directly with non-orientable manifolds, but this gives some extra complications: it may be necessary to cut along projective planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure.
In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.
The eight Thurston geometries
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers.
A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.
A geometric structure on a manifold M is a diffeomorphism from M to X/Γ for some model geometry X, where Γ is a discrete subgroup of G acting freely on X ; this is a special case of a complete (G,X)-structure. If a given manifold admits a geometric structure, then it admits one whose model is maximal.
A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also uncountably many model geometries without compact quotients.)
There is some connection with the Bianchi groups: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However S2 × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.
Spherical geometry S3
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group O(4, R), with 2 components. The corresponding manifolds are exactly the closed 3-manifolds with finite fundamental group. Examples include the 3-sphere, the Poincaré homology sphere, Lens spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type IX. Manifolds with this geometry are all compact, orientable, and have the structure of a Seifert fiber space (often in several ways). The complete list of such manifolds is given in the article on spherical 3-manifolds. Under Ricci flow, manifolds with this geometry collapse to a point in finite time.
Euclidean geometry E3
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group R3 × O(3, R), with 2 components. Examples are the 3-torus, and more generally the mapping torus of a finite-order automorphism of the 2-torus; see torus bundle. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the Bianchi groups of type I or VII0. Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space (sometimes in two ways). The complete list of such manifolds is given in the article on Seifert fiber spaces. Under Ricci flow, manifolds with Euclidean geometry remain invariant.
Hyperbolic geometry H3
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group O+(1, 3, R), with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is the Weeks manifold. Other examples are given by the Seifert–Weber space, or "sufficiently complicated" Dehn surgeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the Bianchi group of type V or VIIh≠0. Under Ricci flow, manifolds with hyperbolic geometry expand.
The geometry of S2 × R
The point stabilizer is O(2, R) × Z/2Z, and the group G is O(3, R) × R × Z/2Z, with 4 components. The four finite volume manifolds with this geometry are: S2 × S1, the mapping torus of the antipode map of S2, the connected sum of two copies of 3-dimensional projective space, and the product of S1 with two-dimensional projective space. The first two are mapping tori of the identity map and antipode map of the 2-sphere, and are the only examples of 3-manifolds that are prime but not irreducible. The third is the only example of a non-trivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a Seifert fiber space (often in several ways). Under normalized Ricci flow manifolds with this geometry converge to a 1-dimensional manifold.
The geometry of H2 × R
The point stabilizer is O(2, R) × Z/2Z, and the group G is O+(1, 2, R) × R × Z/2Z, with 4 components. Examples include the product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with this geometry have the structure of a Seifert fiber space if they are orientable. (If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation"; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori.[1]) The classification of such (oriented) manifolds is given in the article on Seifert fiber spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type III. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.
The geometry of the universal cover of SL(2, "R")
The universal cover of SL(2, R) is denoted ${\widetilde {\rm {SL}}}(2,\mathbf {R} )$. It fibers over H2, and the space is sometimes called "Twisted H2 × R". The group G has 2 components. Its identity component has the structure $(\mathbf {R} \times {\widetilde {\rm {SL}}}_{2}(\mathbf {R} ))/\mathbf {Z} $. The point stabilizer is O(2,R).
Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the Brieskorn homology spheres (excepting the 3-sphere and the Poincare dodecahedral space). This geometry can be modeled as a left invariant metric on the Bianchi group of type VIII or III. Finite volume manifolds with this geometry are orientable and have the structure of a Seifert fiber space. The classification of such manifolds is given in the article on Seifert fiber spaces. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.
Nil geometry
See also: Nilmanifold
This fibers over E2, and so is sometimes known as "Twisted E2 × R". It is the geometry of the Heisenberg group. The point stabilizer is O(2, R). The group G has 2 components, and is a semidirect product of the 3-dimensional Heisenberg group by the group O(2, R) of isometries of a circle. Compact manifolds with this geometry include the mapping torus of a Dehn twist of a 2-torus, or the quotient of the Heisenberg group by the "integral Heisenberg group". This geometry can be modeled as a left invariant metric on the Bianchi group of type II. Finite volume manifolds with this geometry are compact and orientable and have the structure of a Seifert fiber space. The classification of such manifolds is given in the article on Seifert fiber spaces. Under normalized Ricci flow, compact manifolds with this geometry converge to R2 with the flat metric.
Sol geometry
This geometry (also called Solv geometry) fibers over the line with fiber the plane, and is the geometry of the identity component of the group G. The point stabilizer is the dihedral group of order 8. The group G has 8 components, and is the group of maps from 2-dimensional Minkowski space to itself that are either isometries or multiply the metric by −1. The identity component has a normal subgroup R2 with quotient R, where R acts on R2 with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the Bianchi group of type VI0 and the geometry can be modeled as a left invariant metric on this group. All finite volume manifolds with solv geometry are compact. The compact manifolds with solv geometry are either the mapping torus of an Anosov map of the 2-torus (such a map is an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as $\left({\begin{array}{*{20}c}2&1\\1&1\\\end{array}}\right)$), or quotients of these by groups of order at most 8. The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the solv manifolds can be classified in terms of the units and ideal classes of this order.[2] Under normalized Ricci flow compact manifolds with this geometry converge (rather slowly) to R1.
Uniqueness
A closed 3-manifold has a geometric structure of at most one of the 8 types above, but finite volume non-compact 3-manifolds can occasionally have more than one type of geometric structure. (Nevertheless, a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if M is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π1(M):
• If π1(M) is finite then the geometric structure on M is spherical, and M is compact.
• If π1(M) is virtually cyclic but not finite then the geometric structure on M is S2×R, and M is compact.
• If π1(M) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.
• If π1(M) is virtually nilpotent but not virtually abelian then the geometric structure on M is nil geometry, and M is compact.
• If π1(M) is virtually solvable but not virtually nilpotent then the geometric structure on M is solv geometry, and M is compact.
• If π1(M) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H2×R or the universal cover of SL(2, R). The manifold M may be either compact or non-compact. If it is compact, then the 2 geometries can be distinguished by whether or not π1(M) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else. If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples (such as the complement of a trefoil knot) where a manifold may have a finite volume geometric structure of either type.
• If π1(M) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.
Infinite volume manifolds can have many different types of geometric structure: for example, R3 can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.
There can be more than one way to decompose a closed 3-manifold into pieces with geometric structures. For example:
• Taking connected sums with several copies of S3 does not change a manifold.
• The connected sum of two projective 3-spaces has a S2×R geometry, and is also the connected sum of two pieces with S3 geometry.
• The product of a surface of negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. There are many similar examples for Seifert fiber spaces.
It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; in fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.
History
The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.
In 1982, Richard S. Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by Ricci flow with surgery. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries S3 and S2 × R, while what is left at large times should have a thick–thin decomposition into a "thick" piece with hyperbolic geometry and a "thin" graph manifold.
In 2003, Grigori Perelman announced a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above.
One component of Perelman's proof was a novel collapsing theorem in Riemannian geometry. Perelman did not release any details on the proof of this result (Theorem 7.4 in the preprint 'Ricci flow with surgery on three-manifolds'). Beginning with Shioya and Yamaguchi, there are now several different proofs of Perelman's collapsing theorem, or variants thereof.[3][4][5][6] Shioya and Yamaguchi's formulation was used in the first fully detailed formulations of Perelman's work.[7]
A second route to the last part of Perelman's proof of geometrization is the method of Bessières et al.,[8][9] which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov's norm for 3-manifolds.[10][11] A book by the same authors with complete details of their version of the proof has been published by the European Mathematical Society.[12]
Notes
1. Fintushel, Ronald (1976). "Local S1 actions on 3-manifolds". Pacific Journal of Mathematics. 66 (1): 111–118. doi:10.2140/pjm.1976.66.111.
2. Quinn, Joseph; Verjovsky, Alberto (2020-06-01). "Cusp shapes of Hilbert–Blumenthal surfaces". Geometriae Dedicata. 206 (1): 27–42. arXiv:1711.02418. doi:10.1007/s10711-019-00474-w. ISSN 1572-9168. S2CID 55731832.
3. Shioya, T.; Yamaguchi, T. (2005). "Volume collapsed three-manifolds with a lower curvature bound". Math. Ann. 333 (1): 131–155. arXiv:math/0304472. doi:10.1007/s00208-005-0667-x. S2CID 119481.
4. Morgan & Tian 2014.
5. Kleiner, Bruce; Lott, John (2014). "Locally collapsed 3-manifolds". Astérisque. 365 (7–99).
6. Cao, Jianguo; Ge, Jian (2011). "A simple proof of Perelman's collapsing theorem for 3-manifolds". J. Geom. Anal. 21 (4): 807–869. arXiv:1003.2215. doi:10.1007/s12220-010-9169-5. S2CID 514106.
7. Cao & Zhu 2006; Kleiner & Lott 2008.
8. Bessieres, L.; Besson, G.; Boileau, M.; Maillot, S.; Porti, J. (2007). "Weak collapsing and geometrization of aspherical 3-manifolds". arXiv:0706.2065 [math.GT].
9. Bessieres, L.; Besson, G.; Boileau, M.; Maillot, S.; Porti, J. (2010). "Collapsing irreducible 3-manifolds with nontrivial fundamental group". Invent. Math. 179 (2): 435–460. Bibcode:2010InMat.179..435B. doi:10.1007/s00222-009-0222-6. S2CID 119436601.
10. Otal, J.-P. (1998). "Thurston's hyperbolization of Haken manifolds". Surveys in differential geometry. Vol. III. Cambridge, MA: Int. Press. pp. 77–194. ISBN 1-57146-067-5.
11. Gromov, M. (1983). "Volume and bounded cohomology". Inst. Hautes Études Sci. Publ. Math. (56): 5–99.
12. L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. Available at https://www-fourier.ujf-grenoble.fr/~besson/book.pdf
References
• L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010.
• M. Boileau Geometrization of 3-manifolds with symmetries
• F. Bonahon Geometric structures on 3-manifolds Handbook of Geometric Topology (2002) Elsevier.
• Cao, Huai-Dong; Zhu, Xi-Ping (2006). "A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton–Perelman theory of the Ricci flow". Asian Journal of Mathematics. 10 (2): 165–492. doi:10.4310/ajm.2006.v10.n2.a2. MR 2233789. Zbl 1200.53057.
– – (2006). "Erratum". Asian Journal of Mathematics. 10 (4): 663–664. doi:10.4310/AJM.2006.v10.n4.e2. MR 2282358.
– – (2006). "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". arXiv:math/0612069.
• Allen Hatcher: Notes on Basic 3-Manifold Topology 2000
• J. Isenberg, M. Jackson, Ricci flow of locally homogeneous geometries on a Riemannian manifold, J. Diff. Geom. 35 (1992) no. 3 723–741.
• Kleiner, Bruce; Lott, John (2008). Updated for corrections in 2011 & 2013. "Notes on Perelman's papers". Geometry & Topology. 12 (5): 2587–2855. doi:10.2140/gt.2008.12.2587. MR 2460872. Zbl 1204.53033.
• John W. Morgan. Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bulletin Amer. Math. Soc. 42 (2005) no. 1, 57–78 (expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture)
• Morgan, John W.; Fong, Frederick Tsz-Ho (2010). Ricci Flow and Geometrization of 3-Manifolds. University Lecture Series. ISBN 978-0-8218-4963-7. Retrieved 2010-09-26.
• Morgan, John; Tian, Gang (2014). The geometrization conjecture. Clay Mathematics Monographs. Vol. 5. Cambridge, MA: Clay Mathematics Institute. ISBN 978-0-8218-5201-9. MR 3186136.
• Perelman, Grisha (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math/0211159.
• Perelman, Grisha (2003). "Ricci flow with surgery on three-manifolds". arXiv:math/0303109.
• Perelman, Grisha (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math/0307245.
• Scott, Peter The geometries of 3-manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
• Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0. ISSN 0002-9904. MR 0648524. This gives the original statement of the conjecture.
• William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5 (in depth explanation of the eight geometries and the proof that there are only eight)
• William Thurston. The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds.
External links
• "The Geometry of 3-Manifolds (video)". Archived from the original on January 27, 2010. Retrieved January 20, 2010. A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.
| Wikipedia |
Thurston norm
In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces.
Definition
Let $M$ be a differentiable manifold and $c\in H_{2}(M)$. Then $c$ can be represented by a smooth embedding $S\to M$, where $S$ is a (not necessarily connected) surface that is compact and without boundary. The Thurston norm of $c$ is then defined to be[1]
$\|c\|_{T}=\min _{S}\sum _{i=1}^{n}\chi _{-}(S_{i})$,
where the minimum is taken over all embedded surfaces $S=\bigcup _{i}S_{i}$ (the $S_{i}$ being the connected components) representing $c$ as above, and $\chi _{-}(F)=\max(0,-\chi (F))$ is the absolute value of the Euler characteristic for surfaces which are not spheres (and 0 for spheres).
This function satisfies the following properties:
• $\|kc\|_{T}=|k|\cdot \|c\|_{T}$ for $c\in H_{2}(M),k\in \mathbb {Z} $;
• $\|c_{1}+c_{2}\|_{T}\leq \|c_{1}\|_{T}+\|c_{2}\|_{T}$ for $c_{1},c_{2}\in H_{2}(M)$.
These properties imply that $\|\cdot \|$ extends to a function on $H_{2}(M,\mathbb {Q} )$ which can then be extended by continuity to a seminorm $\|\cdot \|_{T}$ on $H_{2}(M,\mathbb {R} )$.[2] By Poincaré duality, one can define the Thurston norm on $H^{1}(M,\mathbb {R} )$.
When $M$ is compact with boundary, the Thurston norm is defined in a similar manner on the relative homology group $H_{2}(M,\partial M,\mathbb {R} )$ and its Poincaré dual $H^{1}(M,\mathbb {R} )$.
It follows from further work of David Gabai[3] that one can also define the Thurston norm using only immersed surfaces. This implies that the Thurston norm is also equal to half the Gromov norm on homology.
Topological applications
The Thurston norm was introduced in view of its applications to fiberings and foliations of 3-manifolds.
The unit ball $B$ of the Thurston norm of a 3-manifold $M$ is a polytope with integer vertices. It can be used to describe the structure of the set of fiberings of $M$ over the circle: if $M$ can be written as the mapping torus of a diffeomorphism $f$ of a surface $S$ then the embedding $S\hookrightarrow M$ represents a class in a top-dimensional (or open) face of $B$: moreover all other integer points on the same face are also fibers in such a fibration.[4]
Embedded surfaces which minimise the Thurston norm in their homology class are exactly the closed leaves of foliations of $M$.[3]
Notes
1. Thurston 1986.
2. Thurston 1986, Theorem 1.
3. Gabai 1983.
4. Thurston 1986, Theorem 5.
References
• Gabai, David (1983). "Foliations and the topology of 3-manifolds". Journal of Differential Geometry. 18 (3): 445–503. doi:10.4310/jdg/1214437784. MR 0723813.
• Thurston, William (1986). "A norm for the homology of 3-manifolds". Memoirs of the American Mathematical Society. 59 (33): i–vi and 99–130. MR 0823443.
| Wikipedia |
Thyra Eibe
Thyra Eibe (3 November 1866 – 4 January 1955)[1] was a Danish mathematician and translator, the first woman to earn a mathematics degree from the University of Copenhagen. She is known for her translation of Euclid's Elements into the Danish Language.[2]
Thyra Eibe
Born(1866-11-03)3 November 1866
Copenhagen, Denmark
Died4 January 1955(1955-01-04) (aged 88)
Copenhagen, Denmark
Alma materUniversity of Copenhagen
Occupation(s)Mathematician, translator
Education and career
Eibe was one of ten children of a Copenhagen bookseller. After completing a degree in historical linguistics in 1889 from N. Zahle's School (then a girls' school), Eibe studied mathematics at the University of Copenhagen, and earned a cand.mag. there in 1895. She returned to Zahle's School as a teacher, also teaching boys at Slomann's School and becoming the first woman to become an advanced mathematics teacher for boys in Denmark. In 1898 she moved to H. Adler Community College, later to become the Sortedam Gymnasium, where she remained until 1934, serving as principal for a year in 1929–1930.[1]
Contributions
In undertaking her translation of Euclid, Eibe was motivated by the earlier work of Danish historian Johan Ludvig Heiberg, who published an edition of Euclid's Elements in its original Greek, with translations into Latin.[2]
As well as her translations, Eibe wrote several widely used Danish mathematics textbooks.[1]
Recognition
In 1942, she was given the Tagea Brandt Rejselegat, an award for Danish woman who have made a significant contribution in science, literature or art.[1]
References
1. Høyrup, Else, "Thyra Eibe (1866 – 1955)", Dansk Biografisk Leksikon (in Danish), retrieved 2018-05-05
2. Andersen, Kirsti (2002), "Scandinavia", in Dauben, Joseph W.; Scriba, Christoph J. (eds.), Writing the History of Mathematics: Its Historical Development, Science Networks: Historical Studies, vol. 27, Birkhäuser, 6.7 Text Editions, p. 157, ISBN 9783764361679
Authority control
International
• ISNI
• VIAF
National
• United States
Academics
• zbMATH
| Wikipedia |
Thyrsa Frazier Svager
Thyrsa Anne Frazier Svager (June 16, 1930 – July 23, 1999) was an American academic who was one of the first African-American woman to gain a PhD in mathematics.[1] Born in Ohio, she graduated from high school at the age of 16, going to Antioch College in Ohio and then doing her postgraduate degrees at Ohio State University.[2] Frazier Svager was the head of the Department of Mathematics at Central State University (CSU) in Ohio for decades, ending her academic career as provost and dean for academic affairs.[3] She and her husband, physics professor Aleksandar Svager, invested one of their salaries during their careers to build a legacy for scholarships.[4][5] After her death, the Thyrsa Frazier Svager Fund was established to provide scholarships for African-American women majoring in mathematics.[6]
Thyrsa Frazier Svager
Born
Thyrsa Anne Frazier
(1930-06-16)June 16, 1930
Wilberforce, Ohio
DiedJuly 23, 1999(1999-07-23) (aged 69)
Alma mater
• Antioch College
• Ohio State University
Known for
• One of the first African-American woman PhDs in mathematics
• Scholarship fund for African-American women in mathematics
Scientific career
Fields
• Mathematics
Institutions
• Wright-Patterson Air Force Base
• Texas Southern University
• Central State University
Early life and education
Frazier Svager was born Thyrsa Anne Frazier on June 16, 1930, in Wilberforce, Ohio.[2] Her mother, Elizabeth Anne Frazier, taught speech at Central State University (CSU), a historically black university in Wilberforce, Ohio.[3] Her father, G. Thuton Frazier, headed the Logistics Department at the Wright-Patterson Air Force Base in Dayton, Ohio.[3] She was a member of the Kappa Alpha Psi fraternity, holding the position of Polemarch in the province.[3][7] Frazier Svager had three sisters, Gail, Constance and Jane, and a brother, William Lafayette.[3][2]
Frazier Svager graduated from Wilberforce University Preparatory Academy in Ohio at the age of 16 in 1947, as class valedictorian.[2][8] She attended Antioch College, a private liberal arts college in Yellow Springs, Ohio, majoring in mathematics, with a minor in chemistry, and placed in the 99th percentile in the Princeton Senior Student Examination.[9] Frazier Svager was one of only four black students at Antioch: one of the others was Coretta Scott King, with whom she was friends.[8]
She gained a Bachelor of Arts degree from Antioch in 1951, going on to gain a master's (1952) and PhD from Ohio State University (OSU) in Columbus in 1965, where Paul Reichelderfer was her doctoral advisor.[10][11] Her dissertation was titled "On the product of absolutely continuous transformations of measure spaces".[12]
Career
Frazier Svager worked for a year at Wright-Patterson Air Force Base in Dayton, before teaching at Texas Southern University in Houston.[10] In 1954, she joined the faculty of CSU in Wilberforce.[10]
In 1967, Frazier Svager was appointed chairman of the department of mathematics.[13] She was awarded tenure in 1970.[14][10] She spent a summer in DC in 1966 as a systems analyst at NASA, as visiting faculty at MIT in 1969, and in 1985, she undertook postdoctoral study at OSU during the summer.[2] She was provost and vice president for academic affairs when she retired in 1993.[8][2] In March 1995, she returned for a short time to CSU as Interim President.[2]
Frazier Svager was active on the issue of scholarships, serving as the president of the local chapter of MOLES, a national association that provided scholarships for college students.[15] She was also a member of Beta Kappa Chi, the National Association of Mathematicians, and the Mathematical Association of America, and was involved with Jack and Jill of America.[15][2] Frazier Svager participated in the meeting that founded the National Association of Mathematics in 1969.[2]
She wrote two books, CSU's Modern Elementary Algebra Workbook (1969),[16] and Essential Mathematics for College Freshmen (1976).[17]
Personal life
While on the CSU faculty, Frazier met Aleksandar Svager, a Holocaust survivor from Yugoslavia and physics professor at CSU.[4][5] They married in June 1968 at her parents' home.[18]
Thyrsa Frazier Svager died on July 23, 1999.[19]
Philanthropy
Both university professors with a strong commitment to furthering education opportunities, the Svagers lived on one income, investing the other to build a scholarship fund.[5] After her death, her husband established the Thyrsa Frazier Svager Fund at the Dayton Foundation, for African-American women who major in mathematics at one of six universities, with a legacy contribution planned.[6][5] As of February 2017, 33 women had received support from the Fund.[5] An annual contribution is also being made to the American Physical Society's Minority Scholarship.[4]
Honors
Frazier Svager was honored with an Honorary Doctor of Humane Letters by CSU on her retirement, and she was inducted into the Hall of Fame in Greene County, Ohio.[2]
References
1. Kenschaft, PD (October 1981). "Black women in mathematics in the United States". The American Mathematical Monthly. 88 (8): 592–604. doi:10.1080/00029890.1981.11995321. JSTOR 2320508.
2. Houston, Johnny L (Summer 2000). "Spotlight on a mathematician: Thyrsa Anne Frazier Svager 1930–1999" (PDF). National Association of Mathematics Newsletter. xxxi (2): 9. Archived from the original (PDF) on April 19, 2017. Retrieved April 18, 2017.
3. Parks, Rena E. Lacey (2010). All the White Folks: A History of a People. Bloomington, IN: Xlibris Corporation. ISBN 978-1465328090. Retrieved April 16, 2017.
4. "APS Member Gives Minority Scholarship Fund a Boost". APS News. No. Vol 12 (2). American Physical Society. February 2003. Archived from the original on May 6, 2017. Retrieved April 16, 2017.
5. "Thrysa Frazier Svager: One of Dayton's Hidden Figures". YouTube. The Dayton Foundation. Retrieved April 18, 2017.
6. "Thyrsa Frazier Svager Scholarship Fund". DaytonFoundation.AcademicWorks.com. The Dayton Foundation. Retrieved April 18, 2017.
7. "Receive key to city". Xenia Daily Gazette. July 2, 1965. Retrieved April 16, 2017.
8. "Thyrsa Frazier Svager". DaytonFoundation.AcademicWorks.com. The Dayton Foundation. Retrieved April 18, 2017.
9. "Miss Frazier receives PhD at Ohio State". Xenia Daily Gazette. September 10, 1965. Retrieved April 16, 2017.
10. Kenschaft, Patricia C. "Thyrsa Anne Frazier Svager". MAA.org. Mathematical Association of America. Retrieved April 16, 2017.
11. Thyrsa Frazier Svager at the Mathematics Genealogy Project
12. Frazier, Thyrsa Anne (1965). On the product of absolutely continuous transformations of measure spaces. Columbus, OH: Ohio State University. Retrieved April 11, 2017.
13. "Several CSU department heads are changing". Xenia Daily Gazette. No. June 12, 1967. Retrieved April 16, 2017.
14. "12 CSU faculty members promoted". Xenia Daily Gazette. July 22, 1970. Retrieved April 16, 2017.
15. "MOLES chapter headed by Thyrsa Svager". Xenia Daily Gazette. October 6, 1971. Retrieved April 16, 2017.
16. Central State University, Department of Mathematics (1972). Modern elementary algebra workbook. Dubuque: W.C. Brown Book Co. Retrieved April 18, 2017.
17. Central State University, Department of Mathematics (August 25, 1976). Essential mathematics for College freshmen. Ohio: Kendall Hunt. Retrieved April 18, 2017.
18. "Marriage solemnized Sunday". Xenia Daily Gazette. June 19, 1968. Retrieved April 16, 2017.
19. "Black women in mathematics: Thyrsa Frazier Svager". Math.Buffalo.Edu. Buffalo University. Retrieved April 18, 2017.
External links
• Thyrsa Frazier Svager at the Mathematics Genealogy Project
• Tribute to Thyrsa Frazier Svager on YouTube (Dayton Foundation).
• Thyrsa Frazier Svager Scholarship Fund.
• Thyrsa Frazier Svager at Find a Grave
Authority control: Academics
• Mathematics Genealogy Project
| Wikipedia |
Thābit ibn Qurra
Thābit ibn Qurra (full name: Abū al-Ḥasan ibn Zahrūn al-Ḥarrānī al-Ṣābiʾ, Arabic: أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, Latin: Thebit/Thebith/Tebit);[2] 826 or 836 – February 19, 901,[3] was a polymath known for his work in mathematics, medicine, astronomy, and translation. He lived in Baghdad in the second half of the ninth century during the time of the Abbasid Caliphate.
Thābit ibn Qurra
Born210-211 AH/220-221 AH / 826 or 836 AD
Harran, the Jazira (Upper Mesopotamia) (now in Şanlıurfa Province, Turkey)
DiedWednesday, 26 Safar, 288 AH / February 19, 901 AD
Baghdad (now Iraq)
Academic background
InfluencesBanu Musa, Archimedes, Apollonius, Nicomachus, Euclid
Academic work
EraIslamic Golden Age
Main interestsMathematics, Mechanics, Astronomy, Astrology, Translation, Number theory
Notable ideas
• Early reformer of the Ptolemaic system
• A founder of statics
• Length of the sidereal year
Influencedal-Khazini, al-Isfizari, Na'im ibn Musa[1]
Thābit ibn Qurra made important discoveries in algebra, geometry, and astronomy. In astronomy, Thābit is considered one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.[4] Thābit also wrote extensively on medicine and produced philosophical treatises.[5]
Biography
Thābit was born in Harran in Upper Mesopotamia, which at the time was part of the Diyar Mudar subdivision of the al-Jazira region of the Abbasid Caliphate. Thābit belonged to the Sabians of Harran, a Hellenized Semitic polytheistic astral religion that still existed in ninth-century Harran.[6]
As a youth, Thābit worked as money changer in a marketplace in Harran until meeting Muḥammad ibn Mūsā, the oldest of three mathematicians and astronomers known as the Banū Mūsā. Thābit displayed such exceptional linguistic skills that ibn Mūsā chose him to come to Baghdad to be trained in mathematics, astronomy, and philosophy under the tutelage of the Banū Mūsā. Here, Thābit was introduced to not only a community of scholars but also to those who had significant power and influence in Baghdad.[7][8]
Thābit and his pupils lived in the midst of the most intellectually vibrant, and probably the largest, city of the time, Baghdad. Thābit came to Baghdad in the first place to work for the Banū Mūsā becoming a part of their circle and helping them translate Greek mathematical texts.[9] What is unknown is how Banū Mūsā and Thābit occupied himself with mathematics, astronomy, astrology, magic, mechanics, medicine, and philosophy. Later in his life, Thābit's patron was the Abbasid Caliph al-Mu'tadid (reigned 892–902), whom he became a court astronomer for.[9] Thābit became the Caliph's personal friend and courtier. Thābit died in Baghdad in 901. His son, Sinan ibn Thabit and grandson, Ibrahim ibn Sinan would also make contributions to the medicine and science.[10] By the end of his life, Thābit had managed to write 150 works on mathematics, astronomy, and medicine.[11] With all the work done by Thābit, most of his work has not lasted time. There are less than a dozen works by him that have survived.[10]
Translation
Thābit's native language was Syriac,[12] which was the Middle Aramaic variety from Edessa, and he was fluent in both Medieval Greek and Arabic.[13] He was the author to multiple treaties. Due to him being trilingual, Thābit was able to have a major role during the Graeco-Arabic translation movement.[10] He would also make a school of translation in Baghdad.[11]
Thābit translated from Greek into Arabic works by Apollonius of Perga, Archimedes, Euclid and Ptolemy. He revised the translation of Euclid's Elements of Hunayn ibn Ishaq. He also rewrote Hunayn's translation of Ptolemy's Almagest and translated Ptolemy's Geography.Thābit's translation of a work by Archimedes which gave a construction of a regular heptagon was discovered in the 20th century, the original having been lost.
Astronomy
Thābit is believed to have been an astronomer of Caliph al-Mu'tadid.[14] Thābit was able to use his mathematical work on the examination of Ptolemaic astronomy.[10] The medieval astronomical theory of the trepidation of the equinoxes is often attributed to Thābit. But it had already been described by Theon of Alexandria in his comments of the Handy Tables of Ptolemy. According to Copernicus, Thābit determined the length of the sidereal year as 365 days, 6 hours, 9 minutes and 12 seconds (an error of 2 seconds). Copernicus based his claim on the Latin text attributed to Thābit. Thābit published his observations of the Sun. In regards to Ptolemy's Planetary Hypotheses, Thābit examined the problems of the motion of the sun and moon, and the theory of sundials.[10] When looking at Ptolemy's Hypotheses, Thābit ibn Qurra found the Sidereal year which is when looking at the Earth and measuring it against the background of fixed stars, it will have a constant value.[15]
Thābit was also an author and wrote De Anno Solis. This book contained and recorded facts about the evolution in astronomy in the ninth century.[14] Thābit mentioned in the book that Ptolemy and Hipparchus believed that the movement of stars is consistent with the movement commonly found in planets. What Thābit believed is that this idea can be broadened to include the Sun and moon.[14] With that in mind, he also thought that the solar year should be calculated by looking at the sun's return to a given star.[14]
Mathematics
See also: Thabit number
In mathematics, Thābit derived an equation for determining amicable numbers. His proof of this rule is presented in the Treatise on the Derivation of the Amicable Numbers in an Easy Way.[16] This was done while writing on the theory of numbers, extending their use to describe the ratios between geometrical quantities, a step which the Greeks did not take. Thābit's work on amicable numbers and number theory helped him to invest more heavily into the Geometrical relations of numbers establishing his Transversal (geometry) theorem.[11][16]
Thābit described a generalized proof of the Pythagorean theorem.[17] He provided a strengthened extension of Pythagoras' proof which included the knowledge of Euclid's fifth postulate.[18] This postulate states that the intersection between two straight line segments combine to create two interior angles which are less than 180 degrees. The method of reduction and composition used by Thābit resulted in a combination and extension of contemporary and ancient knowledge on this famous proof. Thābit believed that geometry was tied with the equality and differences of magnitudes of lines and angles, as well as that ideas of motion (and ideas taken from physics more widely) should be integrated in geometry.[19]
The continued work done on geometric relations and the resulting exponential series allowed Thābit to calculate multiple solutions to chessboard problems. This problem was less to do with the game itself, and more to do with the number of solutions or the nature of solutions possible. In Thābit's case, he worked with combinatorics to work on the permutations needed to win a game of chess.[20]
In addition to Thābit's work on Euclidean geometry there is evidence that he was familiar with the geometry of Archimedes as well. His work with conic sections and the calculation of a paraboloid shape (cupola) show his proficiency as an Archimedean geometer. This is further embossed by Thābit's use of the Archimedean property in order to produce a rudimentary approximation of the volume of a paraboloid. The use of uneven sections, while relatively simple, does show a critical understanding of both Euclidean and Archimedean geometry.[21] Thābit was also responsible for a commentary on Archimedes' Liber Assumpta.[22]
Physics
In physics, Thābit rejected the Peripatetic and Aristotelian notions of a "natural place" for each element. He instead proposed a theory of motion in which both the upward and downward motions are caused by weight, and that the order of the universe is a result of two competing attractions (jadhb): one of these being "between the sublunar and celestial elements", and the other being "between all parts of each element separately".[23] and in mechanics he was a founder of statics.[24] In addition, Thābit's Liber Karatonis contained proof of the law of the lever. This work was the result of combining Aristotelian and Archimedean ideas of dynamics and mechanics.[11]
One of Qurra's most important pieces of text is his work with the Kitab fi 'l-qarastun. This text consists of Arabic mechanical tradition.[25] Another piece of important text is Kitab fi sifat alqazn, which discussed concepts of equal-armed balance. Qurra was reportedly one of the first to write about the concept of equal-armed balance or at least to systematize the treatment.
Qurra sought to establish a relationship between forces of motion and the distance traveled by the mobile.[25]
Medicine
Thābit was well known as a physician and produced a substantial number of medical treatises and commentaries. His works included general reference books such as al-Dhakhira fī ilm al-tibb ("A Treasury of Medicine"), Kitāb al-Rawda fi l–tibb ("Book of the Garden of Medicine"), and al-Kunnash ("Collection"). He also produced specific works on topics such as gallstones; the treatment of diseases such as smallpox, measles, and conditions of the eye; and discussed veterinary medicine and the anatomy of birds. Thābit wrote commentaries on the works of Galen and others, including such works as On Plants (attributed to Aristotle but likely written by the first-century BC philosopher Nicolaus of Damascus).[5]
One account of Thābit's work as a physician is given in Ibn al-Qiftī's Ta’rikh al-hukamā, where Thābit is credited with healing a butcher who was presumed to be certain to die.[5]
Works
Only a few of Thābit's works are preserved in their original form.
• On the Sector-Figure which deals with Menelaus' theorem.[26]
• On the Composition of Ratios[26]
• Kitab fi 'l-qarastun (Book of the Steelyard) [25]
• Kitab fi sifat alwazn (Book on the Description of Weight)[25] - Short text on equal-armed balance
Additional works by Thābit include:
• Kitāb al-Mafrūdāt (Book of Data)
• Maqāla fīistikhrāj al-a‘dād al-mutahābba bi–suhūlat al-maslak ilā dhālika (Book on the Determination of Amicable Numbers)
• Kitāb fi Misāhat qat‘ almakhrūt alladhī yusammaā al-mukāfi’ (Book on the Measurement of the Conic Section Called Parabolic)
• Kitāb fī Sanat al-shams (Book on the Solar Year)
• Qawl fi’l–Sabab alladhī ju‘ilat lahu miyāh al-bahr māliha (Discourse on the Reason Why Seawater Is Salted)
• al-Dhakhira fī ilm al-tibb (A Treasury of Medicine)
• Kitāb fi ‘ilm al-‘ayn . . . (Book on the Science of the Eye…)
• Kitāb fi’l–jadarī wa’l–hasbā (Book on Smallpox and Measles)
• Masā’il su’ila ’anhā Thābit ibn Qurra al-Harrānī (Questions Posed to Thābit. . .)[27]
Eponyms
• Thabit number
• Thebit (crater)
See also
• al-Battani, a contemporary Sabian astronomer and mathematician
References
1. Panza, Marco (2008). "The Role of Algebraic Inferences in Na'īm Ibn Mūsā's Collection of Geometrical Propositions". Arabic Sciences and Philosophy. 18 (2): 165–191. CiteSeerX 10.1.1.491.4854. doi:10.1017/S0957423908000532. S2CID 73620948.
2. For the Arabic name, see Rashed & Morelon 1960–2007; for the nisba al-Ṣābiʾ applied as a family name, see De Blois 1960–2007; for the Latin name, see Latham 2003, p. 403.
3. Rashed 2009d, pp. 23–24; Holme 2010.
4. Holme 2010.
5. Rosenfeld & Grigorian 2008, p. 292.
6. De Blois 1960–2007; Hämeen-Anttila 2006, p. 43, note 112; Van Bladel 2009, p. 65; Rashed 2009b, p. 646; Rashed 2009d, p. 21; Roberts 2017, pp. 253, 261–262. Some scholars have also suggested that he adhered to Mandaeism, a Gnostic baptist sect whose members were likewise called 'Sabians' (see Drower 1960, pp. 111–112; Nasoraia 2012, p. 39).
7. Gingerich 1986; Rashed & Morelon 1960–2007.
8. Rashed 2009c, pp. 3–4.
9. "Thābit ibn Qurrah | Arab mathematician, physician, and philosopher". Encyclopedia Britannica. Retrieved 2020-11-20.
10. "Thabit ibn Qurra". islamsci.mcgill.ca. Retrieved 2020-11-26.
11. Shloming, Robert (1970). "Thabit Ibn Qurra and the Pythagorean Theorem". The Mathematics Teacher. 63 (6): 519–528. doi:10.5951/MT.63.6.0519. ISSN 0025-5769. JSTOR 27958444.
12. Rashed & Morelon 1960–2007; "Thabit biography". www-groups.dcs.st-and.ac.uk. The sect, with strong Greek connections, had in earlier times adopted Greek culture, and it was common for members to speak Greek although after the conquest of the Sabians by Islam, they became Arabic speakers. There was another language spoken in southeastern Turkey, namely Syriac, which was based on the East Aramaic dialect of Edessa. This language was Thābit ibn Qurra's native language, but he was fluent in both Greek and Arabic.
13. Rashed & Morelon 1960–2007.
14. Carmody, Francis J. (1955). "Notes on the Astronomical Works of Thabit b. Qurra". Isis. 46 (3): 235–242. doi:10.1086/348408. ISSN 0021-1753. JSTOR 226342. S2CID 143097606.
15. Cohen, H. Floris (2010). "Greek Nature-Knowledge Transplanted". GREEK NATURE-KNOWLEDGE TRANSPLANTED:: THE ISLAMIC WORLD. pp. 53–76. doi:10.2307/j.ctt45kddd.6. ISBN 978-90-8964-239-4. JSTOR j.ctt45kddd.6. Retrieved 2020-11-27. {{cite book}}: |work= ignored (help)
16. Brentjes, Sonja; Hogendijk, Jan P (1989-11-01). "Notes on Thabit ibn Qurra and his rule for amicable numbers". Historia Mathematica. 16 (4): 373–378. doi:10.1016/0315-0860(89)90084-0. ISSN 0315-0860.
17. Sayili, Aydin (1960-03-01). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem". Isis. 51 (1): 35–37. doi:10.1086/348837. ISSN 0021-1753. S2CID 119868978.
18. "Thabit ibn Qurra". islamsci.mcgill.ca. Retrieved 2022-11-19.
19. Sabra, A. I. (1968). "Thābit Ibn Qurra on Euclid's Parallels Postulate". Journal of the Warburg and Courtauld Institutes. 31: 12–32. doi:10.2307/750634. JSTOR 750634. S2CID 195056568. Retrieved 2022-11-19.
20. Masood, Ehsan (2009). Science & Islam : a history. Library Genesis. London : Icon. ISBN 978-1-84831-040-7.
21. "Wilbur R. Knorr on Thābit ibn Qurra: A Case-Study in the Historiography of Premodern Science | Aestimatio: Sources and Studies in the History of Science". 2021-10-19. {{cite journal}}: Cite journal requires |journal= (help)
22. Shloming, Robert (1970-10-01). "Historically Speaking—: Thabit Qurra and the Pythagorean Theorem". The Mathematics Teacher. 63 (6): 519–528. doi:10.5951/MT.63.6.0519. ISSN 0025-5769.
23. Mohammed Abattouy (2001). "Greek Mechanics in Arabic Context: Thabit ibn Qurra, al-Isfizarı and the Arabic Traditions of Aristotelian and Euclidean Mechanics", Science in Context 14, p. 205-206. Cambridge University Press.
24. Holme 2010.
25. Abattouy, Mohammed (June 2001). "Greek Mechanics in Arabic Context: Thābit ibn Qurra, al-Isfizārī and the Arabic Traditions of Aristotelian and Euclidean Mechanics". Science in Context. 14 (1–2): 179–247. doi:10.1017/s0269889701000084. ISSN 0269-8897. S2CID 145604399.
26. Van Brummelen, Glen (2010-01-26). "Review of "On the Sector-Figure and Related Texts"". MAA Reviews. Retrieved 2017-05-12.
27. Rosenfeld & Grigorian 2008, pp. 292–295.
Sources used
• De Blois, F.C. (1960–2007). "Ṣābiʾ". In Bearman, P.; Bianquis, Th.; Bosworth, C.E.; van Donzel, E.; Heinrichs, W.P. (eds.). Encyclopaedia of Islam, Second Edition. doi:10.1163/1573-3912_islam_COM_0952.
• Drower, E.S. (1960). The Secret Adam: A Study of Nasoraean Gnosis. Oxford: Clarendon Press. OCLC 654318531.
• Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. 254 (4): 74–83. Bibcode:1986SciAm.254d..74G. doi:10.1038/scientificamerican0486-74. ISSN 0036-8733. JSTOR 24975932.
• Hämeen-Anttila, Jaakko (2006). The Last Pagans of Iraq: Ibn Waḥshiyya and His Nabatean Agriculture. Leiden: Brill. ISBN 978-90-04-15010-2.
• Holme, Audun (2010). Geometry : our cultural heritage (2nd ed.). Heidelberg: Springer. p. 188. ISBN 978-3-642-14440-0. Retrieved 2021-03-10.{{cite book}}: CS1 maint: url-status (link)
• Latham, J. D. (2003). "Review of Richard Lorch's 'Thabit ibn Qurran: On the Sector-Figure and Related Texts'". Journal of Semitic Studies. 48 (2): 401–403. doi:10.1093/jss/48.2.401.
• Nasoraia, Brikhah S. (2012). "Sacred Text and Esoteric Praxis in Sabian Mandaean Religion". In Çetinkaya, Bayram (ed.). Religious and Philosophical Texts: Rereading, Understanding and Comprehending Them in the 21st Century. Istanbul: Sultanbeyli Belediyesi. pp. vol. I, pp. 27–53.
• Rashed, Marwan (2009b). "Thabit ibn Qurra sur l'existence et l'infini: les réponses aux questions posées par Ibn Usayyid". In Rashed, Roshdi (ed.). Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad. Scientia Graeco-Arabica. Berlin: De Gruyter. pp. 619–673. doi:10.1515/9783110220797.6.619. ISBN 9783110220780.
• Rashed, Roshdi (2009c). "Thābit ibn Qurra, Scholar and Philosopher (826-901)". In Rashed, Roshdi (ed.). Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad. Scientia Graeco-Arabica. Berlin: De Gruyter. pp. 3–13. doi:10.1515/9783110220797.1.3. ISBN 9783110220780.
• Rashed, Roshdi (2009d). "Thābit ibn Qurra: From Ḥarrān to Baghdad". In Rashed, Roshdi (ed.). Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad. Scientia Graeco-Arabica. Berlin: De Gruyter. pp. 15–24. doi:10.1515/9783110220797.1.15. ISBN 9783110220780.
• Rashed, Roshdi; Morelon, Régis [in French] (1960–2007). "Thābit b. Ḳurra". In Bearman, P.; Bianquis, Th.; Bosworth, C.E.; van Donzel, E.; Heinrichs, W.P. (eds.). Encyclopaedia of Islam, Second Edition. doi:10.1163/1573-3912_islam_SIM_7507.
• Roberts, Alexandre M. (2017). "Being a Sabian at Court in Tenth-Century Baghdad". Journal of the American Oriental Society. 137 (2): 253–277. doi:10.17613/M6GB8Z.
• Rosenfeld, B. A.; Grigorian, A. T. (2008) [1970–80]. "Thābit Ibn Qurra, al-Ṣābiʾ al-Ḥarrānī". Complete Dictionary of Scientific Biography. Detroit, MI: Charles Scribner's Sons. pp. 288–295.
• Van Bladel, Kevin (2009). "Hermes and the Ṣābians of Ḥarrān". The Arabic Hermes: From Pagan Sage to Prophet of Science. Oxford: Oxford University Press. pp. 64–118. doi:10.1093/acprof:oso/9780195376135.003.0003. ISBN 978-0-19-537613-5.
Further reading
• Rashed, Roshdi, ed. (2009a). Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad. Scientia Graeco-Arabica. Berlin: De Gruyter. doi:10.1515/9783110220797. ISBN 9783110220780.
• Francis J. Carmody: The astronomical works of Thābit b. Qurra. 262 pp. Berkeley and Los Angeles: University of California Press, 1960.
• Rashed, Roshdi (1996). Les Mathématiques Infinitésimales du IXe au XIe Siècle 1: Fondateurs et commentateurs: Banū Mūsā, Ibn Qurra, Ibn Sīnān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd. London.{{cite book}}: CS1 maint: location missing publisher (link) Reviews: Seyyed Hossein Nasr (1998) in Isis 89 (1) pp. 112-113; Charles Burnett (1998) in Bulletin of the School of Oriental and African Studies, University of London 61 (2) p. 406.
• Churton, Tobias. The Golden Builders: Alchemists, Rosicrucians, and the First Freemasons. Barnes and Noble Publishing, 2006.
• Hakim S Ayub Ali. Zakhira-i Thābit ibn Qurra (preface by Hakim Syed Zillur Rahman), Aligarh, India, 1987.
External links
• Palmeri, JoAnn (2007). "Thābit ibn Qurra". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 1129–30. ISBN 978-0-387-31022-0. (PDF version)
• O'Connor, John J.; Robertson, Edmund F., "al-Sabi Thabit ibn Qurra al-Harrani", MacTutor History of Mathematics Archive, University of St Andrews
• Thabit ibn Qurra on Astrology & Magic
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| Wikipedia |
Théodore Moutard
Théodore Florentin Moutard (27 July 1827 – 13 March 1901) was a French mining engineer who worked at the École des Mines and contributed to mathematical geometry. The Moutard transformation in inverse geometry is named after him.
Moutard was born in Soultz, Haut-Rhin, to Florentin and Elisabeth Bernon. He was educated at the École Polytechnique and graduated in 1846 and entered the École des Mines and after graduating in 1849 he joined the Mining corps but was discharged in 1852 as he refused to take the oath required following the overthrow of Napoleon III. He joined back in 1870 and became a professor of mechanics at the École des Mines in 1875. He was also an examiner for the École Polytechnique from 1883. Moutard contributed to the La grande encyclopédie and his mathematical work was on algebraic surfaces and differential geometry. He collaborated with Victor Poncelet on elliptic functions.[1]
He was made Commander of the Legion of Honor in 1899. Moutard married twice and had two sons and two daughters. One of his sons was Douard Julien Moutard (1877-1948). One daughter Berthe married the mathematician Hermann Laurent. A daughter from his second marriage, Elisabeth married André Bujeaud, a politician and photographer, in 1868.[2]
References
1. Barbin, Évelyne; Guitart, René (2001). "Algèbre des fonctions elliptiques et géométrie des ovales cartésiennes" (PDF). Revue d'histoire des mathématiques (in French). 7 (2): 161–205.
2. Ferrand, Hervé Le (2014). "Au sujet du mathématicien Hermann Laurent (1841-1908)" (in French). {{cite journal}}: Cite journal requires |journal= (help)
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Théodore Olivier
Théodore Olivier (1793–1853) was a French mathematician.
Théodore Olivier
Born(1793-01-14)January 14, 1793
Lyon, France
DiedAugust 5, 1853(1853-08-05) (aged 60)
Lyon, France
Resting placeMontparnasse Cemetery
48°50′17″N 2°19′37″E
Alma materÉcole Polytechnique
Scientific career
FieldsMathematics
InstitutionsEcole centrale des arts et manufactures
InfluencesGaspard Monge
Life and work
Olivier studied in the Licée Imperial of Lyon where he obtained in 1811 a degree in mathematics with high honours. After this, he went to the École Polytechnique.[1] Olivier looked like Napoleon, but nobody could prove that Olivier was an illegitimate son of the Emperor.[2]
In 1815, he was an adjunct professor in the Artillery School at Metz and, in 1819, he became a full professor. In 1821, at the request of the King of Sweden, Charles XIV John (Jean-Baptiste Bernadotte), he went to Sweden to organize the military school of Mariemberg.[3]
Returning to France, Oliver criticized the pedagogical system in the École Polytechnique and in 1829, jointly with Alphonse Lavallée, Jean-Baptiste Dumas and Jean Claude Eugène Péclet, founded the École Centrale des Arts et Manufactures, where he was professor of geometry and mechanics for the rest of his life.[4] He also was, between 1830 and 1844, a professor at the École Polytechnique and, from 1838, a professor at the École Nationale Supérieure des Arts et Métiers.[5]
Olivier is mainly known for the construction of three-dimensional models of geometry for pedagogical purposes.[4] Most of them were sold to North American institutions such as Union College, the University of Columbia and West Point, where they are preserved.[6]
Olivier also studied the theory of gears, writing an extensive treatise on the subject, and constructing models, preserved in the Musée des Art et Offices in Paris.[7]
Olivier had no children, but he was the uncle of the French explorer Aimé Olivier de Sanderval.
References
1. Nesme, page 4.
2. Hervé, page 294.
3. Nesme, pages 5–6.
4. Nesme, page 7.
5. Hervé, page 296.
6. Hervé, page 298.
7. Hervé, pages 305 and follow.
Bibliography
• Hervé, J.M. (2007). "Théodore Olivier (1793–1853)". In Marco Ceccarelli (ed.). Distinguished Figures in Mechanism and Machine Science. Springer. pp. 294–319. ISBN 978-1-4020-6365-7.
• Jacomy, Bruno (1995). "Du cabinet au Conservatoire. Les instruments scientifiques du Conservatoire des Arts et Métiers à Paris". Journal of the History of Collections (in French). 7 (2): 227–233. doi:10.1093/jhc/7.2.227. ISSN 0954-6650.
• Nesme, Auguste (1858). Notice sur Théodore Olivier (in French). Aimé Vingtrinier.
External links
Wikimedia Commons has media related to Théodore Olivier (mathematician).
• O'Connor, John J.; Robertson, Edmund F., "Théodore Olivier", MacTutor History of Mathematics Archive, University of St Andrews
• Union College Permanent Collection, "Olivier Models".
• Union College (ed.). "The Olivier Models". Retrieved 19 June 2016.
• Canada Science and Technology Museum, ed. (2 June 2011). "Theodore Olivier 3D Geometric Models". Retrieved 19 June 2016.
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Théophile Lepage
Théophile Lepage (24 March 1901 – 1 April 1991) was a Belgian mathematician.
Théophile Lepage
Born(1901-03-24)24 March 1901
Limburg
Died1 April 1991(1991-04-01) (aged 90)
Verviers
NationalityBelgian
Alma materUniversité libre de Bruxelles
Known forCongruence of Lepage
Calculus of variations
Lepagian forms
Scientific career
FieldsMathematics
Doctoral advisorThéophile de Donder
Biography
Théophile Henri Joseph Lepage, better known as Théophile Lepage, was born in Limburg on March 24, 1901. Together with Alfred Errera he founded the seminar for mathematical analysis at the ULB. This seminar played an important role in the flourishing of the department of mathematics at this university.[1] He was professor of mathematics at the University of Liège from 1928 till 1930. He taught differential and integral calculus at the ULB from 1931 till 1956 and higher analysis from 1956 till 1971.
For 43 years he was a member of the Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. On June 5, 1948, he was nominated a corresponding member and on June 9, 1956 an effective member of the Académie. In 1963 he became president of the Académie and director of the Klasse Wetenschappen. He was also active in the Belgisch Wiskundig Genootschap.
He died in Verviers on April 1, 1991.
Mathematical work
At the ULB, the ideas and the enthusiasm of Théophile de Donder formed the foundation of a flourishing mathematical tradition. Thanks to student Théophile Lepage, external differential calculus acquired one of the most helpful methods introduced in mathematics during the 20th century, and one for which De Donder was a pioneer, presenting new applications in the resolution of a classical problem—the partial differential equation of Monge-Ampère—and in the synthesis of the methods of Théophile de Donder, Hermann Weyl and Constantin Carathéodory into a calculus of variations of multipal integrals.
Thanks to the use of differential geometry, it is possible to avoid long and boring calculations. The results of Lepage were named in reference works. His methods are still inspiring contemporary mathematicians: Boener and Sniatycki talked about the congruence of Lepage; not so long ago, Demeter Krupka, introduced—beside the eulerian forms which correspond to the classical equations of the calculus of variations of Euler—the so-called lepagian forms[2] or equivalents of Lepage in equations of variations on fiber spaces.
We also have Lepage to thank for interesting results concerning linear representations of the symplectic group, and more specifically Lepage's dissolution of an outer potency of the product of an even number of duplicates of a complex surface.
References
1. Mawhin, Jean (2001), Robert Halleux; Geert Vanpaemel; Jan Vandersmissen; Andrée Despy-Meyer (eds.), Geschiedenis van de wetenschappen in België 1815-2000, vol. 1, Brussel: Dexia/La Renaissance du livre, p. 71 and p. 75
2. D. Krupka (1977). "A map associated to the Lepagian forms on the calculus of variations in fibred manifolds". Czechoslovak Mathematical Journal. 27 (1): 114–117, 118. doi:10.21136/CMJ.1977.101449.
External links
• Théophile Lepage at the Mathematics Genealogy Project
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Tian Gang
Tian Gang (Chinese: 田刚; born November 24, 1958)[1] is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis.
Tian Gang
Tian at Oberwolfach in 2005
Born (1958-11-24) 24 November 1958
Nanjing, Jiangsu, China
NationalityChinese
Alma materHarvard University
Peking University
Nanjing University
Known forYau-Tian-Donaldson conjecture
K-stability
K-stability of Fano varieties
AwardsVeblen Prize (1996)
Alan T. Waterman Award (1994)
Scientific career
FieldsMathematics
InstitutionsPrinceton University
Peking University
ThesisKähler Metrics on Algebraic Manifolds (1988)
Doctoral advisorShing-Tung Yau
Doctoral studentsNataša Šešum
Wei Dongyi
Chinese name
Traditional Chinese田剛
Simplified Chinese田刚
Transcriptions
Standard Mandarin
Hanyu PinyinTián Gāng
As of 2020, he is the Vice Chairman of the China Democratic League and the President of the Chinese Mathematical Society. From 2017 to 2019 he served as the Vice President of Peking University.
Biography
Tian was born in Nanjing, Jiangsu, China. He qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, under the supervision of Shing-Tung Yau.
In 1998, he was appointed as a Cheung Kong Scholar professor at Peking University. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was a professor of mathematics at the Massachusetts Institute of Technology from 1995 to 2006 (holding the chair of Simons Professor of Mathematics from 1996). His employment at Princeton started from 2003, and was later appointed the Higgins Professor of Mathematics. Starting 2005, he has been the director of the Beijing International Center for Mathematical Research (BICMR);[2] from 2013 to 2017 he was the Dean of School of Mathematical Sciences at Peking University.[3] He and John Milnor are Senior Scholars of the Clay Mathematics Institute (CMI). In 2011, Tian became director of the Sino-French Research Program in Mathematics at the Centre national de la recherche scientifique (CNRS) in Paris. In 2010, he became scientific consultant for the International Center for Theoretical Physics in Trieste, Italy.[4]
Tian has served on many committees, including for the Abel Prize and the Leroy P. Steele Prize.[5] He is a member of the editorial boards of many journals, including Advances in Mathematics and the Journal of Geometric Analysis. In the past he has been on the editorial boards of Annals of Mathematics and the Journal of the American Mathematical Society.
Among his awards and honors:
• Sloan Research Fellowship (1991-1993)
• Alan T. Waterman Award (1994)
• Oswald Veblen Prize in Geometry (1996)
• Elected to the Chinese Academy of Sciences (2001)
• Elected to the American Academy of Arts and Sciences (2004)
Since at least 2013 he has been heavily involved in Chinese politics, serving as the Vice Chairman of the China Democratic League, the second most populous political party in China.
Mathematical contributions
The Kähler-Einstein problem
Tian is well-known for his contributions to Kähler geometry, and in particular to the study of Kähler-Einstein metrics. Shing-Tung Yau, in his renowned resolution of the Calabi conjecture, had settled the case of closed Kähler manifolds with nonpositive first Chern class. His work in applying the method of continuity showed that C0 control of the Kähler potentials would suffice to prove existence of Kähler-Einstein metrics on closed Kähler manifolds with positive first Chern class, also known as "Fano manifolds." Tian and Yau extended Yau's analysis of the Calabi conjecture to noncompact settings, where they obtained partial results.[TY90] They also extended their work to allow orbifold singularities.[TY91]
Tian introduced the "α-invariant," which is essentially the optimal constant in the Moser-Trudinger inequality when applied to Kähler potentials with a supremal value of 0. He showed that if the α-invariant is sufficiently large (i.e. if a sufficiently strong Moser-Trudinger inequality holds), then C0 control in Yau's method of continuity could be achieved.[T87b] This was applied to demonstrate new examples of Kähler-Einstein surfaces. The case of Kähler surfaces was revisited by Tian in 1990, giving a complete resolution of the Kähler-Einstein problem in that context.[T90b] The main technique was to study the possible geometric degenerations of a sequence of Kähler-Einstein metrics, as detectable by the Gromov–Hausdorff convergence. Tian adapted many of the technical innovations of Karen Uhlenbeck, as developed for Yang-Mills connections, to the setting of Kähler metrics. Some similar and influential work in the Riemannian setting was done in 1989 and 1990 by Michael Anderson, Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima.[6][7][8]
Tian's most renowned contribution to the Kähler-Einstein problem came in 1997. Yau had conjectured in the 1980s, based partly in analogy to the Donaldson-Uhlenbeck-Yau theorem, that existence of a Kähler-Einstein metric should correspond to stability of the underlying Kähler manifold in a certain sense of geometric invariant theory. It was generally understood, especially following work of Akito Futaki,[9] that the existence of holomorphic vector fields should act as an obstruction to the existence of Kähler-Einstein metrics. Tian and Wei Yue Ding established that this obstruction is not sufficient within the class of Kähler orbifolds.[DT92] Tian, in his 1997 article, gave concrete examples of Kähler manifolds (rather than orbifolds) which had no holomorphic vector fields and also no Kähler-Einstein metrics, showing that the desired criterion lies deeper.[T97] Yau had proposed that, rather than holomorphic vector fields on the manifold itself, it should be relevant to study the deformations of projective embeddings of Kähler manifolds under holomorphic vector fields on projective space. This idea was modified by Tian, introducing the notion of K-stability and showing that any Kähler-Einstein manifold must be K-stable.[T97]
Simon Donaldson, in 2002, modified and extended Tian's definition of K-stability.[10] The conjecture that K-stability would be sufficient to ensure the existence of a Kähler-Einstein metric became known as the Yau-Tian-Donaldson conjecture. In 2015, Xiuxiong Chen, Donaldson, and Song Sun, published a proof of the conjecture, receiving the Oswald Veblen Prize in Geometry for their work.[11][12][13] Tian published a proof of the conjecture in the same year, although Chen, Donaldson, and Sun have accused Tian of academic and mathematical misconduct over his paper.[T15][14][15]
Kähler geometry
In one of his first articles, Tian studied the space of Calabi-Yau metrics on a Kähler manifold.[T87a] He showed that any infinitesimal deformation of Calabi-Yau structure can be 'integrated' to a one-parameter family of Calabi-Yau metrics; this proves that the "moduli space" of Calabi-Yau metrics on the given manifold has the structure of a smooth manifold. This was also studied earlier by Andrey Todorov, and the result is known as the Tian−Todorov theorem.[16] As an application, Tian found a formula for the Weil-Petersson metric on the moduli space of Calabi-Yau metrics in terms of the period mapping.[T87a][17]
Motivated by the Kähler-Einstein problem and a conjecture of Yau relating to Bergman metrics, Tian studied the following problem. Let L be a line bundle over a Kähler manifold M, and fix a hermitian bundle metric whose curvature form is a Kähler form on M. Suppose that for sufficiently large m, an orthonormal set of holomorphic sections of the line bundle L⊗m defines a projective embedding of M. One can pull back the Fubini-Study metric to define a sequence of metrics on M as m increases. Tian showed that a certain rescaling of this sequence will necessarily converge in the C2 topology to the original Kähler metric.[T90a] The refined asymptotics of this sequence were taken up in a number of influential subsequent papers by other authors, and are particularly important in Simon Donaldson's program on extremal metrics.[18][19][20][21][22] The approximability of a Kähler metric by Kähler metrics induced from projective embeddings is also relevant to Yau's picture of the Yau-Tian-Donaldson conjecture, as indicated above.
In a highly technical article, Xiuxiong Chen and Tian studied the regularity theory of certain complex Monge-Ampère equations, with applications to the study of the geometry of extremal Kähler metrics.[CT08] Although their paper has been very widely cited, Julius Ross and David Witt Nyström found counterexamples to the regularity results of Chen and Tian in 2015.[23] It is not clear which results of Chen and Tian's article remain valid.
Gromov-Witten theory
Pseudoholomorphic curves were shown by Mikhail Gromov in 1985 to be powerful tools in symplectic geometry.[24] In 1991, Edward Witten conjectured a use of Gromov's theory to define enumerative invariants.[25] Tian and Yongbin Ruan found the details of such a construction, proving that the various intersections of the images of pseudo-holomorphic curves is independent of many choices, and in particular gives an associative multilinear mapping on the homology of certain symplectic manifolds.[RT95] This structure is known as quantum cohomology; a contemporaneous and similarly influential approach is due to Dusa McDuff and Dietmar Salamon.[26] Ruan and Tian's results are in a somewhat more general setting.
With Jun Li, Tian gave a purely algebraic adaptation of these results to the setting of algebraic varieties.[LT98b] This was done at the same time as Kai Behrend and Barbara Fantechi, using a different approach.[27]
Li and Tian then adapted their algebro-geometric work back to the analytic setting in symplectic manifolds, extending the earlier work of Ruan and Tian.[LT98a] Tian and Gang Liu made use of this work to prove the well-known Arnold conjecture on the number of fixed points of Hamiltonian diffeomorphisms.[LT98c] However, these papers of Li-Tian and Liu-Tian on symplectic Gromov-Witten theory have been criticized by Dusa McDuff and Katrin Wehrheim as being incomplete or incorrect, saying that Li and Tian's article [LT98a] "lacks almost all detail" on certain points and that Liu and Tian's article [LT98c] has "serious analytic errors."[28]
Geometric analysis
In 1995, Tian and Weiyue Ding studied the harmonic map heat flow of a two-dimensional closed Riemannian manifold into a closed Riemannian manifold N.[DT95] In a seminal 1985 work, following the 1982 breakthrough of Jonathan Sacks and Karen Uhlenbeck, Michael Struwe had studied this problem and showed that there is a weak solution which exists for all positive time. Furthermore, Struwe showed that the solution u is smooth away from finitely many spacetime points; given any sequence of spacetime points at which the solution is smooth and which converge to a given singular point (p, T), one can perform some rescalings to (subsequentially) define a finite number of harmonic maps from the round 2-dimensional sphere into N, called "bubbles." Ding and Tian proved a certain "energy quantization," meaning that the defect between the Dirichlet energy of u(T) and the limit of the Dirichlet energy of u(t) as t approaches T is exactly measured by the sum of the Dirichlet energies of the bubbles. Such results are significant in geometric analysis, following the original energy quantization result of Yum-Tong Siu and Shing-Tung Yau in their proof of the Frankel conjecture.[29] The analogous problem for harmonic maps, as opposed to Ding and Tian's consideration of the harmonic map flow, was considered by Changyou Wang around the same time.[30]
A major paper of Tian's dealt with the Yang–Mills equations.[T00a] In addition to extending much of Karen Uhlenbeck's analysis to higher dimensions, he studied the interaction of Yang-Mills theory with calibrated geometry. Uhlenbeck had shown in the 1980s that, when given a sequence of Yang-Mills connections of uniformly bounded energy, they will converge smoothly on the complement of a subset of codimension at least four, known as the complement of the "singular set". Tian showed that the singular set is a rectifiable set. In the case that the manifold is equipped with a calibration, one can restrict interest to the Yang-Mills connections which are self-dual relative to the calibration. In this case, Tian showed that the singular set is calibrated. For instance, the singular set of a sequence of hermitian Yang-Mills connections of uniformly bounded energy will be a holomorphic cycle. This is a significant geometric feature of the analysis of Yang-Mills connections.
Ricci flow
In 2006, Tian and Zhou Zhang studied the Ricci flow in the special setting of closed Kähler manifolds.[TZ06] Their principal achievement was to show that the maximal time of existence can be characterized in purely cohomological terms. This represents one sense in which the Kähler-Ricci flow is significantly simpler than the usual Ricci flow, where there is no (known) computation of the maximal time of existence from a given geometric context. Tian and Zhang's proof consists of a use of the scalar maximum principle as applied to various geometric evolution equations, in terms of a Kähler potential as parametrized by a linear deformation of forms which is cohomologous to the Kähler-Ricci flow itself. In a notable work with Jian Song, Tian analyzed the Kähler Ricci flow on certain two-dimensional complex manifolds.[ST07]
In 2002 and 2003, Grigori Perelman posted three papers on the arXiv which purported to prove the Poincaré conjecture and Geometrization conjecture in the field of three-dimensional geometric topology.[31][32][33] Perelman's papers were immediately acclaimed for many of their novel ideas and results, although the technical details of many of his arguments were seen as hard to verify. In collaboration with John Morgan, Tian published an exposition of Perelman's papers in 2007, filling in many of the details.[MT07] Other expositions, which have also been widely studied, were written by Huai-Dong Cao and Xi-Ping Zhu, and by Bruce Kleiner and John Lott.[34][35] Morgan and Tian's exposition is the only of the three to deal with Perelman's third paper,[33] which is irrelevant for analysis of the geometrization conjecture but uses curve-shortening flow to provide a simpler argument for the special case of the Poincaré conjecture. Eight years after the publication of Morgan and Tian's book, Abbas Bahri pointed to part of their exposition of this paper to be in error, having relied upon incorrect computations of evolution equations.[36] The error, which dealt with details not present in Perelman's paper, was soon after amended by Morgan and Tian.[37]
In collaboration with Nataša Šešum, Tian also published an exposition of Perelman's work on the Ricci flow of Kähler manifolds, which Perelman did not publish in any form.[38]
Selected publications
Research articles.
T87a.
Tian, Gang (1987). "Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric". In Yau, S.-T. (ed.). Mathematical aspects of string theory. Conference held at the University of California, San Diego (July 21–August 1, 1986). Advanced Series in Mathematical Physics. Vol. 1. Singapore: World Scientific Publishing Co. pp. 629–646. doi:10.1142/9789812798411_0029. ISBN 9971-50-273-9. MR 0915841.
T87b.
Tian, Gang (1987). "On Kähler–Einstein metrics on certain Kähler manifolds with c1(M) > 0". Inventiones Mathematicae. 89 (2): 225–246. doi:10.1007/BF01389077. MR 0894378. S2CID 122352133.
TY87.
Tian, Gang; Yau, Shing-Tung (1987). "Kähler–Einstein metrics on complex surfaces with C1 > 0". Communications in Mathematical Physics. 112 (1): 175–203. doi:10.1007/BF01217685. MR 0904143. S2CID 121216755.
T90a.
Tian, Gang (1990). "On a set of polarized Kähler metrics on algebraic manifolds". Journal of Differential Geometry. 32 (1): 99–130. doi:10.4310/jdg/1214445039. MR 1064867.
T90b.
Tian, G. (1990). "On Calabi's conjecture for complex surfaces with positive first Chern class". Inventiones Mathematicae. 101 (1): 101–172}. Bibcode:1990InMat.101..101T. doi:10.1007/BF01231499. MR 1055713. S2CID 59419559.
TY90.
Tian, G.; Yau, Shing-Tung (1990). "Complete Kähler manifolds with zero Ricci curvature. I". Journal of the American Mathematical Society. 3 (3): 579–609. doi:10.1090/S0894-0347-1990-1040196-6. MR 1040196.
TY91.
Tian, Gang; Yau, Shing-Tung (1991). "Complete Kähler manifolds with zero Ricci curvature. II". Inventiones Mathematicae. 106 (1): 27–60. Bibcode:1991InMat.106...27T. doi:10.1007/BF01243902. MR 1123371. S2CID 122638262.
DT92.
Ding, Wei Yue; Tian, Gang (1992). "Kähler–Einstein metrics and the generalized Futaki invariant". Inventiones Mathematicae. 110: 315–335. Bibcode:1992InMat.110..315D. doi:10.1007/BF01231335. MR 1185586. S2CID 59332400.
DT95.
Ding, Weiyue; Tian, Gang (1995). "Energy identity for a class of approximate harmonic maps from surfaces". Communications in Analysis and Geometry. 3 (3–4): 543–554. doi:10.4310/CAG.1995.v3.n4.a1. MR 1371209.
RT95.
Ruan, Yongbin; Tian, Gang (1995). "A mathematical theory of quantum cohomology". Journal of Differential Geometry. 42 (2): 259–367. doi:10.4310/jdg/1214457234. MR 1366548.
ST97.
Siebert, Bernd; Tian, Gang (1997). "On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator". Asian Journal of Mathematics. 1 (4): 679–695. doi:10.4310/AJM.1997.v1.n4.a2. MR 1621570. S2CID 14494725.
T97.
Tian, Gang (1997). "Kähler–Einstein metrics with positive scalar curvature". Inventiones Mathematicae. 130 (1): 1–37. Bibcode:1997InMat.130....1T. doi:10.1007/s002220050176. MR 1471884. S2CID 122529381.
LT98a.
Li, Jun; Tian, Gang (1998). "Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds". In Stern, Ronald J. (ed.). Topics in symplectic 4-manifolds. 1st International Press Lectures presented at the University of California, Irvine (March 28–30, 1996). First International Press Lecture Series. Vol. I. Cambridge, MA: International Press. pp. 47–83. arXiv:alg-geom/9608032. ISBN 1-57146-019-5. MR 1635695.
LT98b.
Li, Jun; Tian, Gang (1998). "Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties". Journal of the American Mathematical Society. 11 (1): 119–174. doi:10.1090/S0894-0347-98-00250-1. MR 1467172. S2CID 15201721.
LT98c.
Liu, Gang; Tian, Gang (1998). "Floer homology and Arnold conjecture". Journal of Differential Geometry. 49 (1): 1–74. doi:10.4310/jdg/1214460936. MR 1642105.
T00a.
Tian, Gang (2000). "Gauge theory and calibrated geometry. I". Annals of Mathematics. Second Series. 151 (1): 193–268. arXiv:math/0010015. doi:10.2307/121116. JSTOR 121116. MR 1745014.
TZ06.
Tian, Gang; Zhang, Zhou (2006). "On the Kähler–Ricci flow on projective manifolds of general type". Chinese Annals of Mathematics, Series B. 27 (2): 179–192. CiteSeerX 10.1.1.116.5906. doi:10.1007/s11401-005-0533-x. MR 2243679. S2CID 16476473.
ST07.
Song, Jian; Tian, Gang (2007). "The Kähler–Ricci flow on surfaces of positive Kodaira dimension". Inventiones Mathematicae. 17 (3): 609–653. arXiv:math/0602150. Bibcode:2007InMat.170..609S. doi:10.1007/s00222-007-0076-8. MR 2357504. S2CID 735225.
CT08.
Chen, X. X.; Tian, G. (2008). "Geometry of Kähler metrics and foliations by holomorphic discs". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 107: 1–107. arXiv:math/0507148. doi:10.1007/s10240-008-0013-4. MR 2434691. S2CID 119699845.
T15.
Tian, Gang (2015). "K-stability and Kähler–Einstein metrics". Communications on Pure and Applied Mathematics. 68 (7): 1085–1156. arXiv:1211.4669. doi:10.1002/cpa.21578. MR 3352459. S2CID 119303358. (Erratum: doi:10.1002/cpa.21612)
Books.
T00b.
Tian, Gang (2000). Canonical metrics in Kähler geometry. Lectures in Mathematics ETH Zürich. Notes taken by Meike Akveld. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-8389-4. ISBN 3-7643-6194-8. MR 1787650.
MT07.
Morgan, John; Tian, Gang (2007). Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs. Vol. 3. Cambridge, MA: Clay Mathematics Institute. arXiv:math/0607607. ISBN 978-0-8218-4328-4. MR 2334563.
Morgan, John; Tian, Gang (2015). "Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture". arXiv:1512.00699 [math.DG].
MT14.
Morgan, John; Tian, Gang (2014). The geometrization conjecture. Clay Mathematics Monographs. Vol. 5. Cambridge, MA: Clay Mathematics Institute. ISBN 978-0-8218-5201-9. MR 3186136.
References
1. "1996 Oswald Veblen Prize" (PDF). AMS. 1996.
2. Governing Board, Beijing International Center for Mathematical Research, http://www.bicmr.org/content/page/27.html
3. History of School of Mathematical Sciences, Peking University, http://www.math.pku.edu.cn/static/lishiyange.html
4. "ICTP - Governance". www.ictp.it. Retrieved 2018-05-28.
5. http://www.ams.org/notices/201304/rnoti-p480.pdf
6. Anderson, Michael T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), no. 3, 455–490.
7. Bando, Shigetoshi; Kasue, Atsushi; Nakajima, Hiraku. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97 (1989), no. 2, 313–349.
8. Anderson, Michael T. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102 (1990), no. 2, 429–445.
9. Futaki, A. An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73 (1983), no. 3, 437–443.
10. Donaldson, S.K. Scalar curvature and stability of toric varieties. J. Differential Geom. 62 (2002), no. 2, 289–349.
11. Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (2015), no. 1, 183–197.
12. Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. J. Amer. Math. Soc. 28 (2015), no. 1, 199–234.
13. Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof. J. Amer. Math. Soc. 28 (2015), no. 1, 235–278.
14. Xiuxiong Chen, Simon, Donaldson, and Song Sun. On some recent developments in Kähler geometry.
15. Gang Tian. Response to CDS.
16. Todorov, Andrey N. The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds. I. Comm. Math. Phys. 126 (1989), no. 2, 325–346.
17. Huybrechts, Daniel. Complex geometry. An introduction. [Chapter 6.] Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp. ISBN 3-540-21290-6
18. Zelditch, Steve. Szegő kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331.
19. Catlin, David. The Bergman kernel and a theorem of Tian. Analysis and geometry in several complex variables (Katata, 1997), 1–23, Trends Math., Birkhäuser Boston, Boston, MA, 1999.
20. Lu, Zhiqin. On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. 122 (2000), no. 2, 235–273.
21. Donaldson, S.K. Scalar curvature and projective embeddings. I. J. Differential Geom. 59 (2001), no. 3, 479–522.
22. Donaldson, S.K. Lower bounds on the Calabi functional. J. Differential Geom. 70 (2005), no. 3, 453–472.
23. Ross, Julius; Nyström, David Witt. Harmonic discs of solutions to the complex homogeneous Monge-Ampère equation. Publ. Math. Inst. Hautes Études Sci. 122 (2015), 315–335.
24. Gromov, M. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307–347.
25. Witten, Edward. Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.
26. McDuff, Dusa; Salamon, Dietmar. J-holomorphic curves and quantum cohomology. University Lecture Series, 6. American Mathematical Society, Providence, RI, 1994. viii+207 pp. ISBN 0-8218-0332-8
27. Behrend, K.; Fantechi, B. The intrinsic normal cone. Invent. Math. 128 (1997), no. 1, 45–88.
28. McDuff, Dusa; Wehrheim, Katrin. The fundamental class of smooth Kuranishi atlases with trivial isotropy. J. Topol. Anal. 10 (2018), no. 1, 71–243.
29. Siu, Yum Tong; Yau, Shing Tung. Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay. Ann. of Math. (2) 105 (1977), no. 2, 225–264.
30. Wang, Changyou. Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets. Houston J. Math. 22 (1996), no. 3, 559–590.
31. Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
32. Grisha Perelman. Ricci flow with surgery on three-manifolds. arXiv:math/0303109
33. Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245
34. Cao, Huai-Dong; Zhu, Xi-Ping. A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006), no. 2, 165–492.
35. Kleiner, Bruce; Lott, John. Notes on Perelman's papers. Geom. Topol. 12 (2008), no. 5, 2587–2855.
36. Bahri, Abbas. Five gaps in mathematics. Adv. Nonlinear Stud. 15 (2015), no. 2, 289–319.
37. John Morgan and Gang Tian. Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture. arXiv:1512.00699 (2015)
38. Sesum, Natasa; Tian, Gang. Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman). J. Inst. Math. Jussieu 7 (2008), no. 3, 575–587.
External links
• Tian Gang at the Mathematics Genealogy Project
Recipients of the Oswald Veblen Prize in Geometry
• 1964 Christos Papakyriakopoulos
• 1964 Raoul Bott
• 1966 Stephen Smale
• 1966 Morton Brown and Barry Mazur
• 1971 Robion Kirby
• 1971 Dennis Sullivan
• 1976 William Thurston
• 1976 James Harris Simons
• 1981 Mikhail Gromov
• 1981 Shing-Tung Yau
• 1986 Michael Freedman
• 1991 Andrew Casson and Clifford Taubes
• 1996 Richard S. Hamilton and Gang Tian
• 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins
• 2004 David Gabai
• 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó
• 2010 Tobias Colding and William Minicozzi; Paul Seidel
• 2013 Ian Agol and Daniel Wise
• 2016 Fernando Codá Marques and André Neves
• 2019 Xiuxiong Chen, Simon Donaldson and Song Sun
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| Wikipedia |
Tian Ye (mathematician)
Tian Ye or Ye Tian (Chinese: 田野) is a Chinese mathematician known for his research in number theory and arithmetic geometry.
Tian Ye
Alma materColumbia University
Awards
• Morningside Silver Medal (2007)
• Morningside Gold Medal (2013)
• ICTP Ramanujan Prize 2013
Scientific career
FieldsMathematics
InstitutionsChinese Academy of Sciences
ThesisEuler Systems of CM Points on Shimura Curves (2003)
Doctoral advisorShou-Wu Zhang
Career
Tian received his PhD in mathematics under Shou-Wu Zhang at Columbia University in 2003 and is currently a professor at the Chinese Academy of Sciences.[1][2]
He received of the ICTP Ramanujan Prize (2013) and the Morningside Medal (Silver 2007, Gold 2013).[3][4]
Selected publications
• Li, Jian-Shu; Sun, Binyong; Tian, Ye (2011), "The multiplicity one conjecture for local theta correspondences" (PDF), Inventiones Mathematicae, 184 (1): 117–124, Bibcode:2011InMat.184..117L, doi:10.1007/s00222-010-0287-2, MR 2782253, S2CID 122139021.
• Tian, Ye (2014), "Congruent numbers and Heegner points", Cambridge Journal of Mathematics, 2 (1): 117–161, arXiv:1210.8231, doi:10.4310/CJM.2014.v2.n1.a4, MR 3272014, S2CID 55390076.
• Diaconu, Adrian; Tian, Ye (2005), "Twisted Fermat curves over totally real fields", Annals of Mathematics, Second Series, 162 (3): 1353–1376, arXiv:0706.0470, doi:10.4007/annals.2005.162.1353, JSTOR 20159945, MR 2179733, S2CID 15523427.
References
1. Tian Ye at the Mathematics Genealogy Project
2. Home page, Morningside Center of Mathematics, Chinese Academy of Sciences, retrieved 2015-05-06.
3. "Ye Tian Awarded 2013 ICTP/IMU Ramanujan Prize" (PDF), Mathematics People, Notices of the American Mathematical Society, 61 (2): 195, February 2014.
4. Prof.Ye TIAN Honored with the Morningside Medal of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, retrieved 2015-05-06.
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| Wikipedia |
Tiberiu Popoviciu
Tiberiu Popoviciu (February 16, 1906–October 29, 1975) was a Romanian mathematician[1] and the namesake of Popoviciu's inequality and Popoviciu's inequality on variances.
Tiberiu Popoviciu
Born(1906-02-16)February 16, 1906
Arad, Hungary
DiedOctober 29, 1975(1975-10-29) (aged 69)
Cluj-Napoca, Socialist Republic of Romania
Resting placeHajongard Cemetery, Cluj-Napoca
NationalityRomanian
Alma materParis-Sorbonne University
University of Bucharest
Known forPopoviciu's inequality
Popoviciu's inequality on variances
SpouseElena Moldovan Popoviciu
Scientific career
FieldsMathematics
InstitutionsUniversity of Cernăuți
University of Bucharest
University of Iași
University of Cluj
ThesisSur quelques propriétés des fonctions d'une ou de deux variables réelles (1933)
Doctoral advisorPaul Montel
Doctoral studentsElena Moldovan Popoviciu
The Tiberiu Popoviciu High School of Computer Science in Cluj-Napoca is named after him.[2]
In 1951 he founded a research institute which now bears his name: Tiberiu Popoviciu Institute of Numerical Analysis.
Biography
Popoviciu was born in Arad in 1906, and attended high school in his hometown, the school which is now the Moise Nicoară National College. He graduated from the University of Bucharest, and got his doctorate in 1933 under Paul Montel from Paris-Sorbonne University.[3]
He was a lecturer at the Universities of Cernăuți, Bucharest, and Iași. In 1946 he was appointed professor at the University of Cluj. On June 4, 1937 Popoviciu was elected a corresponding member of the Romanian Academy of Sciences. In November 1948 he was elected a corresponding member of the Romanian Academy. He became full member of the mathematical sciences section of the Academy on March 20, 1963.
He married his former student, Elena Moldovan Popoviciu, in 1964; she also became a notable functional analyst.[4] He died in 1975 in Cluj-Napoca, and is buried in the city's Hajongard Cemetery.
References
1. Breckner, Wolfgang, Profesor Tiberiu Popoviciu, Babeș-Bolyai University, Faculty of Mathematics and Informatics, retrieved 2015-04-19.
2. Istoric, Tiberiu Popoviciu High School of Computer Science (in Romanian), retrieved 2015-04-19.
3. Tiberiu Popoviciu at the Mathematics Genealogy Project
4. O'Connor, John J.; Robertson, Edmund F., "Elena Moldovan Popoviciu", MacTutor History of Mathematics Archive, University of St Andrews
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Tiberiu Popoviciu Institute of Numerical Analysis
The Tiberiu Popoviciu Institute of Numerical Analysis Romanian: Institutul de Calcul "Tiberiu Popoviciu" (ICTP) is a mathematics research institute of the Romanian Academy, based in Cluj-Napoca, Romania.
Tiberiu Popoviciu Institute of Numerical Analysis (Romanian Academy)
Founder(s)Tiberiu Popoviciu
Established1951
FocusNumerical Analysis
OwnerRomanian Academy
AddressStr. Fântânele nr. 57, ap. 67-68, 400320
Location
Cluj-Napoca
,
Romania
Coordinates46°45′56″N 23°32′53″E
Websitewww.ictp.acad.ro
ICTP is coordinated by the Mathematical Section and belongs to the Cluj-Napoca Branch of the Romanian Academy. The Institute performs fundamental research mainly in the field of Numerical Analysis.
ICTP was founded in 1951, as the Mathematical Section of the Cluj-Napoca Branch, with residence at 37 Republicii Street.[1]
References
1. "Institute history".
External links
• "Official site".
Authority control
• ISNI
| Wikipedia |
Tibor Szele
Tibor Szele (Debrecen, 21 June 1918 – Szeged, 5 April 1955) Hungarian mathematician, working in combinatorics and abstract algebra. After graduating at the Debrecen University, he became a researcher at the Szeged University in 1946, then he went back at the Debrecen University in 1948 where he became full professor in 1952. He worked especially in the theory of Abelian groups and ring theory. He generalized Hajós's theorem. He founded the Hungarian school of algebra. Tibor Szele received the Kossuth Prize in 1952.
References
A panorama of Hungarian Mathematics in the Twentieth Century, p. 601.
External links
• Grave of Tibor Szele
• O'Connor, John J.; Robertson, Edmund F., "Tibor Szele", MacTutor History of Mathematics Archive, University of St Andrews
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| Wikipedia |
Tietze's graph
In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line – and therefore that graphs that are embedded onto the Möbius strip may require six colors.[1] The boundary segments of the regions of Tietze's subdivision (including the segments along the boundary of the Möbius strip itself) form an embedding of Tietze's graph.
Tietze's graph
The Tietze graph
Vertices12
Edges18
Radius3
Diameter3
Girth3
Automorphisms12 (D6)
Chromatic number3
Chromatic index4
PropertiesCubic
Snark
Table of graphs and parameters
Relation to Petersen graph
Tietze's graph may be formed from the Petersen graph by replacing one of its vertices with a triangle.[2][3] Like the Tietze graph, the Petersen graph forms the boundary of six mutually touching regions, but on the projective plane rather than on the Möbius strip. If one cuts a hole from this subdivision of the projective plane, surrounding a single vertex, the surrounded vertex is replaced by a triangle of region boundaries around the hole, giving the previously described construction of the Tietze graph.
Hamiltonicity
Both Tietze's graph and the Petersen graph are maximally nonhamiltonian: they have no Hamiltonian cycle, but any two non-adjacent vertices can be connected by a Hamiltonian path.[2] Tietze's graph and the Petersen graph are the only 2-vertex-connected cubic non-Hamiltonian graphs with 12 or fewer vertices.[4]
Unlike the Petersen graph, Tietze's graph is not hypohamiltonian: removing one of its three triangle vertices forms a smaller graph that remains non-Hamiltonian.
Edge coloring and perfect matchings
Edge coloring Tietze's graph requires four colors; that is, its chromatic index is 4. Equivalently, the edges of Tietze's graph can be partitioned into four matchings, but no fewer.
Tietze's graph matches part of the definition of a snark: it is a cubic bridgeless graph that is not 3-edge-colorable. However, most authors restrict snarks to graphs without 3-cycles, so Tietze's graph is not generally considered to be a snark. Nevertheless, it is isomorphic to the graph J3, part of an infinite family of flower snarks introduced by R. Isaacs in 1975.[5]
Unlike the Petersen graph, the Tietze graph can be covered by four perfect matchings. This property plays a key role in a proof that testing whether a graph can be covered by four perfect matchings is NP-complete.[6]
Additional properties
Tietze's graph has chromatic number 3, chromatic index 4, girth 3 and diameter 3. The independence number is 5. Its automorphism group has order 12, and is isomorphic to the dihedral group D6, the group of symmetries of a hexagon, including both rotations and reflections. This group has two orbits of size 3 and one of size 6 on vertices, and thus this graph is not vertex-transitive.
Gallery
• The chromatic number of the Tietze graph is 3.
• The chromatic index of the Tietze graph is 4.
• The Tietze graph has crossing number 2 and is 1-planar.
• A three-dimensional embedding of the Tietze graph.
Wikimedia Commons has media related to Tietze's graph.
See also
• Dürer graph and Franklin graph, two other 12-vertex cubic graphs
Notes
1. Tietze, Heinrich (1910), "Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen" [Some remarks on the problem of map coloring on one-sided surfaces] (PDF), DMV Annual Report, 19: 155–159
2. Clark, L.; Entringer, R. (1983), "Smallest maximally nonhamiltonian graphs", Periodica Mathematica Hungarica, 14 (1): 57–68, doi:10.1007/BF02023582
3. Weisstein, Eric W. "Tietze's Graph". MathWorld.
4. Punnim, Narong; Saenpholphat, Varaporn; Thaithae, Sermsri (2007), "Almost Hamiltonian cubic graphs" (PDF), International Journal of Computer Science and Network Security, 7 (1): 83–86
5. Isaacs, R. (1975), "Infinite families of nontrivial trivalent graphs which are not Tait colorable", Amer. Math. Monthly, Mathematical Association of America, 82 (3): 221–239, doi:10.2307/2319844, JSTOR 2319844.
6. Esperet, L.; Mazzuoccolo, G. (2014), "On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings", Journal of Graph Theory, 77 (2): 144–157, arXiv:1301.6926, doi:10.1002/jgt.21778, MR 3246172.
| Wikipedia |
Tietze transformations
In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group. These transformations are named after Heinrich Franz Friedrich Tietze who introduced them in a paper in 1908.
A presentation is in terms of generators and relations; formally speaking the presentation is a pair of a set of named generators, and a set of words in the free group on the generators that are taken to be the relations. Tietze transformations are built up of elementary steps, each of which individually rather evidently takes the presentation to a presentation of an isomorphic group. These elementary steps may operate on generators or relations, and are of four kinds.
Adding a relation
If a relation can be derived from the existing relations then it may be added to the presentation without changing the group. Let G=〈 x | x3=1 〉 be a finite presentation for the cyclic group of order 3. Multiplying x3=1 on both sides by x3 we get x6 = x3 = 1 so x6 = 1 is derivable from x3=1. Hence G=〈 x | x3=1, x6=1 〉 is another presentation for the same group.
Removing a relation
If a relation in a presentation can be derived from the other relations then it can be removed from the presentation without affecting the group. In G = 〈 x | x3 = 1, x6 = 1 〉 the relation x6 = 1 can be derived from x3 = 1 so it can be safely removed. Note, however, that if x3 = 1 is removed from the presentation the group G = 〈 x | x6 = 1 〉 defines the cyclic group of order 6 and does not define the same group. Care must be taken to show that any relations that are removed are consequences of the other relations.
Adding a generator
Given a presentation it is possible to add a new generator that is expressed as a word in the original generators. Starting with G = 〈 x | x3 = 1 〉 and letting y = x2 the new presentation G = 〈 x,y | x3 = 1, y = x2 〉 defines the same group.
Removing a generator
If a relation can be formed where one of the generators is a word in the other generators then that generator may be removed. In order to do this it is necessary to replace all occurrences of the removed generator with its equivalent word. The presentation for the elementary abelian group of order 4, G=〈 x,y,z | x = yz, y2=1, z2=1, x=x−1 〉 can be replaced by G = 〈 y,z | y2 = 1, z2 = 1, (yz) = (yz)−1 〉 by removing x.
Examples
Let G = 〈 x,y | x3 = 1, y2 = 1, (xy)2 = 1 〉 be a presentation for the symmetric group of degree three. The generator x corresponds to the permutation (1,2,3) and y to (2,3). Through Tietze transformations this presentation can be converted to G = 〈 y, z | (zy)3 = 1, y2 = 1, z2 = 1 〉, where z corresponds to (1,2).
G = 〈 x,y | x3 = 1, y2 = 1, (xy)2 = 1 〉(start)
G = 〈 x,y,z| x3 = 1, y2 = 1, (xy)2 = 1, z = xy 〉rule 3 — Add the generator z
G = 〈 x,y,z | x3 = 1, y2 = 1, (xy)2 = 1, x = zy 〉rules 1 and 2 — Add x = zy−1 = zy and remove z = xy
G = 〈 y,z | (zy)3 = 1, y2 = 1, z2 = 1 〉rule 4 - Remove the generator x
See also
• Nielsen Transformation
• Andrews-Curtis Conjecture
References
• Roger C. Lyndon, Paul E. Schupp, Combinatorial Group Theory, Springer, 2001. ISBN 3-540-41158-5.
| Wikipedia |
Truncated great icosahedron
In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices.[1] It is given a Schläfli symbol t{3,5⁄2} or t0,1{3,5⁄2} as a truncated great icosahedron.
Truncated great icosahedron
TypeUniform star polyhedron
ElementsF = 32, E = 90
V = 60 (χ = 2)
Faces by sides12{5/2}+20{6}
Coxeter diagram
Wythoff symbol2 5/2 | 3
2 5/3 | 3
Symmetry groupIh, [5,3], *532
Index referencesU55, C71, W95
Dual polyhedronGreat stellapentakis dodecahedron
Vertex figure
6.6.5/2
Bowers acronymTiggy
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of
(±1, 0, ±3/τ)
(±2, ±1/τ, ±1/τ3)
(±(1+1/τ2), ±1, ±2/τ)
where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2.
Related polyhedra
This polyhedron is the truncation of the great icosahedron:
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.
Name Great
stellated
dodecahedron
Truncated great stellated dodecahedron Great
icosidodecahedron
Truncated
great
icosahedron
Great
icosahedron
Coxeter-Dynkin
diagram
Picture
Great stellapentakis dodecahedron
Great stellapentakis dodecahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 90
V = 32 (χ = 2)
Symmetry groupIh, [5,3], *532
Index referencesDU55
dual polyhedronTruncated great icosahedron
The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "55: great truncated icosahedron". MathConsult.
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Truncated great icosahedron". MathWorld.
• Weisstein, Eric W. "Great stellapentakis dodecahedron". MathWorld.
• Uniform polyhedra and duals
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
| Wikipedia |
Taut submanifold
In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every $q\in M$ the distance function
$L_{q}:N\to \mathbf {R} ,\qquad L_{q}(x)=\operatorname {dist} (x,q)^{2}$
is a perfect Morse function.
If N is not compact, one needs to consider the restriction of the $L_{q}$ to any of their sublevel sets.
References
• Kuiper, N.H. (2001) [1994], "Tight and taut immersions", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
Tight closure
In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Melvin Hochster and Craig Huneke (1988, 1990).
Let $R$ be a commutative noetherian ring containing a field of characteristic $p>0$. Hence $p$ is a prime number.
Let $I$ be an ideal of $R$. The tight closure of $I$, denoted by $I^{*}$, is another ideal of $R$ containing $I$. The ideal $I^{*}$ is defined as follows.
$z\in I^{*}$ if and only if there exists a $c\in R$, where $c$ is not contained in any minimal prime ideal of $R$, such that $cz^{p^{e}}\in I^{[p^{e}]}$ for all $e\gg 0$. If $R$ is reduced, then one can instead consider all $e>0$.
Here $I^{[p^{e}]}$ is used to denote the ideal of $R$ generated by the $p^{e}$'th powers of elements of $I$, called the $e$th Frobenius power of $I$.
An ideal is called tightly closed if $I=I^{*}$. A ring in which all ideals are tightly closed is called weakly $F$-regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of $F$-regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.
Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly $F$-regular ring is $F$-regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?
References
• Brenner, Holger; Monsky, Paul (2010), "Tight closure does not commute with localization", Annals of Mathematics, Second Series, 171 (1): 571–588, arXiv:0710.2913, doi:10.4007/annals.2010.171.571, ISSN 0003-486X, MR 2630050
• Hochster, Melvin; Huneke, Craig (1988), "Tightly closed ideals", Bulletin of the American Mathematical Society, New Series, 18 (1): 45–48, doi:10.1090/S0273-0979-1988-15592-9, ISSN 0002-9904, MR 0919658
• Hochster, Melvin; Huneke, Craig (1990), "Tight closure, invariant theory, and the Briançon–Skoda theorem", Journal of the American Mathematical Society, 3 (1): 31–116, doi:10.2307/1990984, ISSN 0894-0347, JSTOR 1990984, MR 1017784
| Wikipedia |
Tightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity".
Definitions
Let $(X,T)$ be a Hausdorff space, and let $\Sigma $ be a σ-algebra on $X$ that contains the topology $T$. (Thus, every open subset of $X$ is a measurable set and $\Sigma $ is at least as fine as the Borel σ-algebra on $X$.) Let $M$ be a collection of (possibly signed or complex) measures defined on $\Sigma $. The collection $M$ is called tight (or sometimes uniformly tight) if, for any $\varepsilon >0$, there is a compact subset $K_{\varepsilon }$ of $X$ such that, for all measures $\mu \in M$,
$|\mu |(X\setminus K_{\varepsilon })<\varepsilon .$
where $|\mu |$ is the total variation measure of $\mu $. Very often, the measures in question are probability measures, so the last part can be written as
$\mu (K_{\varepsilon })>1-\varepsilon .\,$
If a tight collection $M$ consists of a single measure $\mu $, then (depending upon the author) $\mu $ may either be said to be a tight measure or to be an inner regular measure.
If $Y$ is an $X$-valued random variable whose probability distribution on $X$ is a tight measure then $Y$ is said to be a separable random variable or a Radon random variable.
Examples
Compact spaces
If $X$ is a metrisable compact space, then every collection of (possibly complex) measures on $X$ is tight. This is not necessarily so for non-metrisable compact spaces. If we take $[0,\omega _{1}]$ with its order topology, then there exists a measure $\mu $ on it that is not inner regular. Therefore, the singleton $\{\mu \}$ is not tight.
Polish spaces
If $X$ is a Polish space, then every probability measure on $X$ is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on $X$ is tight if and only if it is precompact in the topology of weak convergence.
A collection of point masses
Consider the real line $\mathbb {R} $ with its usual Borel topology. Let $\delta _{x}$ denote the Dirac measure, a unit mass at the point $x$ in $\mathbb {R} $. The collection
$M_{1}:=\{\delta _{n}|n\in \mathbb {N} \}$
is not tight, since the compact subsets of $\mathbb {R} $ are precisely the closed and bounded subsets, and any such set, since it is bounded, has $\delta _{n}$-measure zero for large enough $n$. On the other hand, the collection
$M_{2}:=\{\delta _{1/n}|n\in \mathbb {N} \}$
is tight: the compact interval $[0,1]$ will work as $K_{\varepsilon }$ for any $\varepsilon >0$. In general, a collection of Dirac delta measures on $\mathbb {R} ^{n}$ is tight if, and only if, the collection of their supports is bounded.
A collection of Gaussian measures
Consider $n$-dimensional Euclidean space $\mathbb {R} ^{n}$ with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures
$\Gamma =\{\gamma _{i}|i\in I\},$
where the measure $\gamma _{i}$ has expected value (mean) $m_{i}\in \mathbb {R} ^{n}$ and covariance matrix $C_{i}\in \mathbb {R} ^{n\times n}$. Then the collection $\Gamma $ is tight if, and only if, the collections $\{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}$ and $\{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}$ are both bounded.
Tightness and convergence
Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension. See
• Finite-dimensional distribution
• Prokhorov's theorem
• Lévy–Prokhorov metric
• Weak convergence of measures
• Tightness in classical Wiener space
• Tightness in Skorokhod space
Exponential tightness
A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures $(\mu _{\delta })_{\delta >0}$ on a Hausdorff topological space $X$ is said to be exponentially tight if, for any $\varepsilon >0$, there is a compact subset $K_{\varepsilon }$ of $X$ such that
$\limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(X\setminus K_{\varepsilon })<-\varepsilon .$
References
• Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
• Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015 (See chapter 2)
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Beal conjecture
The Beal conjecture is the following conjecture in number theory:
Unsolved problem in mathematics:
If $A^{x}+B^{y}=C^{z}$ where A, B, C, x, y, z are positive integers and x, y, z are ≥ 3, do A, B, and C have a common prime factor?
(more unsolved problems in mathematics)
If
$A^{x}+B^{y}=C^{z},$
where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor.
Equivalently,
The equation $A^{x}+B^{y}=C^{z}$ has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3.
The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.[1][2] Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.[3] The value of the prize has increased several times and is currently $1 million.[4]
In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation,[5] the Mauldin conjecture,[6] and the Tijdeman-Zagier conjecture.[7][8][9]
Related examples
To illustrate, the solution $3^{3}+6^{3}=3^{5}$ has bases with a common factor of 3, the solution $7^{3}+7^{4}=14^{3}$ has bases with a common factor of 7, and $2^{n}+2^{n}=2^{n+1}$ has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively
$3^{3n}+[2(3^{n})]^{3}=3^{3n+2},\quad \quad n\geq 1;$
$[b(a^{n}-b^{n})^{k}]^{n}+(a^{n}-b^{n})^{kn+1}=[a(a^{n}-b^{n})^{k}]^{n},\quad \quad a>b,\quad b\geq 1,\quad k\geq 1,\quad n\geq 3;$
and
$[a(a^{n}+b^{n})^{k}]^{n}+[b(a^{n}+b^{n})^{k}]^{n}=(a^{n}+b^{n})^{kn+1},\quad \quad a\geq 1,\quad b\geq 1,\quad k\geq 1,\quad n\geq 3.$
Furthermore, for each solution (with or without coprime bases), there are infinitely many solutions with the same set of exponents and an increasing set of non-coprime bases. That is, for solution
$A_{1}^{x}+B_{1}^{y}=C_{1}^{z}$
we additionally have
$A_{n}^{x}+B_{n}^{y}=C_{n}^{z};$ $n\geq 2$
where
$A_{n}=(A_{n-1}^{yz+1})(B_{n-1}^{yz})(C_{n-1}^{yz})$
$B_{n}=(A_{n-1}^{xz})(B_{n-1}^{xz+1})(C_{n-1}^{xz})$
$C_{n}=(A_{n-1}^{xy})(B_{n-1}^{xy})(C_{n-1}^{xy+1})$
Any solutions to the Beal conjecture will necessarily involve three terms all of which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers;[10] however, such sums are rare. The smallest two examples are:
${\begin{aligned}271^{3}+2^{3}\ 3^{5}\ 73^{3}=919^{3}&=776{,}151{,}559\\3^{4}\ 29^{3}\ 89^{3}+7^{3}\ 11^{3}\ 167^{3}=2^{7}\ 5^{4}\ 353^{3}&=3{,}518{,}958{,}160{,}000\\\end{aligned}}$
What distinguishes Beal's conjecture is that it requires each of the three terms to be expressible as a single power.
Relation to other conjectures
Fermat's Last Theorem established that $A^{n}+B^{n}=C^{n}$ has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C coprime. Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.
The Fermat–Catalan conjecture is that $A^{x}+B^{y}=C^{z}$ has only finitely many solutions with A, B, and C being positive integers with no common prime factor and x, y, and z being positive integers satisfying ${\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}<1.$ Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent".
The abc conjecture would imply that there are at most finitely many counterexamples to Beal's conjecture.
Partial results
In the cases below where n is an exponent, multiples of n are also proven, since a kn-th power is also an n-th power. Where solutions involving a second power are alluded to below, they can be found specifically at Fermat–Catalan conjecture#Known solutions. All cases of the form (2, 3, n) or (2, n, 3) have the solution 23 + 1n = 32 which is referred below as the Catalan solution.
• The case x = y = z ≥ 3 (and thus the case gcd(x, y, z) ≥ 3) is Fermat's Last Theorem, proven to have no solutions by Andrew Wiles in 1994.[11]
• The case (x, y, z) = (2, 3, 7) and all its permutations were proven to have only four non-Catalan solutions, none of them contradicting Beal conjecture, by Bjorn Poonen, Edward F. Schaefer, and Michael Stoll in 2005.[12]
• The case (x, y, z) = (2, 3, 8) and all its permutations were proven to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.[13]
• The case (x, y, z) = (2, 3, 9) and all its permutations are known to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.[14][15][9]
• The case (x, y, z) = (2, 3, 10) and all its permutations were proven by David Zureick-Brown in 2009 to have only the Catalan solution.[16]
• The case (x, y, z) = (2, 3, 11) and all its permutations were proven by Freitas, Naskręcki and Stoll to have only the Catalan solution.[17]
• The case (x, y, z) = (2, 3, 15) and all its permutations were proven by Samir Siksek and Michael Stoll in 2013.[18]
• The case (x, y, z) = (2, 4, 4) and all its permutations were proven to have no solutions by combined work of Pierre de Fermat in the 1640s and Euler in 1738. (See one proof here and another here)
• The case (x, y, z) = (2, 4, 5) and all its permutations are known to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.[14]
• The case (x, y, z) = (2, 4, n) and all its permutations were proven for n ≥ 6 by Michael Bennett, Jordan Ellenberg, and Nathan Ng in 2009.[19]
• The case (x, y, z) = (2, 6, n) and all its permutations were proven for n ≥ 3 by Michael Bennett and Imin Chen in 2011 and by Bennett, Chen, Dahmen and Yazdani in 2014.[20][5]
• The case (x, y, z) = (2, 2n, 3) and all its permutations were proven for 3 ≤ n ≤ 107 except n = 7 and various modulo congruences when n is prime to have no non-Catalan solution by Bennett, Chen, Dahmen and Yazdani.[21][5]
• The cases (x, y, z) = (2, 2n, 9), (2, 2n, 10), (2, 2n, 15) and all their permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.[5]
• The case (x, y, z) = (3, 3, n) and all its permutations have been proven for 3 ≤ n ≤ 109 and various modulo congruences when n is prime.[15]
• The case (x, y, z) = (3, 4, 5) and all its permutations were proven by Siksek and Stoll in 2011.[22]
• The case (x, y, z) = (3, 5, 5) and all its permutations were proven by Bjorn Poonen in 1998.[23]
• The case (x, y, z) = (3, 6, n) and all its permutations were proven for n ≥ 3 by Bennett, Chen, Dahmen and Yazdani in 2014.[5]
• The case (x, y, z) = (2n, 3, 4) and all its permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.[5]
• The cases (5, 5, 7), (5, 5, 19), (7, 7, 5) and all their permutations were proven by Sander R. Dahmen and Samir Siksek in 2013.[24]
• The cases (x, y, z) = (n, n, 2) and all its permutations were proven for n ≥ 4 by Darmon and Merel in 1995 following work from Euler and Poonen.[25][23]
• The cases (x, y, z) = (n, n, 3) and all its permutations were proven for n ≥ 3 by Édouard Lucas, Bjorn Poonen, and Darmon and Merel.[25]
• The case (x, y, z) = (2n, 2n, 5) and all its permutations were proven for n ≥ 2 by Bennett in 2006.[26]
• The case (x, y, z) = (2l, 2m, n) and all its permutations were proven for l, m ≥ 5 primes and n = 3, 5, 7, 11 by Anni and Siksek.[27]
• The case (x, y, z) = (2l, 2m, 13) and all its permutations were proven for l, m ≥ 5 primes by Billerey, Chen, Dembélé, Dieulefait, Freitas.[28]
• The case (x, y, z) = (3l, 3m, n) is direct for l, m ≥ 2 and n ≥ 3 from work by Kraus.[29]
• The Darmon–Granville theorem uses Faltings's theorem to show that for every specific choice of exponents (x, y, z), there are at most finitely many coprime solutions for (A, B, C).[30][7]: p. 64
• The impossibility of the case A = 1 or B = 1 is implied by Catalan's conjecture, proven in 2002 by Preda Mihăilescu. (Notice C cannot be 1, or one of A and B must be 0, which is not permitted.)
• A potential class of solutions to the equation, namely those with A, B, C also forming a Pythagorean triple, were considered by L. Jesmanowicz in the 1950s. J. Jozefiak proved that there are an infinite number of primitive Pythagorean triples that cannot satisfy the Beal equation. Further results are due to Chao Ko.[31]
• Peter Norvig, Director of Research at Google, reported having conducted a series of numerical searches for counterexamples to Beal's conjecture. Among his results, he excluded all possible solutions having each of x, y, z ≤ 7 and each of A, B, C ≤ 250,000, as well as possible solutions having each of x, y, z ≤ 100 and each of A, B, C ≤ 10,000.[32]
• If A, B are odd and x, y are even, Beal's conjecture has no counterexample.[33]
• By assuming the validity of Beal's conjecture, there exists an upper bound for any common divisor of x, y and z in the expression $ax^{m}+by^{n}=z^{r}$.[34]
Prize
For a published proof or counterexample, banker Andrew Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years,[3] but has since raised it to US $1,000,000.[4]
The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved.[35] It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.[36]
Variants
The counterexamples $7^{3}+13^{2}=2^{9}$ and $1^{m}+2^{3}=3^{2}$ show that the conjecture would be false if one of the exponents were allowed to be 2. The Fermat–Catalan conjecture is an open conjecture dealing with such cases (the condition of this conjecture is that the sum of the reciprocals is less than 1). If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case $1^{m}+2^{3}=3^{2}$).
If A, B, C can have a common prime factor then the conjecture is not true; a classic counterexample is $2^{10}+2^{10}=2^{11}$.
A variation of the conjecture asserting that x, y, z (instead of A, B, C) must have a common prime factor is not true. A counterexample is $27^{4}+162^{3}=9^{7},$ in which 4, 3, and 7 have no common prime factor. (In fact, the maximum common prime factor of the exponents that is valid is 2; a common factor greater than 2 would be a counterexample to Fermat's Last Theorem.)
The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided $(-2+i)^{3}+(-2-i)^{3}=(1+i)^{4}$.[37]
See also
• Euler's sum of powers conjecture
• Jacobi–Madden equation
• Prouhet–Tarry–Escott problem
• Taxicab number
• Pythagorean quadruple
• Sums of powers, a list of related conjectures and theorems
• Distributed computing
• BOINC
References
1. "Beal Conjecture". American Mathematical Society. Retrieved 21 August 2016.
2. "Beal Conjecture". Bealconjecture.com. Retrieved 2014-03-06.
3. R. Daniel Mauldin (1997). "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem" (PDF). Notices of the AMS. 44 (11): 1436–1439.
4. "Beal Prize". Ams.org. Retrieved 2014-03-06.
5. Bennett, Michael A.; Chen, Imin; Dahmen, Sander R.; Yazdani, Soroosh (June 2014). "Generalized Fermat Equations: A Miscellany" (PDF). Simon Fraser University. Retrieved 1 October 2016.
6. "Mauldin / Tijdeman-Zagier Conjecture". Prime Puzzles. Retrieved 1 October 2016.
7. Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1).
8. Michel Waldschmidt (2004). "Open Diophantine Problems". Moscow Mathematical Journal. 4: 245–305. arXiv:math/0312440. doi:10.17323/1609-4514-2004-4-1-245-305. S2CID 11845578.
9. Crandall, Richard; Pomerance, Carl (2000). Prime Numbers: A Computational Perspective. Springer. p. 417. ISBN 978-0387-25282-7.
10. Nitaj, Abderrahmane (1995). "On A Conjecture of Erdos on 3-Powerful Numbers". Bulletin of the London Mathematical Society. 27 (4): 317–318. CiteSeerX 10.1.1.24.563. doi:10.1112/blms/27.4.317.
11. "Billionaire Offers $1 Million to Solve Math Problem | ABC News Blogs – Yahoo". Gma.yahoo.com. 2013-06-06. Retrieved 2014-03-06.
12. Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2005). "Twists of X(7) and primitive solutions to x2 + y3 = z7". Duke Mathematical Journal. 137: 103–158. arXiv:math/0508174. Bibcode:2005math......8174P. doi:10.1215/S0012-7094-07-13714-1. S2CID 2326034.
13. Bruin, Nils (2003-01-09). "Chabauty methods using elliptic curves". Journal für die reine und angewandte Mathematik. 2003 (562). doi:10.1515/crll.2003.076. ISSN 0075-4102.
14. Bruin, Nils (2005-03-01). "The primitive solutions to x^3 + y^9 = z^2". Journal of Number Theory. 111 (1): 179–189. arXiv:math/0311002. doi:10.1016/j.jnt.2004.11.008. ISSN 0022-314X. S2CID 9704470.
15. Frits Beukers (January 20, 2006). "The generalized Fermat equation" (PDF). Staff.science.uu.nl. Retrieved 2014-03-06.
16. Brown, David (2009). "Primitive Integral Solutions to x2 + y3 = z10". arXiv:0911.2932 [math.NT].
17. Freitas, Nuno; Naskręcki, Bartosz; Stoll, Michael (January 2020). "The generalized Fermat equation with exponents 2, 3, n". Compositio Mathematica. 156 (1): 77–113. doi:10.1112/S0010437X19007693. ISSN 0010-437X. S2CID 15030869.
18. Siksek, Samir; Stoll, Michael (2013). "The Generalised Fermat Equation x2 + y3 = z15". Archiv der Mathematik. 102 (5): 411–421. arXiv:1309.4421. doi:10.1007/s00013-014-0639-z. S2CID 14582110.
19. "The Diophantine Equation" (PDF). Math.wisc.edu. Retrieved 2014-03-06.
20. Bennett, Michael A.; Chen, Imin (2012-07-25). "Multi-Frey $\mathbb {Q} $-curves and the Diophantine equation a^2 + b^6 = c^n". Algebra & Number Theory. 6 (4): 707–730. doi:10.2140/ant.2012.6.707. ISSN 1944-7833.
21. Chen, Imin (2007-10-23). "On the equation $s^2+y^{2p} = \alpha^3$". Mathematics of Computation. 77 (262): 1223–1228. doi:10.1090/S0025-5718-07-02083-2. ISSN 0025-5718.
22. Siksek, Samir; Stoll, Michael (2012). "Partial descent on hyperelliptic curves and the generalized Fermat equation x^3 + y^4 + z^5 = 0". Bulletin of the London Mathematical Society. 44 (1): 151–166. arXiv:1103.1979. doi:10.1112/blms/bdr086. ISSN 1469-2120. S2CID 12565749.
23. Poonen, Bjorn (1998). "Some diophantine equations of the form x^n + y^n = z^m". Acta Arithmetica (in Polish). 86 (3): 193–205. doi:10.4064/aa-86-3-193-205. ISSN 0065-1036.
24. Dahmen, Sander R.; Siksek, Samir (2013). "Perfect powers expressible as sums of two fifth or seventh powers". arXiv:1309.4030 [math.NT].
25. H. Darmon and L. Merel. Winding quotients and some variants of Fermat's Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.
26. Bennett, Michael A. (2006). "The equation x^{2n} + y^{2n} = z^5" (PDF). Journal de Théorie des Nombres de Bordeaux. 18 (2): 315–321. doi:10.5802/jtnb.546. ISSN 1246-7405.
27. Anni, Samuele; Siksek, Samir (2016-08-30). "Modular elliptic curves over real abelian fields and the generalized Fermat equation x^{2ℓ} + y^{2m} = z^p". Algebra & Number Theory. 10 (6): 1147–1172. arXiv:1506.02860. doi:10.2140/ant.2016.10.1147. ISSN 1944-7833. S2CID 118935511.
28. Billerey, Nicolas; Chen, Imin; Dembélé, Lassina; Dieulefait, Luis; Freitas, Nuno (2019-03-05). "Some extensions of the modular method and Fermat equations of signature (13, 13, n)". arXiv:1802.04330 [math.NT].
29. Kraus, Alain (1998-01-01). "Sur l'équation a^3 + b^3 = c^p". Experimental Mathematics. 7 (1): 1–13. doi:10.1080/10586458.1998.10504355. ISSN 1058-6458.
30. Darmon, H.; Granville, A. (1995). "On the equations zm = F(x, y) and Axp + Byq = Czr". Bulletin of the London Mathematical Society. 27 (6): 513–43. doi:10.1112/blms/27.6.513.
31. Wacław Sierpiński, Pythagorean Triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).
32. Norvig, Peter. "Beal's Conjecture: A Search for Counterexamples". Norvig.com. Retrieved 2014-03-06.
33. "Sloane's A261782 (see the Theorem and its proof in the comment from May 08 2021)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-06-19.
34. Rahimi, Amir M. (2017). "An Elementary Approach to the Diophantine Equation $ax^{m}+by^{n}=z^{r}$ Using Center of Mass". Missouri J. Math. Sci. 29 (2): 115–124. doi:10.35834/mjms/1513306825.
35. Walter Hickey (5 June 2013). "If You Can Solve This Math Problem, Then A Texas Banker Will Give You $1 Million". Business Insider. Retrieved 8 July 2016.
36. "$1 Million Math Problem: Banker D. Andrew Beal Offers Award To Crack Conjecture Unsolved For 30 Years". International Science Times. 5 June 2013. Archived from the original on 29 September 2017.
37. "Neglected Gaussians". Mathpuzzle.com. Retrieved 2014-03-06.
External links
• The Beal Prize office page
• Bealconjecture.com
• Math.unt.edu
• Beal Conjecture at PlanetMath.
• Mathoverflow.net discussion about the name and date of origin of the theorem
| Wikipedia |
Tijdeman's theorem
In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation
$y^{m}=x^{n}+1,$
for exponents n and m greater than one, is finite.[1][2]
History
The theorem was proven by Dutch number theorist Robert Tijdeman in 1976,[3] making use of Baker's method in transcendental number theory to give an effective upper bound for x,y,m,n. Michel Langevin computed a value of exp exp exp exp 730 for the bound.[1][4][5]
Tijdeman's theorem provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu.[6] Mihăilescu's theorem states that there is only one member to the set of consecutive power pairs, namely 9=8+1.[7]
Generalized Tijdeman problem
That the powers are consecutive is essential to Tijdeman's proof; if we replace the difference of 1 by any other difference k and ask for the number of solutions of
$y^{m}=x^{n}+k$
with n and m greater than one we have an unsolved problem,[8] called the generalized Tijdeman problem. It is conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Subbayya Sivasankaranarayana Pillai (1931), see Catalan's conjecture, stating that the equation $Ay^{m}=Bx^{n}+k$ only has a finite number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture.[9]
References
1. Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, p. 352, ISBN 978-0-857-29531-6
2. Schmidt, Wolfgang M. (1996), Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467 (2nd ed.), Springer-Verlag, p. 207, ISBN 978-3-540-54058-8, Zbl 0754.11020
3. Tijdeman, Robert (1976), "On the equation of Catalan", Acta Arithmetica, 29 (2): 197–209, doi:10.4064/aa-29-2-197-209, Zbl 0286.10013
4. Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 978-0-387-90432-0, Zbl 0456.10006
5. Langevin, Michel (1977), "Quelques applications de nouveaux résultats de Van der Poorten", Séminaire Delange-Pisot-Poitou, 17e Année (1975/76), Théorie des Nombres, 2 (G12), MR 0498426
6. Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved" (PDF), Bulletin of the American Mathematical Society, 41 (1): 43–57, doi:10.1090/S0273-0979-03-00993-5
7. Mihăilescu, Preda (2004), "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", Journal für die reine und angewandte Mathematik, 2004 (572): 167–195, doi:10.1515/crll.2004.048, MR 2076124
8. Shorey, Tarlok N.; Tijdeman, Robert (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. Vol. 87. Cambridge University Press. p. 202. ISBN 978-0-521-26826-4. MR 0891406. Zbl 0606.10011.
9. Narkiewicz (2011), pp. 253–254
| Wikipedia |
Tikhonov's theorem (dynamical systems)
In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics.[1][2] The theorem is named after Andrey Nikolayevich Tikhonov.
Statement
Consider this system of differential equations:
${\begin{aligned}{\frac {d\mathbf {x} }{dt}}&=\mathbf {f} (\mathbf {x} ,\mathbf {z} ,t),\\\mu {\frac {d\mathbf {z} }{dt}}&=\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t).\end{aligned}}$
Taking the limit as $\mu \to 0$, this becomes the "degenerate system":
${\begin{aligned}{\frac {d\mathbf {x} }{dt}}&=\mathbf {f} (\mathbf {x} ,\mathbf {z} ,t),\\\mathbf {z} &=\varphi (\mathbf {x} ,t),\end{aligned}}$
where the second equation is the solution of the algebraic equation
$\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t)=0.$
Note that there may be more than one such function $\varphi $.
Tikhonov's theorem states that as $\mu \to 0,$ the solution of the system of two differential equations above approaches the solution of the degenerate system if $\mathbf {z} =\varphi (\mathbf {x} ,t)$ is a stable root of the "adjoined system"
${\frac {d\mathbf {z} }{dt}}=\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t).$
References
1. Klonowski, Wlodzimierz (1983). "Simplifying Principles for Chemical and Enzyme Reaction Kinetics". Biophysical Chemistry. 18 (2): 73–87. doi:10.1016/0301-4622(83)85001-7. PMID 6626688.
2. Roussel, Marc R. (October 19, 2005). "Singular perturbation theory" (PDF). Lecture Notes.
| Wikipedia |
Substitution tiling
In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
Introduction
A tile substitution is described by a set of prototiles (tile shapes) $T_{1},T_{2},\dots ,T_{m}$, an expanding map $Q$ and a dissection rule showing how to dissect the expanded prototiles $QT_{i}$ to form copies of some prototiles $T_{j}$. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system. The tilings by these marked tiles are necessarily aperiodic.[1][2]
A simple example that produces a periodic tiling has only one prototile, namely a square:
By iterating this tile substitution, larger and larger regions of the plane are covered with a square grid. A more sophisticated example with two prototiles is shown below, with the two steps of blowing up and dissecting merged into one step.
One may intuitively get an idea how this procedure yields a substitution tiling of the entire plane. A mathematically rigorous definition is given below. Substitution tilings are notably useful as ways of defining aperiodic tilings, which are objects of interest in many fields of mathematics, including automata theory, combinatorics, discrete geometry, dynamical systems, group theory, harmonic analysis and number theory, as well as crystallography and chemistry. In particular, the celebrated Penrose tiling is an example of an aperiodic substitution tiling.
History
In 1973 and 1974, Roger Penrose discovered a family of aperiodic tilings, now called Penrose tilings. The first description was given in terms of 'matching rules' treating the prototiles as jigsaw puzzle pieces. The proof that copies of these prototiles can be put together to form a tiling of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings; in particular, he rediscovered Penrose's first example. This work gave an impact to scientists working in crystallography, eventually leading to the discovery of quasicrystals. In turn, the interest in quasicrystals led to the discovery of several well-ordered aperiodic tilings. Many of them can be easily described as substitution tilings.
Mathematical definition
We will consider regions in ${\mathbb {R} }^{d}$ that are well-behaved, in the sense that a region is a nonempty compact subset that is the closure of its interior.
We take a set of regions $\mathbf {P} =\{T_{1},T_{2},\dots ,T_{m}\}$ as prototiles. A placement of a prototile $T_{i}$ is a pair $(T_{i},\varphi )$ where $\varphi $is an isometry of ${\mathbb {R} }^{d}$. The image $\varphi (T_{i})$ is called the placement's region. A tiling T is a set of prototile placements whose regions have pairwise disjoint interiors. We say that the tiling T is a tiling of W where W is the union of the regions of the placements in T.
A tile substitution is often loosely defined in the literature. A precise definition is as follows.[3]
A tile substitution with respect to the prototiles P is a pair $(Q,\sigma )$, where $Q:{\mathbb {R} }^{d}\to {\mathbb {R} }^{d}$ is a linear map, all of whose eigenvalues are larger than one in modulus, together with a substitution rule $\sigma $ that maps each $T_{i}$ to a tiling of $QT_{i}$. The substitution rule $\sigma $ induces a map from any tiling T of a region W to a tiling $\sigma (\mathbf {T} )$ of $Q_{\sigma }(\mathbf {W} )$, defined by
$\sigma (\mathbf {T} )=\bigcup _{(T_{i},\varphi )\in \mathbf {T} }\{(T_{j},Q\circ \varphi \circ Q^{-1}\circ \rho ):(T_{j},\rho )\in \sigma (T_{i})\}.$
Note, that the prototiles can be deduced from the tile substitution. Therefore it is not necessary to include them in the tile substitution $(Q,\sigma )$.[4]
Every tiling of ${\mathbb {R} }^{d}$, where any finite part of it is congruent to a subset of some $\sigma ^{k}(T_{i})$ is called a substitution tiling (for the tile substitution $(Q,\sigma )$).
See also
• Pinwheel tiling
• Photographic mosaic
References
1. C. Goodman-Strauss, Matching Rules and Substitution Tilings, Annals Math., 147 (1998), 181-223.
2. Th. Fernique and N. Ollinger, Combinatorial substitutions and sofic tilings, Journees Automates Cellulaires 2010, J. Kari ed., TUCS Lecture Notes 13 (2010), 100-110.
3. D. Frettlöh, Duality of Model Sets Generated by Substitutions, Romanian Journal of Pure and Applied Math. 50, 2005
4. A. Vince, Digit Tiling of Euclidean Space, in: Directions in Mathematical Quasicrystals, eds: M. Baake, R.V. Moody, AMS, 2000
Further reading
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
External links
• Dirk Frettlöh's and Edmund Harriss's Encyclopedia of Substitution Tilings
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Merge sort
In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the relative order of equal elements is the same in the input and output. Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945.[2] A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948.[3]
Merge sort
An example of merge sort. First, divide the list into the smallest unit (1 element), then compare each element with the adjacent list to sort and merge the two adjacent lists. Finally, all the elements are sorted and merged.
ClassSorting algorithm
Data structureArray
Worst-case performance$O(n\log n)$
Best-case performance$\Omega (n\log n)$ typical, $\Omega (n)$ natural variant
Average performance$\Theta (n\log n)$
Worst-case space complexity$O(n)$ total with $O(n)$ auxiliary, $O(1)$ auxiliary with linked lists[1]
Algorithm
Conceptually, a merge sort works as follows:
1. Divide the unsorted list into n sublists, each containing one element (a list of one element is considered sorted).
2. Repeatedly merge sublists to produce new sorted sublists until there is only one sublist remaining. This will be the sorted list.
Top-down implementation
Example C-like code using indices for top-down merge sort algorithm that recursively splits the list (called runs in this example) into sublists until sublist size is 1, then merges those sublists to produce a sorted list. The copy back step is avoided with alternating the direction of the merge with each level of recursion (except for an initial one-time copy, that can be avoided too). To help understand this, consider an array with two elements. The elements are copied to B[], then merged back to A[]. If there are four elements, when the bottom of the recursion level is reached, single element runs from A[] are merged to B[], and then at the next higher level of recursion, those two-element runs are merged to A[]. This pattern continues with each level of recursion.
// Array A[] has the items to sort; array B[] is a work array.
void TopDownMergeSort(A[], B[], n)
{
CopyArray(A, 0, n, B); // one time copy of A[] to B[]
TopDownSplitMerge(A, 0, n, B); // sort data from B[] into A[]
}
// Split A[] into 2 runs, sort both runs into B[], merge both runs from B[] to A[]
// iBegin is inclusive; iEnd is exclusive (A[iEnd] is not in the set).
void TopDownSplitMerge(B[], iBegin, iEnd, A[])
{
if (iEnd - iBegin <= 1) // if run size == 1
return; // consider it sorted
// split the run longer than 1 item into halves
iMiddle = (iEnd + iBegin) / 2; // iMiddle = mid point
// recursively sort both runs from array A[] into B[]
TopDownSplitMerge(A, iBegin, iMiddle, B); // sort the left run
TopDownSplitMerge(A, iMiddle, iEnd, B); // sort the right run
// merge the resulting runs from array B[] into A[]
TopDownMerge(B, iBegin, iMiddle, iEnd, A);
}
// Left source half is A[ iBegin:iMiddle-1].
// Right source half is A[iMiddle:iEnd-1 ].
// Result is B[ iBegin:iEnd-1 ].
void TopDownMerge(B[], iBegin, iMiddle, iEnd, A[])
{
i = iBegin, j = iMiddle;
// While there are elements in the left or right runs...
for (k = iBegin; k < iEnd; k++) {
// If left run head exists and is <= existing right run head.
if (i < iMiddle && (j >= iEnd || A[i] <= A[j])) {
B[k] = A[i];
i = i + 1;
} else {
B[k] = A[j];
j = j + 1;
}
}
}
void CopyArray(A[], iBegin, iEnd, B[])
{
for (k = iBegin; k < iEnd; k++)
B[k] = A[k];
}
Sorting the entire array is accomplished by TopDownMergeSort(A, B, length(A)).
Bottom-up implementation
Example C-like code using indices for bottom-up merge sort algorithm which treats the list as an array of n sublists (called runs in this example) of size 1, and iteratively merges sub-lists back and forth between two buffers:
// array A[] has the items to sort; array B[] is a work array
void BottomUpMergeSort(A[], B[], n)
{
// Each 1-element run in A is already "sorted".
// Make successively longer sorted runs of length 2, 4, 8, 16... until the whole array is sorted.
for (width = 1; width < n; width = 2 * width)
{
// Array A is full of runs of length width.
for (i = 0; i < n; i = i + 2 * width)
{
// Merge two runs: A[i:i+width-1] and A[i+width:i+2*width-1] to B[]
// or copy A[i:n-1] to B[] ( if (i+width >= n) )
BottomUpMerge(A, i, min(i+width, n), min(i+2*width, n), B);
}
// Now work array B is full of runs of length 2*width.
// Copy array B to array A for the next iteration.
// A more efficient implementation would swap the roles of A and B.
CopyArray(B, A, n);
// Now array A is full of runs of length 2*width.
}
}
// Left run is A[iLeft :iRight-1].
// Right run is A[iRight:iEnd-1 ].
void BottomUpMerge(A[], iLeft, iRight, iEnd, B[])
{
i = iLeft, j = iRight;
// While there are elements in the left or right runs...
for (k = iLeft; k < iEnd; k++) {
// If left run head exists and is <= existing right run head.
if (i < iRight && (j >= iEnd || A[i] <= A[j])) {
B[k] = A[i];
i = i + 1;
} else {
B[k] = A[j];
j = j + 1;
}
}
}
void CopyArray(B[], A[], n)
{
for (i = 0; i < n; i++)
A[i] = B[i];
}
Top-down implementation using lists
Pseudocode for top-down merge sort algorithm which recursively divides the input list into smaller sublists until the sublists are trivially sorted, and then merges the sublists while returning up the call chain.
function merge_sort(list m) is
// Base case. A list of zero or one elements is sorted, by definition.
if length of m ≤ 1 then
return m
// Recursive case. First, divide the list into equal-sized sublists
// consisting of the first half and second half of the list.
// This assumes lists start at index 0.
var left := empty list
var right := empty list
for each x with index i in m do
if i < (length of m)/2 then
add x to left
else
add x to right
// Recursively sort both sublists.
left := merge_sort(left)
right := merge_sort(right)
// Then merge the now-sorted sublists.
return merge(left, right)
In this example, the merge function merges the left and right sublists.
function merge(left, right) is
var result := empty list
while left is not empty and right is not empty do
if first(left) ≤ first(right) then
append first(left) to result
left := rest(left)
else
append first(right) to result
right := rest(right)
// Either left or right may have elements left; consume them.
// (Only one of the following loops will actually be entered.)
while left is not empty do
append first(left) to result
left := rest(left)
while right is not empty do
append first(right) to result
right := rest(right)
return result
Bottom-up implementation using lists
Pseudocode for bottom-up merge sort algorithm which uses a small fixed size array of references to nodes, where array[i] is either a reference to a list of size 2i or nil. node is a reference or pointer to a node. The merge() function would be similar to the one shown in the top-down merge lists example, it merges two already sorted lists, and handles empty lists. In this case, merge() would use node for its input parameters and return value.
function merge_sort(node head) is
// return if empty list
if head = nil then
return nil
var node array[32]; initially all nil
var node result
var node next
var int i
result := head
// merge nodes into array
while result ≠ nil do
next := result.next;
result.next := nil
for (i = 0; (i < 32) && (array[i] ≠ nil); i += 1) do
result := merge(array[i], result)
array[i] := nil
// do not go past end of array
if i = 32 then
i -= 1
array[i] := result
result := next
// merge array into single list
result := nil
for (i = 0; i < 32; i += 1) do
result := merge(array[i], result)
return result
Analysis
In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence relation T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list, and add the n steps taken to merge the resulting two lists).[4] The closed form follows from the master theorem for divide-and-conquer recurrences.
The number of comparisons made by merge sort in the worst case is given by the sorting numbers. These numbers are equal to or slightly smaller than (n ⌈lg n⌉ − 2⌈lg n⌉ + 1), which is between (n lg n − n + 1) and (n lg n + n + O(lg n)).[5] Merge sort's best case takes about half as many iterations as its worst case.[6]
For large n and a randomly ordered input list, merge sort's expected (average) number of comparisons approaches α·n fewer than the worst case, where $\alpha =-1+\sum _{k=0}^{\infty }{\frac {1}{2^{k}+1}}\approx 0.2645.$
In the worst case, merge sort uses approximately 39% fewer comparisons than quicksort does in its average case, and in terms of moves, merge sort's worst case complexity is O(n log n) - the same complexity as quicksort's best case.[6]
Merge sort is more efficient than quicksort for some types of lists if the data to be sorted can only be efficiently accessed sequentially, and is thus popular in languages such as Lisp, where sequentially accessed data structures are very common. Unlike some (efficient) implementations of quicksort, merge sort is a stable sort.
Merge sort's most common implementation does not sort in place;[7] therefore, the memory size of the input must be allocated for the sorted output to be stored in (see below for variations that need only n/2 extra spaces).
Natural merge sort
A natural merge sort is similar to a bottom-up merge sort except that any naturally occurring runs (sorted sequences) in the input are exploited. Both monotonic and bitonic (alternating up/down) runs may be exploited, with lists (or equivalently tapes or files) being convenient data structures (used as FIFO queues or LIFO stacks).[8] In the bottom-up merge sort, the starting point assumes each run is one item long. In practice, random input data will have many short runs that just happen to be sorted. In the typical case, the natural merge sort may not need as many passes because there are fewer runs to merge. In the best case, the input is already sorted (i.e., is one run), so the natural merge sort need only make one pass through the data. In many practical cases, long natural runs are present, and for that reason natural merge sort is exploited as the key component of Timsort. Example:
Start : 3 4 2 1 7 5 8 9 0 6
Select runs : (3 4)(2)(1 7)(5 8 9)(0 6)
Merge : (2 3 4)(1 5 7 8 9)(0 6)
Merge : (1 2 3 4 5 7 8 9)(0 6)
Merge : (0 1 2 3 4 5 6 7 8 9)
Formally, the natural merge sort is said to be Runs-optimal, where ${\mathtt {Runs}}(L)$ is the number of runs in $L$, minus one.
Tournament replacement selection sorts are used to gather the initial runs for external sorting algorithms.
Ping-pong merge sort
Instead of merging two blocks at a time, a ping-pong merge merges four blocks at a time. The four sorted blocks are merged simultaneously to auxiliary space into two sorted blocks, then the two sorted blocks are merged back to main memory. Doing so omits the copy operation and reduces the total number of moves by half. An early public domain implementation of a four-at-once merge was by WikiSort in 2014, the method was later that year described as an optimization for patience sorting and named a ping-pong merge.[9][10] Quadsort implemented the method in 2020 and named it a quad merge.[11]
In-place merge sort
One drawback of merge sort, when implemented on arrays, is its O(n) working memory requirement. Several methods to reduce memory or make merge sort fully in-place have been suggested:
• Kronrod (1969) suggested an alternative version of merge sort that uses constant additional space.
• Katajainen et al. present an algorithm that requires a constant amount of working memory: enough storage space to hold one element of the input array, and additional space to hold O(1) pointers into the input array. They achieve an O(n log n) time bound with small constants, but their algorithm is not stable.[12]
• Several attempts have been made at producing an in-place merge algorithm that can be combined with a standard (top-down or bottom-up) merge sort to produce an in-place merge sort. In this case, the notion of "in-place" can be relaxed to mean "taking logarithmic stack space", because standard merge sort requires that amount of space for its own stack usage. It was shown by Geffert et al. that in-place, stable merging is possible in O(n log n) time using a constant amount of scratch space, but their algorithm is complicated and has high constant factors: merging arrays of length n and m can take 5n + 12m + o(m) moves.[13] This high constant factor and complicated in-place algorithm was made simpler and easier to understand. Bing-Chao Huang and Michael A. Langston[14] presented a straightforward linear time algorithm practical in-place merge to merge a sorted list using fixed amount of additional space. They both have used the work of Kronrod and others. It merges in linear time and constant extra space. The algorithm takes little more average time than standard merge sort algorithms, free to exploit O(n) temporary extra memory cells, by less than a factor of two. Though the algorithm is much faster in a practical way but it is unstable also for some lists. But using similar concepts, they have been able to solve this problem. Other in-place algorithms include SymMerge, which takes O((n + m) log (n + m)) time in total and is stable.[15] Plugging such an algorithm into merge sort increases its complexity to the non-linearithmic, but still quasilinear, O(n (log n)2).
• Many applications of external sorting use a form of merge sorting where the input get split up to a higher number of sublists, ideally to a number for which merging them still makes the currently processed set of pages fit into main memory.
• A modern stable linear and in-place merge variant is block merge sort which creates a section of unique values to use as swap space.
• The space overhead can be reduced to sqrt(n) by using binary searches and rotations.[16] This method is employed by the C++ STL library and quadsort.[11]
• An alternative to reduce the copying into multiple lists is to associate a new field of information with each key (the elements in m are called keys). This field will be used to link the keys and any associated information together in a sorted list (a key and its related information is called a record). Then the merging of the sorted lists proceeds by changing the link values; no records need to be moved at all. A field which contains only a link will generally be smaller than an entire record so less space will also be used. This is a standard sorting technique, not restricted to merge sort.
• A simple way to reduce the space overhead to n/2 is to maintain left and right as a combined structure, copy only the left part of m into temporary space, and to direct the merge routine to place the merged output into m. With this version it is better to allocate the temporary space outside the merge routine, so that only one allocation is needed. The excessive copying mentioned previously is also mitigated, since the last pair of lines before the return result statement (function merge in the pseudo code above) become superfluous.
Use with tape drives
An external merge sort is practical to run using disk or tape drives when the data to be sorted is too large to fit into memory. External sorting explains how merge sort is implemented with disk drives. A typical tape drive sort uses four tape drives. All I/O is sequential (except for rewinds at the end of each pass). A minimal implementation can get by with just two record buffers and a few program variables.
Naming the four tape drives as A, B, C, D, with the original data on A, and using only two record buffers, the algorithm is similar to the bottom-up implementation, using pairs of tape drives instead of arrays in memory. The basic algorithm can be described as follows:
1. Merge pairs of records from A; writing two-record sublists alternately to C and D.
2. Merge two-record sublists from C and D into four-record sublists; writing these alternately to A and B.
3. Merge four-record sublists from A and B into eight-record sublists; writing these alternately to C and D
4. Repeat until you have one list containing all the data, sorted—in log2(n) passes.
Instead of starting with very short runs, usually a hybrid algorithm is used, where the initial pass will read many records into memory, do an internal sort to create a long run, and then distribute those long runs onto the output set. The step avoids many early passes. For example, an internal sort of 1024 records will save nine passes. The internal sort is often large because it has such a benefit. In fact, there are techniques that can make the initial runs longer than the available internal memory. One of them, the Knuth's 'snowplow' (based on a binary min-heap), generates runs twice as long (on average) as a size of memory used.[17]
With some overhead, the above algorithm can be modified to use three tapes. O(n log n) running time can also be achieved using two queues, or a stack and a queue, or three stacks. In the other direction, using k > two tapes (and O(k) items in memory), we can reduce the number of tape operations in O(log k) times by using a k/2-way merge.
A more sophisticated merge sort that optimizes tape (and disk) drive usage is the polyphase merge sort.
Optimizing merge sort
On modern computers, locality of reference can be of paramount importance in software optimization, because multilevel memory hierarchies are used. Cache-aware versions of the merge sort algorithm, whose operations have been specifically chosen to minimize the movement of pages in and out of a machine's memory cache, have been proposed. For example, the tiled merge sort algorithm stops partitioning subarrays when subarrays of size S are reached, where S is the number of data items fitting into a CPU's cache. Each of these subarrays is sorted with an in-place sorting algorithm such as insertion sort, to discourage memory swaps, and normal merge sort is then completed in the standard recursive fashion. This algorithm has demonstrated better performance on machines that benefit from cache optimization. (LaMarca & Ladner 1997)
Parallel merge sort
Merge sort parallelizes well due to the use of the divide-and-conquer method. Several different parallel variants of the algorithm have been developed over the years. Some parallel merge sort algorithms are strongly related to the sequential top-down merge algorithm while others have a different general structure and use the K-way merge method.
Merge sort with parallel recursion
The sequential merge sort procedure can be described in two phases, the divide phase and the merge phase. The first consists of many recursive calls that repeatedly perform the same division process until the subsequences are trivially sorted (containing one or no element). An intuitive approach is the parallelization of those recursive calls.[18] Following pseudocode describes the merge sort with parallel recursion using the fork and join keywords:
// Sort elements lo through hi (exclusive) of array A.
algorithm mergesort(A, lo, hi) is
if lo+1 < hi then // Two or more elements.
mid := ⌊(lo + hi) / 2⌋
fork mergesort(A, lo, mid)
mergesort(A, mid, hi)
join
merge(A, lo, mid, hi)
This algorithm is the trivial modification of the sequential version and does not parallelize well. Therefore, its speedup is not very impressive. It has a span of $\Theta (n)$, which is only an improvement of $\Theta (\log n)$ compared to the sequential version (see Introduction to Algorithms). This is mainly due to the sequential merge method, as it is the bottleneck of the parallel executions.
Merge sort with parallel merging
Better parallelism can be achieved by using a parallel merge algorithm. Cormen et al. present a binary variant that merges two sorted sub-sequences into one sorted output sequence.[18]
In one of the sequences (the longer one if unequal length), the element of the middle index is selected. Its position in the other sequence is determined in such a way that this sequence would remain sorted if this element were inserted at this position. Thus, one knows how many other elements from both sequences are smaller and the position of the selected element in the output sequence can be calculated. For the partial sequences of the smaller and larger elements created in this way, the merge algorithm is again executed in parallel until the base case of the recursion is reached.
The following pseudocode shows the modified parallel merge sort method using the parallel merge algorithm (adopted from Cormen et al.).
/**
* A: Input array
* B: Output array
* lo: lower bound
* hi: upper bound
* off: offset
*/
algorithm parallelMergesort(A, lo, hi, B, off) is
len := hi - lo + 1
if len == 1 then
B[off] := A[lo]
else let T[1..len] be a new array
mid := ⌊(lo + hi) / 2⌋
mid' := mid - lo + 1
fork parallelMergesort(A, lo, mid, T, 1)
parallelMergesort(A, mid + 1, hi, T, mid' + 1)
join
parallelMerge(T, 1, mid', mid' + 1, len, B, off)
In order to analyze a recurrence relation for the worst case span, the recursive calls of parallelMergesort have to be incorporated only once due to their parallel execution, obtaining
$T_{\infty }^{\text{sort}}(n)=T_{\infty }^{\text{sort}}\left({\frac {n}{2}}\right)+T_{\infty }^{\text{merge}}(n)=T_{\infty }^{\text{sort}}\left({\frac {n}{2}}\right)+\Theta \left(\log(n)^{2}\right).$
For detailed information about the complexity of the parallel merge procedure, see Merge algorithm.
The solution of this recurrence is given by
$T_{\infty }^{\text{sort}}=\Theta \left(\log(n)^{3}\right).$
This parallel merge algorithm reaches a parallelism of $ \Theta \left({\frac {n}{(\log n)^{2}}}\right)$, which is much higher than the parallelism of the previous algorithm. Such a sort can perform well in practice when combined with a fast stable sequential sort, such as insertion sort, and a fast sequential merge as a base case for merging small arrays.[19]
Parallel multiway merge sort
It seems arbitrary to restrict the merge sort algorithms to a binary merge method, since there are usually p > 2 processors available. A better approach may be to use a K-way merge method, a generalization of binary merge, in which $k$ sorted sequences are merged. This merge variant is well suited to describe a sorting algorithm on a PRAM.[20][21]
Basic Idea
Given an unsorted sequence of $n$ elements, the goal is to sort the sequence with $p$ available processors. These elements are distributed equally among all processors and sorted locally using a sequential Sorting algorithm. Hence, the sequence consists of sorted sequences $S_{1},...,S_{p}$ of length $ \lceil {\frac {n}{p}}\rceil $. For simplification let $n$ be a multiple of $p$, so that $ \left\vert S_{i}\right\vert ={\frac {n}{p}}$ for $i=1,...,p$.
These sequences will be used to perform a multisequence selection/splitter selection. For $j=1,...,p$, the algorithm determines splitter elements $v_{j}$ with global rank $ k=j{\frac {n}{p}}$. Then the corresponding positions of $v_{1},...,v_{p}$ in each sequence $S_{i}$ are determined with binary search and thus the $S_{i}$ are further partitioned into $p$ subsequences $S_{i,1},...,S_{i,p}$ with $ S_{i,j}:=\{x\in S_{i}|rank(v_{j-1})<rank(x)\leq rank(v_{j})\}$.
Furthermore, the elements of $S_{1,i},...,S_{p,i}$ are assigned to processor $i$, means all elements between rank $ (i-1){\frac {n}{p}}$ and rank $ i{\frac {n}{p}}$, which are distributed over all $S_{i}$. Thus, each processor receives a sequence of sorted sequences. The fact that the rank $k$ of the splitter elements $v_{i}$ was chosen globally, provides two important properties: On the one hand, $k$ was chosen so that each processor can still operate on $ n/p$ elements after assignment. The algorithm is perfectly load-balanced. On the other hand, all elements on processor $i$ are less than or equal to all elements on processor $i+1$. Hence, each processor performs the p-way merge locally and thus obtains a sorted sequence from its sub-sequences. Because of the second property, no further p-way-merge has to be performed, the results only have to be put together in the order of the processor number.
Multi-sequence selection
In its simplest form, given $p$ sorted sequences $S_{1},...,S_{p}$ distributed evenly on $p$ processors and a rank $k$, the task is to find an element $x$ with a global rank $k$ in the union of the sequences. Hence, this can be used to divide each $S_{i}$ in two parts at a splitter index $l_{i}$, where the lower part contains only elements which are smaller than $x$, while the elements bigger than $x$ are located in the upper part.
The presented sequential algorithm returns the indices of the splits in each sequence, e.g. the indices $l_{i}$ in sequences $S_{i}$ such that $S_{i}[l_{i}]$ has a global rank less than $k$ and $\mathrm {rank} \left(S_{i}[l_{i}+1]\right)\geq k$.[22]
algorithm msSelect(S : Array of sorted Sequences [S_1,..,S_p], k : int) is
for i = 1 to p do
(l_i, r_i) = (0, |S_i|-1)
while there exists i: l_i < r_i do
// pick Pivot Element in S_j[l_j], .., S_j[r_j], chose random j uniformly
v := pickPivot(S, l, r)
for i = 1 to p do
m_i = binarySearch(v, S_i[l_i, r_i]) // sequentially
if m_1 + ... + m_p >= k then // m_1+ ... + m_p is the global rank of v
r := m // vector assignment
else
l := m
return l
For the complexity analysis the PRAM model is chosen. If the data is evenly distributed over all $p$, the p-fold execution of the binarySearch method has a running time of ${\mathcal {O}}\left(p\log \left(n/p\right)\right)$. The expected recursion depth is ${\mathcal {O}}\left(\log \left(\textstyle \sum _{i}|S_{i}|\right)\right)={\mathcal {O}}(\log(n))$ as in the ordinary Quickselect. Thus the overall expected running time is ${\mathcal {O}}\left(p\log(n/p)\log(n)\right)$.
Applied on the parallel multiway merge sort, this algorithm has to be invoked in parallel such that all splitter elements of rank $ i{\frac {n}{p}}$ for $i=1,..,p$ are found simultaneously. These splitter elements can then be used to partition each sequence in $p$ parts, with the same total running time of ${\mathcal {O}}\left(p\,\log(n/p)\log(n)\right)$.
Pseudocode
Below, the complete pseudocode of the parallel multiway merge sort algorithm is given. We assume that there is a barrier synchronization before and after the multisequence selection such that every processor can determine the splitting elements and the sequence partition properly.
/**
* d: Unsorted Array of Elements
* n: Number of Elements
* p: Number of Processors
* return Sorted Array
*/
algorithm parallelMultiwayMergesort(d : Array, n : int, p : int) is
o := new Array[0, n] // the output array
for i = 1 to p do in parallel // each processor in parallel
S_i := d[(i-1) * n/p, i * n/p] // Sequence of length n/p
sort(S_i) // sort locally
synch
v_i := msSelect([S_1,...,S_p], i * n/p) // element with global rank i * n/p
synch
(S_i,1, ..., S_i,p) := sequence_partitioning(si, v_1, ..., v_p) // split s_i into subsequences
o[(i-1) * n/p, i * n/p] := kWayMerge(s_1,i, ..., s_p,i) // merge and assign to output array
return o
Analysis
Firstly, each processor sorts the assigned $n/p$ elements locally using a sorting algorithm with complexity ${\mathcal {O}}\left(n/p\;\log(n/p)\right)$. After that, the splitter elements have to be calculated in time ${\mathcal {O}}\left(p\,\log(n/p)\log(n)\right)$. Finally, each group of $p$ splits have to be merged in parallel by each processor with a running time of ${\mathcal {O}}(\log(p)\;n/p)$ using a sequential p-way merge algorithm. Thus, the overall running time is given by
${\mathcal {O}}\left({\frac {n}{p}}\log \left({\frac {n}{p}}\right)+p\log \left({\frac {n}{p}}\right)\log(n)+{\frac {n}{p}}\log(p)\right)$.
Practical adaption and application
The multiway merge sort algorithm is very scalable through its high parallelization capability, which allows the use of many processors. This makes the algorithm a viable candidate for sorting large amounts of data, such as those processed in computer clusters. Also, since in such systems memory is usually not a limiting resource, the disadvantage of space complexity of merge sort is negligible. However, other factors become important in such systems, which are not taken into account when modelling on a PRAM. Here, the following aspects need to be considered: Memory hierarchy, when the data does not fit into the processors cache, or the communication overhead of exchanging data between processors, which could become a bottleneck when the data can no longer be accessed via the shared memory.
Sanders et al. have presented in their paper a bulk synchronous parallel algorithm for multilevel multiway mergesort, which divides $p$ processors into $r$ groups of size $p'$. All processors sort locally first. Unlike single level multiway mergesort, these sequences are then partitioned into $r$ parts and assigned to the appropriate processor groups. These steps are repeated recursively in those groups. This reduces communication and especially avoids problems with many small messages. The hierarchical structure of the underlying real network can be used to define the processor groups (e.g. racks, clusters,...).[21]
Further variants
Merge sort was one of the first sorting algorithms where optimal speed up was achieved, with Richard Cole using a clever subsampling algorithm to ensure O(1) merge.[23] Other sophisticated parallel sorting algorithms can achieve the same or better time bounds with a lower constant. For example, in 1991 David Powers described a parallelized quicksort (and a related radix sort) that can operate in O(log n) time on a CRCW parallel random-access machine (PRAM) with n processors by performing partitioning implicitly.[24] Powers further shows that a pipelined version of Batcher's Bitonic Mergesort at O((log n)2) time on a butterfly sorting network is in practice actually faster than his O(log n) sorts on a PRAM, and he provides detailed discussion of the hidden overheads in comparison, radix and parallel sorting.[25]
Comparison with other sort algorithms
Although heapsort has the same time bounds as merge sort, it requires only Θ(1) auxiliary space instead of merge sort's Θ(n). On typical modern architectures, efficient quicksort implementations generally outperform merge sort for sorting RAM-based arrays. On the other hand, merge sort is a stable sort and is more efficient at handling slow-to-access sequential media. Merge sort is often the best choice for sorting a linked list: in this situation it is relatively easy to implement a merge sort in such a way that it requires only Θ(1) extra space, and the slow random-access performance of a linked list makes some other algorithms (such as quicksort) perform poorly, and others (such as heapsort) completely impossible.
As of Perl 5.8, merge sort is its default sorting algorithm (it was quicksort in previous versions of Perl).[26] In Java, the Arrays.sort() methods use merge sort or a tuned quicksort depending on the datatypes and for implementation efficiency switch to insertion sort when fewer than seven array elements are being sorted.[27] The Linux kernel uses merge sort for its linked lists.[28] Python uses Timsort, another tuned hybrid of merge sort and insertion sort, that has become the standard sort algorithm in Java SE 7 (for arrays of non-primitive types),[29] on the Android platform,[30] and in GNU Octave.[31]
Notes
1. Skiena (2008, p. 122)
2. Knuth (1998, p. 158)
3. Katajainen, Jyrki; Träff, Jesper Larsson (March 1997). "Algorithms and Complexity". Proceedings of the 3rd Italian Conference on Algorithms and Complexity. Italian Conference on Algorithms and Complexity. Lecture Notes in Computer Science. Vol. 1203. Rome. pp. 217–228. CiteSeerX 10.1.1.86.3154. doi:10.1007/3-540-62592-5_74. ISBN 978-3-540-62592-6.
4. Cormen et al. (2009, p. 36)
5. The worst case number given here does not agree with that given in Knuth's Art of Computer Programming, Vol 3. The discrepancy is due to Knuth analyzing a variant implementation of merge sort that is slightly sub-optimal
6. Jayalakshmi, N. (2007). Data structure using C++. Firewall Media. ISBN 978-81-318-0020-1. OCLC 849900742.
7. Cormen et al. (2009, p. 151)
8. Powers, David M. W.; McMahon, Graham B. (1983). "A compendium of interesting prolog programs". DCS Technical Report 8313 (Report). Department of Computer Science, University of New South Wales.
9. "WikiSort. Fast and stable sort algorithm that uses O(1) memory. Public domain". GitHub. 14 Apr 2014.
10. Chandramouli, Badrish; Goldstein, Jonathan (2014). Patience is a Virtue: Revisiting Merge and Sort on Modern Processors (PDF). SIGMOD/PODS.
11. "Quadsort is a branchless stable adaptive merge sort". GitHub. 8 Jun 2022.
12. Katajainen, Pasanen & Teuhola (1996)
13. Geffert, Viliam; Katajainen, Jyrki; Pasanen, Tomi (2000). "Asymptotically efficient in-place merging". Theoretical Computer Science. 237 (1–2): 159–181. doi:10.1016/S0304-3975(98)00162-5.
14. Huang, Bing-Chao; Langston, Michael A. (March 1988). "Practical In-Place Merging". Communications of the ACM. 31 (3): 348–352. doi:10.1145/42392.42403. S2CID 4841909.
15. Kim, Pok-Son; Kutzner, Arne (2004). "Stable Minimum Storage Merging by Symmetric Comparisons". Algorithms – ESA 2004. European Symp. Algorithms. Lecture Notes in Computer Science. Vol. 3221. pp. 714–723. CiteSeerX 10.1.1.102.4612. doi:10.1007/978-3-540-30140-0_63. ISBN 978-3-540-23025-0.
16. Kim, Pok-Son; Kutzner, Arne (1 Sep 2003). "A New Method for Efficient in-Place Merging". Proceedings of the Korean Institute of Intelligent Systems Conference: 392–394.
17. Ferragina, Paolo (2009–2019), "5. Sorting Atomic Items" (PDF), The magic of Algorithms!, p. 5-4, archived (PDF) from the original on 2021-05-12
18. Cormen et al. (2009, pp. 797–805)
19. Victor J. Duvanenko "Parallel Merge Sort" Dr. Dobb's Journal & blog and GitHub repo C++ implementation
20. Peter Sanders; Johannes Singler (2008). "Lecture Parallel algorithms" (PDF). Retrieved 2020-05-02.
21. Axtmann, Michael; Bingmann, Timo; Sanders, Peter; Schulz, Christian (2015). "Practical Massively Parallel Sorting". Proceedings of the 27th ACM symposium on Parallelism in Algorithms and Architectures. pp. 13–23. doi:10.1145/2755573.2755595. ISBN 9781450335881. S2CID 18249978.
22. Peter Sanders (2019). "Lecture Parallel algorithms" (PDF). Retrieved 2020-05-02.
23. Cole, Richard (August 1988). "Parallel merge sort". SIAM J. Comput. 17 (4): 770–785. CiteSeerX 10.1.1.464.7118. doi:10.1137/0217049. S2CID 2416667.
24. Powers, David M. W. (1991). "Parallelized Quicksort and Radixsort with Optimal Speedup". Proceedings of International Conference on Parallel Computing Technologies, Novosibirsk. Archived from the original on 2007-05-25.
25. Powers, David M. W. (January 1995). Parallel Unification: Practical Complexity (PDF). Australasian Computer Architecture Workshop Flinders University.
26. "Sort – Perl 5 version 8.8 documentation". Retrieved 2020-08-23.
27. coleenp (22 Feb 2019). "src/java.base/share/classes/java/util/Arrays.java @ 53904:9c3fe09f69bc". OpenJDK.
28. linux kernel /lib/list_sort.c
29. jjb (29 Jul 2009). "Commit 6804124: Replace "modified mergesort" in java.util.Arrays.sort with timsort". Java Development Kit 7 Hg repo. Archived from the original on 2018-01-26. Retrieved 24 Feb 2011.
30. "Class: java.util.TimSort<T>". Android JDK Documentation. Archived from the original on January 20, 2015. Retrieved 19 Jan 2015.
31. "liboctave/util/oct-sort.cc". Mercurial repository of Octave source code. Lines 23-25 of the initial comment block. Retrieved 18 Feb 2013. Code stolen in large part from Python's, listobject.c, which itself had no license header. However, thanks to Tim Peters for the parts of the code I ripped-off.
References
• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) [1990]. Introduction to Algorithms (3rd ed.). MIT Press and McGraw-Hill. ISBN 0-262-03384-4.
• Katajainen, Jyrki; Pasanen, Tomi; Teuhola, Jukka (1996). "Practical in-place mergesort". Nordic Journal of Computing. 3 (1): 27–40. CiteSeerX 10.1.1.22.8523. ISSN 1236-6064. Archived from the original on 2011-08-07. Retrieved 2009-04-04.. Also Practical In-Place Mergesort. Also
• Knuth, Donald (1998). "Section 5.2.4: Sorting by Merging". Sorting and Searching. The Art of Computer Programming. Vol. 3 (2nd ed.). Addison-Wesley. pp. 158–168. ISBN 0-201-89685-0.
• Kronrod, M. A. (1969). "Optimal ordering algorithm without operational field". Soviet Mathematics - Doklady. 10: 744.
• LaMarca, A.; Ladner, R. E. (1997). "The influence of caches on the performance of sorting". Proc. 8th Ann. ACM-SIAM Symp. On Discrete Algorithms (SODA97): 370–379. CiteSeerX 10.1.1.31.1153.
• Skiena, Steven S. (2008). "4.5: Mergesort: Sorting by Divide-and-Conquer". The Algorithm Design Manual (2nd ed.). Springer. pp. 120–125. ISBN 978-1-84800-069-8.
• Sun Microsystems. "Arrays API (Java SE 6)". Retrieved 2007-11-19.
• Oracle Corp. "Arrays (Java SE 10 & JDK 10)". Retrieved 2018-07-23.
External links
The Wikibook Algorithm implementation has a page on the topic of: Merge sort
• Animated Sorting Algorithms: Merge Sort at the Wayback Machine (archived 6 March 2015) – graphical demonstration
• Open Data Structures - Section 11.1.1 - Merge Sort, Pat Morin
Sorting algorithms
Theory
• Computational complexity theory
• Big O notation
• Total order
• Lists
• Inplacement
• Stability
• Comparison sort
• Adaptive sort
• Sorting network
• Integer sorting
• X + Y sorting
• Transdichotomous model
• Quantum sort
Exchange sorts
• Bubble sort
• Cocktail shaker sort
• Odd–even sort
• Comb sort
• Gnome sort
• Proportion extend sort
• Quicksort
Selection sorts
• Selection sort
• Heapsort
• Smoothsort
• Cartesian tree sort
• Tournament sort
• Cycle sort
• Weak-heap sort
Insertion sorts
• Insertion sort
• Shellsort
• Splaysort
• Tree sort
• Library sort
• Patience sorting
Merge sorts
• Merge sort
• Cascade merge sort
• Oscillating merge sort
• Polyphase merge sort
Distribution sorts
• American flag sort
• Bead sort
• Bucket sort
• Burstsort
• Counting sort
• Interpolation sort
• Pigeonhole sort
• Proxmap sort
• Radix sort
• Flashsort
Concurrent sorts
• Bitonic sorter
• Batcher odd–even mergesort
• Pairwise sorting network
• Samplesort
Hybrid sorts
• Block merge sort
• Kirkpatrick–Reisch sort
• Timsort
• Introsort
• Spreadsort
• Merge-insertion sort
Other
• Topological sorting
• Pre-topological order
• Pancake sorting
• Spaghetti sort
Impractical sorts
• Stooge sort
• Slowsort
• Bogosort
| Wikipedia |
Tilemann Stella
Tilemann Stella (c. 1525-1589) was a German mathematician.[1][2]
Biography
Tilemann was born as Tilemann Stoltz in 1525 in Siegen.[3] He studied at the Latin school in Siegen before attending the Martin Luther University of Halle-Wittenberg, the University of Marburg, and the University of Cologne.[3] He was taught by mathematicians such as Johannes Dryander and Erasmus Reinhold.[1] Stella became a close student of Philipp Melanchthon, who entrusted him with creating cartographic representations for biblical studies. He created maps of the Holy Land, Exodus route, and Germany, with plans for further maps that were never completed.[1]
In 1552, John Albert I, Duke of Mecklenburg started funding his work.[1] He began mapping Mecklenburg and made a celestial globe in 1555.[1] In 1560, he was employed as a court mathematician and librarian in Schwerin.[1] He documented his travels with Duke Johann Albrecht and created further maps in Mecklenburg, including planning and supervising the construction of a canal between Dömitz and Wismar.[1] He also charted Mansfeld and Luxembourg for the Counts of Mansfeld and created maps for Duke Wolfgang von Pfalz-Zweibrücken.[1]
Following Duke Johann Albrecht's death in 1576, Stella's connection with Mecklenburg weakened, and he worked for the courts of Saxony and Brandenburg-Ansbach.[1] Around 1582, he entered the service of Duke John I of Palatinate-Zweibrücken, becoming the head of the court library and planning a canal between the Rhine and Saar. Stella died in 1589, and his estate was eventually sold to the "Bibliotheca Bipontina."[1]
References
1. "Stella, Tillmann - Deutsche Biographie".
2. "Tilemann Stella (1525-1589)".
3. Hangard, Brunnenpfad. "Brunnenpfad Hangard - Premiumwanderweg".
| Wikipedia |
Dividing a square into similar rectangles
Dividing a square into similar rectangles (or, equivalently, tiling a square with similar rectangles) is a problem in mathematics.
Three rectangles
See also: Plastic number
There is only one way (up to rotation and reflection) to divide a square into two similar rectangles.
However, there are three distinct ways of partitioning a square into three similar rectangles:[1][2]
1. The trivial solution given by three congruent rectangles with aspect ratio 3:1.
2. The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2.
3. The solution in which the three rectangles are all of different sizes and where they have aspect ratio ρ2, where ρ is the plastic number.
The fact that a rectangle of aspect ratio ρ2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ2 related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.[3][4]
Generalization to n rectangles
In 2022, the mathematician John Baez brought the problem of generalizing this problem to n rectangles to the attention of the Mathstodon online mathematics community.[5][6]
The problem has two parts: what aspect ratios are possible, and how many different solutions are there for a given n.[7] Frieling and Rinne had previously published a result in 1994 that states that the aspect ratio of rectangles in these dissections must be an algebraic number and that each of its conjugates must have a positive real part.[3] However, their proof was not a constructive proof.
Numerous participants have attacked the problem of finding individual dissections using exhaustive computer search of possible solutions. One approach is to exhaustively enumerate possible coarse-grained placements of rectangles, then convert these to candidate topologies of connected rectangles. Given the topology of a potential solution, the determination of the rectangle's aspect ratio can then trivially be expressed as a set of simultaneous equations, thus either determining the solution exactly, or eliminating it from possibility.[8]
As of March 2023, the following results (sequence A359146 in the OEIS) have been obtained for the number of distinct valid dissections for different values of n:[7][9][10]
n # of dissections
1 1
2 1
3 3
4 11
5 51
6 245
7 1372
8 8522
See also
• Squaring the square
References
1. Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
2. de Spinadel, Vera W.; Antonia, Redondo Buitrago (2009), "Towards van der Laan's plastic number in the plane" (PDF), Journal for Geometry and Graphics, 13 (2): 163–175.
3. Freiling, C.; Rinne, D. (1994), "Tiling a square with similar rectangles", Mathematical Research Letters, 1 (5): 547–558, doi:10.4310/MRL.1994.v1.n5.a3, MR 1295549
4. Laczkovich, M.; Szekeres, G. (1995), "Tilings of the square with similar rectangles", Discrete & Computational Geometry, 13 (3–4): 569–572, doi:10.1007/BF02574063, MR 1318796
5. Baez, John (2022-12-22). "Dividing a Square into Similar Rectangles". golem.ph.utexas.edu. Retrieved 2023-03-09.
6. "John Carlos Baez (@[email protected])". Mathstodon. 2022-12-15. Retrieved 2023-03-09.
7. Roberts, Siobhan (2023-02-07). "The Quest to Find Rectangles in a Square". The New York Times. ISSN 0362-4331. Retrieved 2023-03-09.
8. "cutting squares into similar rectangles using a computer program". ianhenderson.org. Retrieved 2023-03-09.
9. Baez, John Carlos (2023-03-06). "Dividing a Square into 7 Similar Rectangles". Azimuth. Retrieved 2023-03-09.
10. "A359146: Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible". On-Line Encyclopedia of Integer Sequences. Retrieved 2023-03-09.
External links
• Python code for dissection of a square into n similar rectangles via "guillotine cuts" by Rahul Narain
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Tiling with rectangles
A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 side ratio. The tilings with straight polyominoes of shapes such as 1 × 3, 1 × 4 and tilings with polyominoes of shapes such as 2 × 3 fall also into this category.
Congruent rectangles
Some tiling of rectangles include:
Stacked bond
Running bond
Basket weave
Basket weave
Herringbone pattern
Tilings with non-congruent rectangles
The smallest square that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 11 × 11 square, and the tiling uses five rectangles.[1]
The smallest rectangle that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, and the tiling uses five rectangles.[1][2]
See also
• Squaring the square
• Tessellation
• Tiling puzzle
Notes
1. Madachy, Joseph S. (1998). "Solutions to Problems and Conjectures". Journal of Recreational Mathematics. 29 (1): 73. ISSN 0022-412X.
2. Herringbone Tiles on a Bathroom Wall
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Tilla Weinstein
Tilla Weinstein (1934–2002, née Savanuck, also published as Tilla Klotz and Tilla K. Milnor) was an American mathematician known for her mentorship of younger women in mathematics. Her research concerned differential geometry, including conformal structures, harmonic maps, and Lorentz surfaces. She taught for many years at Rutgers University, where she headed the mathematics department in the Douglass Residential College.[1]
Early life and education
Weinstein was born as Tilla Savanuck, in 1934.[1] Her father was a Russian immigrant and lawyer in New York City; her mother was a legal secretary. She began her undergraduate studies in 1951 as an English major at the University of Michigan, in part to get away from her parents' rocky marriage and to live near relatives in Detroit. There, her courses included calculus from Hans Samelson and a course in the foundations of mathematics from Raymond Louis Wilder.[2]
After her first year in Michigan, she became engaged to an English student she knew in New York, and after her second year (in 1953), she married him and returned to New York. Not wishing to repeat her earlier coursework (as she would if she had transferred to the City College of New York), she became a mathematics undergraduate student at New York University. She was the only woman in the program at the time.[2] After her marriage, she used the name Tilla Klotz for her publications, a name she would continue to use until the late 1960s.[1]
At NYU, her calculus instructor, Jean van Heijenoort, noted her ability, encouraged her to participate in the school's team for the William Lowell Putnam Mathematical Competition, and led her to take advanced classes in mathematics, including a class in complex analysis from Lipman Bers.[2] In her first meeting with Bers, she announced her pregnancy, but Bers was supportive and helped her to complete her bachelor's degree "without undue delay",[2] despite opposition from the dean of the school.[3] She completed her Ph.D. in 1959 at NYU. Her dissertation, On G. Bol's Proof of Caratheodory's Conjecture, was supervised by Bers.[4]
Career and later life
Although Bers found a position for Weinstein at the University of California, Berkeley, she was unable to find a matching position for her husband, and declined the offer. With the assistance of Bers, she instead became a faculty member at the University of California, Los Angeles,[2] and earned tenure there in 1966,[1] the first woman to earn tenure in the mathematics department there.[2]
After a "sudden and unexpected" divorce in the late 1960s, she married Princeton mathematician John Milnor (also recently divorced) in 1968,[2] and changed her name on her publications to Tilla K. Milnor, a name she continued to use until 1991.[1] She moved in 1969 to Boston College,[1] in order to be less far from her husband, who was working in Princeton, New Jersey.[2]
In 1970, Weinstein was hired by Rutgers University to become the chair of the mathematics department in the Douglass Residential College. She served as chair from 1970 to 1973, and a second term from 1978 to 1981, after which that department was merged into the main mathematics department at Rutgers.[5]
In 1992, she married Kive Weinstein and changed her name again to Tilla Weinstein. She retired from Rutgers in 2000,[1] and died on January 21, 2002.[6]
Selected publications
• Klotz, Tilla; Sario, Leo (1965), "Existence of complete minimal surfaces of arbitrary connectivity and genus", Proceedings of the National Academy of Sciences, 54 (1): 42–44, Bibcode:1965PNAS...54...42K, doi:10.1073/pnas.54.1.42, MR 0178408, PMC 285793, PMID 16591288
• Milnor, Tilla Klotz (1972), "Efimov's theorem about complete immersed surfaces of negative curvature", Advances in Mathematics, 8 (3): 474–543, doi:10.1016/0001-8708(72)90007-2, MR 0301679
• Milnor, Tilla Klotz (1982), "Characterizing harmonic immersions of surfaces with indefinite metric", Proceedings of the National Academy of Sciences, 79 (6): 2143–2144, Bibcode:1982PNAS...79.2143K, doi:10.1073/pnas.79.6.2143, MR 0648910, PMC 346142, PMID 16593177
• Milnor, Tilla Klotz (1983), "Harmonic maps and classical surface theory in Minkowski 3-space", Transactions of the American Mathematical Society, 280 (1): 161–185, doi:10.2307/1999607, JSTOR 1999607, MR 0712254
• Weinstein, Tilla (1996), An introduction to Lorentz surfaces, Berlin: De Gruyter, doi:10.1515/9783110821635, ISBN 3-11-014333-X[7]
Recognition
The Rutgers mathematics department offers an annual award for outstanding undergraduate achievement, the Tilla Weinstein Award, in her honor.[1]
References
1. Women in Rutgers Mathematics, Rutgers Mathematics Department, retrieved 2019-09-07
2. Murray, Margaret A. M. (2001), Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America, MIT Press, pp. 53, 87–91, 104–105, 186–189, ISBN 9780262632461
3. Hersh, Reuben; John-Steiner, Vera (2010), Loving and Hating Mathematics: Challenging the Myths of Mathematical Life, Princeton University Press, p. 37, ISBN 9781400836116
4. Tilla Weinstein at the Mathematics Genealogy Project
5. Weibel, Charles (1995), A History of Mathematics at Rutgers, Rutgers Mathematics Department, retrieved 2019-09-07
6. "Weinstein, Tilla (née Savanuck)", Paid death notices, The New York Times, January 27, 2002 – via Legacy.com
7. Reviews of An Introduction to Lorentz Surfaces:
• Hertrich-Jeromin, U., zbMATH, Zbl 0881.53001{{citation}}: CS1 maint: untitled periodical (link)
• Ehrlich, Paul E. (1998), Mathematical Reviews, MR 1405166{{citation}}: CS1 maint: untitled periodical (link)
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Tilting theory
In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.
It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis for a fixed root-system — a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone. … For this reason, and because the word 'tilt' inflects easily, we call our functors tilting functors or simply tilts.
Brenner & Butler (1980, p. 103)
Tilting theory was motivated by the introduction of reflection functors by Joseph Bernšteĭn, Israel Gelfand, and V. A. Ponomarev (1973); these functors were used to relate representations of two quivers. These functors were reformulated by Maurice Auslander, María Inés Platzeck, and Idun Reiten (1979), and generalized by Sheila Brenner and Michael C. R. Butler (1980) who introduced tilting functors. Dieter Happel and Claus Michael Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this.
Definitions
Suppose that A is a finite-dimensional unital associative algebra over some field. A finitely-generated right A-module T is called a tilting module if it has the following three properties:
• T has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule.
• Ext1
A
(T,T ) = 0.
• The right A-module A is the kernel of a surjective morphism between finite direct sums of direct summands of T.
Given such a tilting module, we define the endomorphism algebra B = EndA(T ). This is another finite-dimensional algebra, and T is a finitely-generated left B-module. The tilting functors HomA(T,−), Ext1
A
(T,−), −⊗BT and TorB
1
(−,T) relate the category mod-A of finitely-generated right A-modules to the category mod-B of finitely-generated right B-modules.
In practice one often considers hereditary finite-dimensional algebras A because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite-dimensional algebra is called a tilted algebra.
Facts
Suppose A is a finite-dimensional algebra, T is a tilting module over A, and B = EndA(T ). Write F = HomA(T,−), F′ = Ext1
A
(T,−), G = −⊗BT, and G′ = TorB
1
(−,T). F is right adjoint to G and F′ is right adjoint to G′.
Brenner & Butler (1980) showed that tilting functors give equivalences between certain subcategories of mod-A and mod-B. Specifically, if we define the two subcategories ${\mathcal {F}}=\ker(F)$ and ${\mathcal {T}}=\ker(F')$ of A-mod, and the two subcategories ${\mathcal {X}}=\ker(G)$ and ${\mathcal {Y}}=\ker(G')$ of B-mod, then $({\mathcal {T}},{\mathcal {F}})$ is a torsion pair in A-mod (i.e. ${\mathcal {T}}$ and ${\mathcal {F}}$ are maximal subcategories with the property $\operatorname {Hom} ({\mathcal {T}},{\mathcal {F}})=0$; this implies that every M in A-mod admits a natural short exact sequence $0\to U\to M\to V\to 0$ with U in ${\mathcal {T}}$ and V in ${\mathcal {F}}$) and $({\mathcal {X}},{\mathcal {Y}})$ is a torsion pair in B-mod. Further, the restrictions of the functors F and G yield inverse equivalences between ${\mathcal {T}}$ and ${\mathcal {Y}}$, while the restrictions of F′ and G′ yield inverse equivalences between ${\mathcal {F}}$ and ${\mathcal {X}}$. (Note that these equivalences switch the order of the torsion pairs $({\mathcal {T}},{\mathcal {F}})$ and $({\mathcal {X}},{\mathcal {Y}})$.)
Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator; in that case ${\mathcal {T}}=\operatorname {mod} -A$ and ${\mathcal {Y}}=\operatorname {mod} -B$.
If A has finite global dimension, then B also has finite global dimension, and the difference of F and F' induces an isometry between the Grothendieck groups K0(A) and K0(B).
In case A is hereditary (i.e. B is a tilted algebra), the global dimension of B is at most 2, and the torsion pair $({\mathcal {X}},{\mathcal {Y}})$ splits, i.e. every indecomposable object of B-mod is either in ${\mathcal {X}}$ or in ${\mathcal {Y}}$.
Happel (1988) and Cline, Parshall & Scott (1986) showed that in general A and B are derived equivalent (i.e. the derived categories Db(A-mod) and Db(B-mod) are equivalent as triangulated categories).
Generalizations and extensions
A generalized tilting module over the finite-dimensional algebra A is a right A-module T with the following three properties:
• T has finite projective dimension.
• Exti
A
(T,T) = 0 for all i > 0.
• There is an exact sequence $0\to A\to T_{1}\to \dots \to T_{n}\to 0$ where the Ti are finite direct sums of direct summands of T.
These generalized tilting modules also yield derived equivalences between A and B, where B = EndA(T ).
Rickard (1989) extended the results on derived equivalence by proving that two finite-dimensional algebras R and S are derived equivalent if and only if S is the endomorphism algebra of a "tilting complex" over R. Tilting complexes are generalizations of generalized tilting modules. A version of this theorem is valid for arbitrary rings R and S.
Happel, Reiten & Smalø (1996) defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some algebraically closed field k. The endomorphism algebras of these tilting objects are the quasi-tilted algebras, a generalization of tilted algebras. The quasi-tilted algebras over k are precisely the finite-dimensional algebras over k of global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1. Happel (2001) classified the hereditary abelian categories that can appear in the above construction.
Colpi & Fuller (2007) defined tilting objects T in an arbitrary abelian category C; their definition requires that C contain the direct sums of arbitrary (possibly infinite) numbers of copies of T, so this is not a direct generalization of the finite-dimensional situation considered above. Given such a tilting object with endomorphism ring R, they establish tilting functors that provide equivalences between a torsion pair in C and a torsion pair in R-Mod, the category of all R-modules.
From the theory of cluster algebras came the definition of cluster category (from Buan et al. (2006)) and cluster tilted algebra (Buan, Marsh & Reiten (2007)) associated to a hereditary algebra A. A cluster tilted algebra arises from a tilted algebra as a certain semidirect product, and the cluster category of A summarizes all the module categories of cluster tilted algebras arising from A.
References
• Angeleri Hügel, Lidia; Happel, Dieter; Krause, Henning, eds. (2007), Handbook of tilting theory (PDF), London Mathematical Society Lecture Note Series, vol. 332, Cambridge University Press, doi:10.1017/CBO9780511735134, ISBN 978-0-521-68045-5, MR 2385175
• Assem, Ibrahim (1990). "Tilting theory–an introduction" (PDF). In Balcerzyk, Stanisław; Józefiak, Tadeusz; Krempa, Jan; Simson, Daniel; Vogel, Wolfgang (eds.). Topics in algebra, Part 1 (Warsaw, 1988). Banach Center Publications. Vol. 26. Warsaw: PWN. pp. 127–180. doi:10.4064/-26-1-127-180. MR 1171230.
• Auslander, Maurice; Platzeck, María Inés; Reiten, Idun (1979), "Coxeter functors without diagrams", Transactions of the American Mathematical Society, 250: 1–46, doi:10.2307/1998978, ISSN 0002-9947, JSTOR 1998978, MR 0530043
• Bernšteĭn, Iosif N.; Gelfand, Izrail M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem", Russian Mathematical Surveys, 28 (2): 17–32, Bibcode:1973RuMaS..28...17B, CiteSeerX 10.1.1.642.2527, doi:10.1070/RM1973v028n02ABEH001526, ISSN 0042-1316, MR 0393065
• Brenner, Sheila; Butler, Michael C. R. (1980), "Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors", Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Berlin, New York: Springer-Verlag, pp. 103–169, doi:10.1007/BFb0088461, ISBN 978-3-540-10264-9, MR 0607151
• Buan, Aslak; Marsh, Robert; Reineke, Markus; Reiten, Idun; Todorov, Gordana (2006), "Tilting theory and cluster combinatorics", Advances in Mathematics, 204 (2): 572–618, arXiv:math/0402054, doi:10.1016/j.aim.2005.06.003, MR 2249625, S2CID 15318919
• Buan, Aslak; Marsh, Robert; Reiten, Idun (2007), "Cluster-tilted algebras", Transactions of the American Mathematical Society, 359 (1): 323–332, doi:10.1090/s0002-9947-06-03879-7, MR 2247893
• Cline, Edward; Parshall, Brian; Scott, Leonard (1986), "Derived categories and Morita theory", Algebra, 104 (2): 397–409, doi:10.1016/0021-8693(86)90224-3, MR 0866784
• Colpi, Riccardo; Fuller, Kent R. (February 2007), "Tilting Objects in Abelian Categories and Quasitilted Rings" (PDF), Transactions of the American Mathematical Society, 359 (2): 741–765, doi:10.1090/s0002-9947-06-03909-2
• Happel, Dieter; Reiten, Idun; Smalø, Sverre O. (1996), "Tilting in abelian categories and quasitilted algebras", Memoirs of the American Mathematical Society, 575
• Happel, Dieter; Ringel, Claus Michael (1982), "Tilted algebras", Transactions of the American Mathematical Society, 274 (2): 399–443, doi:10.2307/1999116, ISSN 0002-9947, JSTOR 1999116, MR 0675063
• Happel, Dieter (1988), Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Notes Series, vol. 119, Cambridge University Press, doi:10.1017/CBO9780511629228, ISBN 9780521339223
• Happel, Dieter (2001), "A characterization of hereditary categories with tilting object", Invent. Math., 144 (2): 381–398, Bibcode:2001InMat.144..381H, doi:10.1007/s002220100135, S2CID 120437744
• Rickard, Jeremy (1989), "Morita theory for derived categories", Journal of the London Mathematical Society, 39 (2): 436–456, doi:10.1112/jlms/s2-39.3.436
• Unger, L. (2001) [1994], "Tilting theory", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
Tilted large deviation principle
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i.e. integration against an exponential functional. It can be seen as an alternative formulation of Varadhan's lemma.
Statement of the theorem
Let X be a Polish space (i.e., a separable, completely metrizable topological space), and let (με)ε>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let F : X → R be a continuous function that is bounded from above. For each Borel set S ⊆ X, let
$J_{\varepsilon }(S)=\int _{S}e^{F(x)/\varepsilon }\,\mathrm {d} \mu _{\varepsilon }(x)$
and define a new family of probability measures (νε)ε>0 on X by
$\nu _{\varepsilon }(S)={\frac {J_{\varepsilon }(S)}{J_{\varepsilon }(X)}}.$
Then (νε)ε>0 satisfies the large deviation principle on X with rate function IF : X → [0, +∞] given by
$I^{F}(x)=\sup _{y\in X}{\big [}F(y)-I(y){\big ]}-{\big [}F(x)-I(x){\big ]}.$
References
• den Hollander, Frank (2000). Large deviations. Fields Institute Monographs 14. Providence, RI: American Mathematical Society. pp. x+143. ISBN 0-8218-1989-5. MR1739680
| Wikipedia |
Tim Chartier
Timothy P. Chartier (born 1969)[1] is Joseph R. Morton Professor of Mathematics and Computer Science at Davidson College,[2] known for his expertise in sports analytics and bracketology,[3][4][5] for his popular mathematics books, and for the "mime-matics" shows combining mime and mathematics that he and his wife Tanya have staged.[6]
The National Museum of Mathematics announced him as 2022-23 Distinguished Visiting Professor for the Public Dissemination of Mathematics, in June 2021.
Education and career
Chartier majored in applied mathematics at Western Michigan University, graduating in 1993, and stayed at Western Michigan for a master's degree in computational mathematics in 1996.[7] He completed a PhD at the University of Colorado Boulder in 2001, with the dissertation Algebraic Multigrid Based on Element Interpolation (AMGe) and Spectral AMGe supervised by Steve McCormick.[8] He has also studied mime, at the Centre du Silence in Colorado, at the Dell'Arte International School of Physical Theatre in California, and with Marcel Marceau.[9]
After postdoctoral research at the University of Washington, he joined the Davidson College faculty in 2003.[7] As well as his academic work, he is also a frequent consultant on sports analytics for ESPN, NASCAR, the National Basketball Association, and other groups.[10]
Books
Chartier is the author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing (2014),[11] which won the Euler Book Prize in 2020,[12] and of When Life is Linear: From Computer Graphics to Bracketology (2015),[13] which won the Beckenbach Book Prize in 2017.[14]
He is also the author of X Games In Mathematics: Sports Training That Counts! (2020) and the coauthor, with Anne Greenbaum, of Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms (2012).[15]
References
1. Birth year from WorldCat Identities, retrieved 2021-10-16
2. Tim Chartier, Davidson College, retrieved 2021-04-16
3. Drape, Joe (31 March 2014), "Mathematician and Matildas Humbled by Coin Flips", The New York Times, archived from the original on 2014-04-01
4. Strauss, Robert (1 March 2016), "March Math-ness: Tim Chartier shines during the NCAA Division I College Basketball Tournament, AKA March Madness: He's one of the nation's top bracketologists", Coloradan Alumni Magazine, University of Colorado Boulder Alumni Magazine
5. Bennett, Jay (21 March 2019), "The Mathematical Madness Behind a Perfect N.C.A.A. Basketball Bracket", Smithsonian
6. Strauss, Robert (15 April 2015), "Performing Math and Mime, for Fun and Profit", The New York Times, archived from the original on 2021-10-01
7. "Mime-Matics", Stories & News, WMU Alumni Association, retrieved 2021-10-14
8. Tim Chartier at the Mathematics Genealogy Project
9. Publica, Liz (4 December 2020), "Interview With the Mathematical Mime Tim Chartier: Proof that math and performance art add up", Art Publika Magazine
10. "Tim Chartier", Governance, Mathematical Association of America, retrieved 2021-10-15
11. Reviews of Math Bytes: John Tucker Bane , SIGACT News, doi:10.1145/2852040.2852045; Alexander Bogomolny, Cut the knot, ; Adhemar Bultheel, EMS Reviews, ; Brie Finegold, Math Horizons, ; Mark Hunacek, MAA Reviews, ; Martin Jones, Mathematics in School, JSTOR 24767733; Alasdair McAndrew, Australian Mathematical Society Gazette, ; Ilia Nouretdinov, MR3183704; Anne Quinn, The Mathematics Teacher, doi:10.5951/mathteacher.109.3.0236, JSTOR 10.5951/mathteacher.109.3.0236; Robert Schaefer, New York Journal of Books, ; Rachael Skyner, Science, JSTOR 24917414; Vincent Ting, The Mathematical Gazette, doi:10.1017/mag.2017.158
12. "Euler Book Prize", MAA Awards, Mathematical Association of America
13. Reviews of When Life is Linear: Christopher S. Brownell, The Mathematics Teacher, doi:10.5951/mathteacher.109.9.0717, JSTOR 10.5951/mathteacher.109.9.0717 Mark Hunacek, MAA Reviews, ; Dieter Riebesehl, Zbl 1309.15003
14. "Beckenbach Book Prize", MAA Awards, Mathematical Association of America
15. Reviews of Numerical Methods: Octavian Pastravanu, Zbl 1247.65001; William J. Satzer, MAA Reviews,
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Tim Cochran
Thomas "Tim" Daniel Cochran (April 7, 1955 – December 16, 2014) was a professor of mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra.
Tim Cochran
Tim Cochran at Multnomah Falls in 2012
Born(1955-04-07)April 7, 1955
DiedDecember 16, 2014(2014-12-16) (aged 59)
NationalityAmerican
Alma materUniversity of California
Known forCochran–Orr–Teichner (solvable) filtration
Scientific career
FieldsMathematics
InstitutionsRice University
Doctoral advisorRobion Kirby
Doctoral studentsShelly Harvey
Education and career
Tim Cochran was a valedictorian for the Severna Park High School Class of 1973. Later, he was an undergraduate at the Massachusetts Institute of Technology, and received his Ph.D. from the University of California, Berkeley in 1982 (Embedding 4-manifolds in S5).[1] He then returned to MIT as a C.L.E. Moore Postdoctoral Instructor from 1982 to 1984. He was an NSF postdoctoral fellow from 1985 to 1987. Following brief appointments at Berkeley and Northwestern University, he started at Rice University as an associate professor in 1990. He became a full professor at Rice University in 1998. He died unexpectedly, aged 59, on December 16, 2014,[2] while on a year-long sabbatical leave supported by a fellowship from the Simons Foundation.[3]
Research contributions
With his coauthors Kent Orr and Peter Teichner, Cochran defined the solvable filtration of the knot concordance group, whose lower levels encapsulate many classical knot concordance invariants.
Cochran was also responsible for naming the slam-dunk move for surgery diagrams in low-dimensional topology.
Awards and honors
While at Rice, he was named an Outstanding Faculty Associate (1992–93), and received the Faculty Teaching and Mentoring Award from the Rice Graduate Student Association (2014)[4]
He was named a fellow of the American Mathematical Society[5] in 2014, for contributions to low-dimensional topology, specifically knot and link concordance, and for mentoring numerous junior mathematicians.
Selected publications
• Four-manifolds which embed in R6 but not in R5 and Seifert manifolds for fibered knots, Inventiones Mathematicae 77 (1984), 173–184
• Geometric invariants of link cobordism, Commentarii Mathematici Helvetici 60 (1985), 291–311.
• Derivatives of links: Massey products and Milnor's concordance invariants, Memoirs of the American Mathematical Society 84 (1990), no. 427.
• Not all links are concordant to boundary links (with Kent Orr), Annals of Mathematics 138 (1993), 519–554.
• Knot Concordance, Whitney Towers and L2-signatures (with Kent Orr and Peter Teichner), Annals of Mathematics 157 (2003), 433–519.
• Structure in the Classical Knot Concordance Group (with Kent Orr and Peter Teichner), Commentarii Mathematici Helvetici 79 (2004), no. 1, 105–123.
• Noncommutative Knot Theory, Algebraic and Geometric Topology 4 (2004), 347–398.
• Knot Concordance and von Neumann rho invariants (with Peter Teichner), Duke Math. Journal, 137 (2007), no.2, 337–379.
• Homology and Derived Series of Groups II: Dwyer's Theorem, (with Shelly Harvey), Geometry and Topology, 12(2008), 199–232.
• Knot concordance and Higher-order Blanchfield duality (with Shelly Harvey and Constance Leidy), Geometry and Topology, 13 (2009), 1419–1482.
• Primary decomposition and the fractal nature of knot concordance (with Shelly Harvey and Constance Leidy), Math. Annalen, 351 (2011), no. 2, 443–508.
• Counterexamples to Kauffman's conjectures on slice knots (with Christopher Davis), Advances in Mathematics 274 (2015), 263–284.
References
1. Tim Cochran at the Mathematics Genealogy Project
2. "Rice mourns loss of mathematician Tim Cochran". Retrieved December 20, 2014.
3. "2 Rice mathematicians honored". Retrieved December 20, 2014.
4. "GSA honors those who support grad students at Rice". Retrieved December 19, 2014.
5. "List of Fellows of the American Mathematical Society". Retrieved December 18, 2014.
External links
• Tim Cochran's home page.
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Timothy Gowers
Sir William Timothy Gowers, FRS (/ˈɡaʊ.ərz/; born 20 November 1963)[1] is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and director of research at the University of Cambridge and Fellow of Trinity College, Cambridge. In 1998, he received the Fields Medal for research connecting the fields of functional analysis and combinatorics.[3][4][5]
Sir
Timothy Gowers
FRS
Born
William Timothy Gowers
(1963-11-20) 20 November 1963[1]
Marlborough, Wiltshire, England, UK
EducationKing's College School, Cambridge
Eton College
Alma materUniversity of Cambridge (BA, PhD)
Known forFunctional analysis, combinatorics
Awards
• Fields Medal (1998)
• Knight Bachelor (2012)[1]
• Nature's 10 (2012)[2]
• De Morgan Medal (2016)
• Sylvester Medal (2016)
Scientific career
InstitutionsUniversity of Cambridge
University College London
ThesisSymmetric Structures in Banach Spaces (1990)
Doctoral advisorBéla Bollobás[3]
Doctoral studentsDavid Conlon
Ben Green
Tom Sanders[3]
InfluencesNorman Routledge
Websitegowers.wordpress.com
www.dpmms.cam.ac.uk/~wtg10
Education
Gowers attended King's College School, Cambridge, as a choirboy in the King's College choir, and then Eton College[1] as a King's Scholar, where he was taught mathematics by Norman Routledge.[6] In 1981, Gowers won a gold medal at the International Mathematical Olympiad with a perfect score.[7] He completed his PhD, with a dissertation on Symmetric Structures in Banach Spaces[8] at Trinity College, Cambridge in 1990, supervised by Béla Bollobás.[8][3]
Career and research
After his PhD, Gowers was elected to a Junior Research Fellowship at Trinity College. From 1991 until his return to Cambridge in 1995 he was lecturer at University College London. He was elected to the Rouse Ball Professorship at Cambridge in 1998. During 2000–2 he was visiting professor at Princeton University. In May 2020 it was announced[9] that he would be assuming the title chaire de combinatoire at the College de France beginning in October 2020, though he intends[10] to continue to reside in Cambridge and maintain a part-time affiliation at the university, as well as enjoy the privileges of his life fellowship of Trinity College.
Gowers initially worked on Banach spaces. He used combinatorial tools in proving several of Stefan Banach's conjectures in the subject, in particular constructing a Banach space with almost no symmetry, serving as a counterexample to several other conjectures.[11] With Bernard Maurey he resolved the "unconditional basic sequence problem" in 1992, showing that not every infinite-dimensional Banach space has an infinite-dimensional subspace that admits an unconditional Schauder basis.[12]
After this, Gowers turned to combinatorics and combinatorial number theory. In 1997 he proved[13] that the Szemerédi regularity lemma necessarily comes with tower-type bounds.
In 1998, Gowers proved[14] the first effective bounds for Szemerédi's theorem, showing that any subset $A\subset \{1,\dots ,N\}$ free of k-term arithmetic progressions has cardinality $O(N(\log \log N)^{-c_{k}})$ for an appropriate $c_{k}>0$. One of the ingredients in Gowers's argument is a tool now known as the Balog–Szemerédi–Gowers theorem, which has found many further applications. He also introduced the Gowers norms, a tool in arithmetic combinatorics, and provided the basic techniques for analysing them. This work was further developed by Ben Green and Terence Tao, leading to the Green–Tao theorem.
In 2003, Gowers established a regularity lemma for hypergraphs,[15] analogous to the Szemerédi regularity lemma for graphs.
In 2005, he introduced[16] the notion of a quasirandom group.
More recently, Gowers has worked on Ramsey theory in random graphs and random sets with David Conlon, and has turned his attention[17] to other problems such as the P versus NP problem. He has also developed an interest, in joint work with Mohan Ganesalingam,[18] in automated problem solving.
Gowers has an Erdős number of three.[19]
Popularisation work
Gowers has written several works popularising mathematics, including Mathematics: A Very Short Introduction (2002),[20] which describes modern mathematical research for the general reader. He was consulted about the 2005 film Proof, starring Gwyneth Paltrow and Anthony Hopkins. He edited The Princeton Companion to Mathematics (2008), which traces the development of various branches and concepts of modern mathematics.[21] For his work on this book, he won the 2011 Euler Book Prize of the Mathematical Association of America.[22] In May 2020 he was made a professor at the Collège de France, a historic institution dedicated to popularising science.[23]
Blogging
Main article: Polymath Project
After asking on his blog whether "massively collaborative mathematics" was possible,[24] he solicited comments on his blog from people who wanted to try to solve mathematical problems collaboratively.[25] The first problem in what is called the Polymath Project, Polymath1, was to find a new combinatorial proof to the density version of the Hales–Jewett theorem. After seven weeks, Gowers wrote on his blog that the problem was "probably solved".[26]
In 2009, with Olof Sisask and Alex Frolkin, he invited people to post comments to his blog to contribute to a collection of methods of mathematical problem solving.[27] Contributors to this Wikipedia-style project, called Tricki.org, include Terence Tao and Ben Green.[28]
Elsevier boycott
In 2012, Gowers posted to his blog to call for a boycott of the publishing house Elsevier.[29][2] A petition ensued, branded the Cost of Knowledge project, in which researchers commit to stop supporting Elsevier journals. Commenting on the petition in The Guardian, Alok Jha credited Gowers with starting an Academic Spring.[30][31]
In 2016, Gowers started Discrete Analysis to demonstrate that a high-quality mathematics journal could be inexpensively produced outside of the traditional academic publishing industry.[32]
Awards and honours
In 1994, Gowers was an invited speaker at the International Congress of Mathematicians in Zürich where he discussed the theory of infinite-dimensional Banach spaces.[33] In 1996, Gowers received the Prize of the European Mathematical Society, and in 1998 the Fields Medal for research on functional analysis and combinatorics.[34][35] In 1999 he became a Fellow of the Royal Society and a member of the American Philosophical Society in 2010.[36] In 2012 he was knighted by the British monarch for services to mathematics.[37][38] He also sits on the selection committee for the Mathematics award, given under the auspices of the Shaw Prize. He was listed in Nature's 10 people who mattered in 2012.[2]
Personal life
Timothy Gowers was born on November 20, 1963 in Marlborough, Wiltshire, England.[33]
Gowers's father was Patrick Gowers, a composer; his great-grandfather was Sir Ernest Gowers, a British civil servant who was best known for guides to English usage; and his great-great-grandfather was Sir William Gowers, a neurologist. He has two siblings, the writer Rebecca Gowers, and the violinist Katharine Gowers. He has five children[39] and plays jazz piano.[1]
In November 2012, Gowers opted to undergo catheter ablation to treat a sporadic atrial fibrillation, after performing a mathematical risk–benefit analysis to decide whether to have the treatment.[40]
In 1988, Gowers married Emily Thomas, a classicist and Cambridge academic: they divorced in 2007. Together they had three children. In 2008, he married for a second time, to Julie Barrau, a University Lecturer in British Medieval History at the University of Cambridge. They have two children together.[41]
Publications
Selected research articles
• Gowers, W. T.; Maurey, Bernard (6 May 1992). "The unconditional basic sequence problem". arXiv:math/9205204.
• Gowers, W. T. (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. CiteSeerX 10.1.1.145.2961. doi:10.1007/s00039-001-0332-9. S2CID 124324198.
• Gowers, W. T. (2007). "Hypergraph regularity and the multidimensional Szemerédi theorem". Ann. of Math. 166 (3): 897–946. arXiv:0710.3032. Bibcode:2007arXiv0710.3032G. doi:10.4007/annals.2007.166.897. S2CID 56118006.
• Gowers, Timothy, ed. (2008). The Princeton Companion to Mathematics. Princeton University Press. ISBN 978-0-691-11880-2.
Popular mathematics books
• Gowers, Timothy (2002). Mathematics: A Very Short Introduction. Oxford University Press. ISBN 978-0-19-285361-5.[42]
References
1. Anon (2013). "Gowers, Sir (William) Timothy". Who's Who (online Oxford University Press ed.). Oxford: A & C Black. doi:10.1093/ww/9780199540884.013.U17733. (Subscription or UK public library membership required.)
2. Brumfiel, G.; Tollefson, J.; Hand, E.; Baker, M.; Cyranoski, D.; Shen, H.; Van Noorden, R.; Nosengo, N.; et al. (2012). "366 days: Nature's 10". Nature. 492 (7429): 335–343. Bibcode:2012Natur.492..335.. doi:10.1038/492335a. PMID 23257862. S2CID 4418086.
3. Timothy Gowers at the Mathematics Genealogy Project
4. Timothy Gowers's results at International Mathematical Olympiad
5. O'Connor, John J.; Robertson, Edmund F., "Timothy Gowers", MacTutor History of Mathematics Archive, University of St Andrews
6. Dalyell, Tam (29 May 2013). "Norman Arthur Routledge: Inspirational teacher and mathematician". The Independent. Archived from the original on 21 June 2022. Retrieved 15 April 2020.
7. Mohammed Aassila - Olympiades Internationales de Mathématiques. Ed ellipses, Paris 2003 p. 156
8. Gowers, William Timothy (1990). Symmetric structures in Banach spaces. cam.ac.uk (PhD thesis). University of Cambridge. doi:10.17863/CAM.16243. OCLC 556577304. EThOS uk.bl.ethos.333263.
9. "Décret du 18 mai 2020 portant nomination et titularisation (enseignements supérieurs)".
10. "Timothy Gowers twitter feed".
11. 1998 Fields Medalist William Timothy Gowers from the American Mathematical Society
12. Gowers, William Timothy; Maurey, Bernard (1993). "The unconditional basic sequence problem". Journal of the American Mathematical Society. 6 (4): 851–874. arXiv:math/9205204. doi:10.1090/S0894-0347-1993-1201238-0. S2CID 5963081.
13. Gowers, W. Timothy (1997). "A lower bound of tower type for Szemeredi's uniformity lemma". Geometric and Functional Analysis. 7 (2): 322–337. doi:10.1007/PL00001621. MR 1445389. S2CID 115242956.
14. Gowers, W. Timothy (2001). "A new proof of Szemeréi's theorem". Geometric and Functional Analysis. 11 (3): 465–588. CiteSeerX 10.1.1.145.2961. doi:10.1007/s00039-001-0332-9. MR 1844079. S2CID 124324198.
15. Gowers, W. Timothy (2007). "Hypergraph regularity and the multidimensional Szemeredi theorem". Annals of Mathematics. 166 (3): 897–946. arXiv:0710.3032. doi:10.4007/annals.2007.166.897. MR 2373376. S2CID 56118006.
16. Gowers, W.Timothy (2008). "Quasirandom groups". Combinatorics, Probability and Computing. 17 (3): 363–387. arXiv:0710.3877. doi:10.1017/S0963548307008826. MR 2410393. S2CID 45356584.
17. "What I did in my summer holidays". 24 October 2013.
18. Ganesalingam, Mohan; Gowers, W. Timothy (2013). "A fully automatic problem solver with human-style output". arXiv:1309.4501 [cs.AI].
19. "Mathematical Reviews: Collaboration Distance". mathscinet.ams.org. Retrieved 22 March 2018.
20. Gowers, Timothy (2002). Mathematics: A Very Short Introduction. Very Short Introductions. Vol. 66. Oxford: Oxford University Press. ISBN 978-0-19-285361-5. MR 2147526.
21. Roberts, David P. (2009). "Review: The Princeton Companion to Mathematics". Mathematical Association of America. Retrieved 1 July 2020.
22. January 2011 Prizes and Awards, American Mathematical Society, retrieved 1 February 2011.
23. "Sonia Garel et Timothy Gowers nommés professeurs au Collège de France".
24. Gowers, T.; Nielsen, M. (2009). "Massively collaborative mathematics". Nature. 461 (7266): 879–881. Bibcode:2009Natur.461..879G. doi:10.1038/461879a. PMID 19829354. S2CID 205050360.
25. Gowers, Timothy (27 January 2009). Is massively collaborative mathematics possible?. Gowers's Weblog. Retrieved 30 March 2009.
26. Nielsen, Michael (20 March 2009). "The Polymath project: scope of participation". Retrieved 30 March 2009.
27. Gowers, Timothy (16 April 2009). "Tricki now fully live". Retrieved 16 April 2009.
28. Tao, Terence (16 April 2009). "Tricki now live". What's new. Retrieved 16 April 2009.
29. Whitfield, J. (2012). "Elsevier boycott gathers pace:Rebel academics ponder how to break free of commercial publishers". Nature. doi:10.1038/nature.2012.10010. S2CID 153496298.
30. Grant, Bob (7 February 2012). "Occupy Elsevier?". The Scientist. Retrieved 12 February 2012.
31. Jha, Alok (9 April 2012). "Academic spring: how an angry maths blog sparked a scientific revolution". The Guardian.
32. Gowers, Timothy (10 September 2015). "Discrete Analysis — an arXiv overlay journal". Gower's Weblog. Retrieved 2 March 2016.
33. "Timothy Gowers - Biography". Maths History. Retrieved 6 March 2023.
34. Lepowsky, James; Lindenstrauss, Joram; Manin, Yuri I.; Milnor, John (January 1999). "The Mathematical Work of the 1998 Fields Medalists" (PDF). Notices of the AMS. 46 (1): 17–26.
35. Gowers, W. T. (1998). "Fourier analysis and Szemerédi's theorem". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. I. pp. 617–629.
36. "APS Member History". search.amphilsoc.org. Retrieved 19 April 2021.
37. "No. 60173". The London Gazette (Supplement). 16 June 2012. p. 1.
38. "Queens Birthday Honors list" (PDF). Archived from the original (PDF) on 6 September 2012. Retrieved 16 June 2012.
39. "Status update". Gowers's Weblog. Timothy Gowers. 30 November 2010. Retrieved 1 December 2010.
40. Mathematics meets real life, by Tim Gowers, 5 November 2012.
41. "Gowers, Sir (William) Timothy, (born 20 Nov. 1963), Rouse Ball Professor of Mathematics, University of Cambridge, since 1998; Fellow of Trinity College, Cambridge, since 1995; Royal Society 2010 Anniversary Research Professor, since 2010 | WHO'S WHO & WHO WAS WHO". Who's Who 2019. Oxford University Press. 1 December 2018. doi:10.1093/ww/9780199540884.013.U17733. ISBN 978-0-19-954088-4. Retrieved 28 July 2019.
42. Gouvêa, Fernando Q. (23 May 2003). "Review of Mathematics: A Very Short Introduction by Timothy Gowers". MAA Reviews, Mathematical Association of America website.
External links
• 1998 Fields Medalist William Timothy Gowers from the American Mathematical Society
• Video lectures by Timothy Gowers on Computational Complexity and Quantum Computation
• Timothy Gowers – Faces of Mathematics
• BBC News (1998): British academics Tim Gowers and Richard Borcherds win top maths awards
• "Multiplying and dividing by whole numbers: Why it is more difficult than you might think", lecture by Timothy Gowers at Gresham College, 22 May 2007 (available for download as video and audio files)
• Körner, Tom (September 1999). "Interview with Tim Gowers (Cambridge)" (PDF). Newsletter of the European Mathematical Society (33): 8–9.
• "William Timothy Gowers". PlanetMath.
• Listen to Timothy Gowers on The Forum, BBC World Service Radio
• Timothy Gowers's results at International Mathematical Olympiad
• 'What can learned societies do to improve scholarly communication?' Audio recording of Sir Timothy Gower's presentation at State Library of Queensland, 29 June 2017
Fields Medalists
• 1936 Ahlfors
• Douglas
• 1950 Schwartz
• Selberg
• 1954 Kodaira
• Serre
• 1958 Roth
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• 1962 Hörmander
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• 1966 Atiyah
• Cohen
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• 1970 Baker
• Hironaka
• Novikov
• Thompson
• 1974 Bombieri
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• 1978 Deligne
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• 1982 Connes
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• 1986 Donaldson
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• 1990 Drinfeld
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• 1994 Bourgain
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De Morgan Medallists
• Arthur Cayley (1884)
• James Joseph Sylvester (1887)
• Lord Rayleigh (1890)
• Felix Klein (1893)
• S. Roberts (1896)
• William Burnside (1899)
• A. G. Greenhill (1902)
• H. F. Baker (1905)
• J. W. L. Glaisher (1908)
• Horace Lamb (1911)
• J. Larmor (1914)
• W. H. Young (1917)
• E. W. Hobson (1920)
• P. A. MacMahon (1923)
• A. E. H. Love (1926)
• Godfrey Harold Hardy (1929)
• Bertrand Russell (1932)
• E. T. Whittaker (1935)
• J. E. Littlewood (1938)
• Louis Mordell (1941)
• Sydney Chapman (1944)
• George Neville Watson (1947)
• A. S. Besicovitch (1950)
• E. C. Titchmarsh (1953)
• G. I. Taylor (1956)
• W. V. D. Hodge (1959)
• Max Newman (1962)
• Philip Hall (1965)
• Mary Cartwright (1968)
• Kurt Mahler (1971)
• Graham Higman (1974)
• C. Ambrose Rogers (1977)
• Michael Atiyah (1980)
• K. F. Roth (1983)
• J. W. S. Cassels (1986)
• D. G. Kendall (1989)
• Albrecht Fröhlich (1992)
• W. K. Hayman (1995)
• R. A. Rankin (1998)
• J. A. Green (2001)
• Roger Penrose (2004)
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• Keith William Morton (2010)
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Fellows of the Royal Society elected in 1999
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Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 treatment, the theory is based on two postulates:[p 1][1][2]
1. The laws of physics are invariant (identical) in all inertial frames of reference (that is, frames of reference with no acceleration).
2. The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer.
Special relativity
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Origins and significance
Special relativity was described by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".[p 1] Maxwell's equations of electromagnetism appeared to be incompatible with Newtonian mechanics, and the Michelson–Morley null result failed to detect the Earth's motion against the hypothesized luminiferous aether. These led to the development of the Lorentz transformations, which adjust distances and times for moving objects. Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible.[3][4] Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth.
Special relativity has a wide range of consequences that have been experimentally verified.[5] They include the relativity of simultaneity, length contraction, time dilation, the relativistic velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and the Thomas precession.[1][2] It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula $E=mc^{2}$, where $c$ is the speed of light in vacuum.[6][7] It also explains how the phenomena of electricity and magnetism are related.[1][2]
A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into a single continuum known as "spacetime". Events that occur at the same time for one observer can occur at different times for another.
Until several years later when Einstein developed general relativity, which introduced a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity"; "special" really means "special case".[p 2][p 3][p 4][note 1] Some of the work of Albert Einstein in special relativity is built on the earlier work by Hendrik Lorentz and Henri Poincaré. The theory became essentially complete in 1907, with Hermann Minkowski's papers on spacetime.[4]
The theory is "special" in that it only applies in the special case where the spacetime is "flat", that is, where the curvature of spacetime (a consequence of the energy–momentum tensor and representing gravity) is negligible.[8][note 2] To correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate accelerations as well as accelerating frames of reference.[9][10]
Just as Galilean relativity is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, that is, at a sufficiently small scale (e.g., when tidal forces are negligible) and in conditions of free fall. But general relativity incorporates non-Euclidean geometry to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as Minkowski space. As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime.
Galileo Galilei had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light,[11] a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics.[12]
Traditional "two postulates" approach to special relativity
"Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results ... How, then, could such a universal principle be found?"
Albert Einstein: Autobiographical Notes[p 5]
Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:[p 1]
• The principle of relativity – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.[p 1]
• The principle of invariant light speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface).[p 1] That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.
The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.[13][14] In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.
The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.[p 6]
Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.[15] But the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the principle of relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:
Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K′ moving in uniform translation relatively to K.[16]
Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.
Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.[p 7]
Principle of relativity
Reference frames and relative motion
Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a "clock" (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in spacetime. Since the speed of light is constant in relativity irrespective of the reference frame, pulses of light can be used to unambiguously measure distances and refer back to the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.
In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called transformation equations.
Standard configuration
To gain insight into how the spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration.[17]: 107 With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, two Galilean reference frames (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime" or "S dash") belongs to a second observer O′.
• The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.
• Frame S′ moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.
• The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.
Since there is no absolute reference frame in relativity theory, a concept of "moving" does not strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.
Lack of an absolute reference frame
The principle of relativity, which states that physical laws have the same form in each inertial reference frame, dates back to Galileo, and was incorporated into Newtonian physics. But in the late 19th century the existence of electromagnetic waves led some physicists to suggest that the universe was filled with a substance they called "aether", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an absolute reference frame against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether was supposed to be sufficiently elastic to support electromagnetic waves, while those waves could interact with matter, yet offering no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the Michelson–Morley experiment in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist.[18] Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.
Relativity without the second postulate
From the principle of relativity alone without assuming the constancy of the speed of light (i.e., using the isotropy of space and the symmetry implied by the principle of special relativity) it can be shown that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.[p 8][19]
Lorentz invariance as the essential core of special relativity
Alternative approaches to special relativity
Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:
The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events ... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws ...[p 5]
Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.[p 9][p 10]
Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations.[20] Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler[21] and by Callahan.[22] This is also the approach followed by the Wikipedia articles Spacetime and Minkowski diagram.
Lorentz transformation and its inverse
Define an event to have spacetime coordinates (t, x, y, z) in system S and (t′, x′, y′, z′) in a reference frame moving at a velocity v with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:
${\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}$
where
$\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$
is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′, relative to S, is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.
Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:
${\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}$
This shows that the unprimed frame is moving with the velocity −v, as measured in the primed frame.[23]
There is nothing special about the x-axis. The transformation can apply to the y- or z-axis, or indeed in any direction parallel to the motion (which are warped by the γ factor) and perpendicular; see the article Lorentz transformation for details.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x1, t1) and (x′1, t′1), another event has coordinates (x2, t2) and (x′2, t′2), and the differences are defined as
• Eq. 1: $\Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .$
• Eq. 2: $\Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .$
we get
• Eq. 3: $\Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ $ $\Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .$
• Eq. 4: $\Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ $ $\Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .$
If we take differentials instead of taking differences, we get
• Eq. 5: $dx'=\gamma \ (dx-v\,dt)\ ,\ \ $ $dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .$
• Eq. 6: $dx=\gamma \ (dx'+v\,dt')\ ,\ $ $dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .$
Graphical representation of the Lorentz transformation
Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.
Spacetime diagrams (Minkowski diagrams) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.[19]
To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1.[19][24]: 155–199
Fig. 3-1a. Draw the $x$ and $t$ axes of frame S. The $x$ axis is horizontal and the $t$ (actually $ct$) axis is vertical, which is the opposite of the usual convention in kinematics. The $ct$ axis is scaled by a factor of $c$ so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the worldlines of two photons passing through the origin at time $t=0.$ The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, ${\text{A}}$ and ${\text{B}},$ have been plotted on this graph so that their coordinates may be compared in the S and S' frames.
Fig. 3-1b. Draw the $x'$ and $ct'$ axes of frame S'. The $ct'$ axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, $v=c/2.$ Both the $ct'$ and $x'$ axes are tilted from the unprimed axes by an angle $\alpha =\tan ^{-1}(\beta ),$ where $\beta =v/c.$ The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that $t=0$ when $t'=0.$
Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that $(x',ct')$ coordinates of $(0,1)$ in the primed coordinate system transform to $(\beta \gamma ,\gamma )$ in the unprimed coordinate system. Likewise, $(x',ct')$ coordinates of $(1,0)$ in the primed coordinate system transform to $(\gamma ,\beta \gamma )$ in the unprimed system. Draw gridlines parallel with the $ct'$ axis through points $(k\gamma ,k\beta \gamma )$ as measured in the unprimed frame, where $k$ is an integer. Likewise, draw gridlines parallel with the $x'$ axis through $(k\beta \gamma ,k\gamma )$ as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between $ct'$ units equals $ {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}$ times the spacing between $ct$ units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as $\beta \to 1.$
Fig. 3-1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time $t'=0$ still plot as 45° diagonal lines. The primed coordinates of ${\text{A}}$ and ${\text{B}}$ are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space.
While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane, but the frames are actually equivalent.
Consequences derived from the Lorentz transformation
The consequences of special relativity can be derived from the Lorentz transformation equations.[25] These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.
Invariant interval
In Galilean relativity, an object's length ($\Delta r$)[note 3] and the temporal separation between two events ($\Delta t$) are independent invariants, the values of which do not change when observed from different frames of reference.[note 4][note 5]
In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as $\Delta s^{2}$:[note 6]
$\Delta s^{2}\;{\overset {\text{def}}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})$
The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.
The form of $\Delta s^{2},$ being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances.[note 7] The invariance of this interval is a property of the general Lorentz transform (also called the Poincaré transformation), making it an isometry of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is, Lorentz boosts, in the x-direction) with all other translations, reflections, and rotations between any Cartesian inertial frame.[29]: 33–34
In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:
$\Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}$
Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:[19]
${\begin{aligned}c^{2}\Delta t^{2}-\Delta x^{2}&=c^{2}\gamma ^{2}\left(\Delta t'+{\dfrac {v\Delta x'}{c^{2}}}\right)^{2}-\gamma ^{2}\ (\Delta x'+v\Delta t')^{2}\\&=\gamma ^{2}\left(c^{2}\Delta t'^{\,2}+2v\Delta x'\Delta t'+{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\right)-\gamma ^{2}\ (\Delta x'^{\,2}+2v\Delta x'\Delta t'+v^{2}\Delta t'^{\,2})\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}-\gamma ^{2}v^{2}\Delta t'^{\,2}-\gamma ^{2}\Delta x'^{\,2}+\gamma ^{2}{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)-\gamma ^{2}\Delta x'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)\\&=c^{2}\Delta t'^{\,2}-\Delta x'^{\,2}\end{aligned}}$
The value of $\Delta s^{2}$ is hence independent of the frame in which it is measured.
In considering the physical significance of $\Delta s^{2}$, there are three cases to note:[19][30]: 25–39
• Δs2 > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that $|\Delta x/\Delta t|<c,$ and given the Lorentz transformation $\Delta x'=\gamma \ (\Delta x-v\,\Delta t),$ it is evident that there exists a $v$ less than $c$ for which $\Delta x'=0$ (in particular, $v=\Delta x/\Delta t$). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, $\Delta s/c,$ is called the proper time.
• Δs2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that $|\Delta x/\Delta t|>c,$ and given the Lorentz transformation $\Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2}),$ there exists a $v$ less than $c$ for which $\Delta t'=0$ (in particular, $v=c^{2}\Delta t/\Delta x$). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, ${\sqrt {-\Delta s^{2}}},$ is called the proper distance, or proper length. For values of $v$ greater than and less than $c^{2}\Delta t/\Delta x,$ the sign of $\Delta t'$ changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that $v$ could be greater than $c^{2}\Delta t/\Delta x$ would be if $v>c.$
• Δs2 = 0: In this case, the two events are said to be lightlike separated. This implies that $|\Delta x/\Delta t|=c,$ and this relationship is frame independent due to the invariance of $s^{2}.$ From this, we observe that the speed of light is $c$ in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.
Relativity of simultaneity
Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).
From Equation 3 (the forward Lorentz transformation in terms of coordinate differences)
$\Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)$
It is clear that the two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are additionally co-local in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′.
The Sagnac effect can be considered a manifestation of the relativity of simultaneity.[31] Since relativity of simultaneity is a first order effect in $v$,[19] instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber optic gyroscopes, are capable of extreme levels of sensitivity.[p 14]
Time dilation
The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames.
Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, Equation 3 can be used to find:
$\Delta t'=\gamma \,\Delta t$ for events satisfying $\Delta x=0\ .$
This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed muons created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.[32]
Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at the same point.[33]: 149–152
The measurements that we would get if we actually looked at a moving clock would, in general, not at all be the same thing, because the time that would see would be delayed by the finite speed of light, i.e the times that we see would be distorted by the Doppler effect. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.[33]: 149–152
Langevin's light-clock
Paul Langevin, an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass-energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909[34]) which he used to perform a novel derivation of the Lorentz transformation.[35]
A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.[36][37][38][39] (Langevin himself made use of a light-clock oriented parallel to its line of motion.[35])
Consider the scenario illustrated in Fig. 4-3A. Observer A holds a light-clock of length $L$ as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds $t_{A}=2L/c.$ A time interval measured using a single clock which is motionless in a particular reference frame is called a proper time interval.[40]
Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of $v.$ Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same $c$ that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as $\pm {\sqrt {c^{2}-v^{2}}},$ so that the total round-trip time of the pulses is $t_{B}=2L/{\sqrt {c^{2}-v^{2}}}\,$ $=t_{A}/{\sqrt {1-v^{2}/c^{2}}}.$ Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore not a proper time interval because he did not measure it with a single resting clock.[40]
Reciprocal time dilation
In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of $v$ to the left, in which case observer A would perceive B's clock as running slower than her local clock.
There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with both A's and B's measurement of time.[41]
Twin paradox
The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called twin paradox, articulated in its present form by Langevin in 1911.[42] Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older.
This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less. In reality, there is no paradox at all, because in order for the two observers to compare their proper times, the symmetry of the situation must be broken: At least one of the two observers must change their state of motion to match that of the other.[43]
Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in the Twin paradox article. We will examine in the following one such solution to the paradox.
Our basic aim will be to demonstrate that, after the trip, both twins are in perfect agreement about who aged by how much, regardless of their different experiences. Fig 4-4 illustrates a scenario where the traveling twin flies at 0.6 c to and from a star 3 ly distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of $f'=f{\sqrt {(1-\beta )/(1+\beta )}}.$ Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip, she receives signals at the enhanced rate of $f''=f{\sqrt {(1+\beta )/(1-\beta )}}.$ The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox.[33]: 152–159
ItemMeasured by the
stay-at-home
Fig 4-4Measured by
the traveler
Fig 4-4
Total time of trip $T={\frac {2L}{v}}$ 10 yr $T'={\frac {2L}{\gamma v}}$ 8 yr
Total number of pulses sent $fT={\frac {2fL}{v}}$ 10 $fT'={\frac {2fL}{\gamma v}}$ 8
Time when traveler's turnaround is detected $t_{1}={\frac {L}{v}}+{\frac {L}{c}}$ 8 yr $t_{1}'={\frac {L}{\gamma v}}$ 4 yr
Number of pulses received at initial $f'$ rate $f't_{1}$ $={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}$
$={\frac {fL}{v}}(1-\beta ^{2})^{1/2}$
4 $f't_{1}'$ $={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}$
$={\frac {fL}{v}}(1-\beta )$
2
Time for remainder of trip $t_{2}={\frac {L}{v}}-{\frac {L}{c}}$ 2 yr $t_{2}'={\frac {L}{\gamma v}}$ 4 yr
Number of signals received at final $f''$ rate $f''t_{2}$ $={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}$ $={\frac {fL}{v}}(1-\beta ^{2})^{1/2}$ 4 $f''t_{2}'$ $={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}$ $={\frac {fL}{v}}(1+\beta )$ 8
Total number of received pulses ${\frac {2fL}{v}}(1-\beta ^{2})^{1/2}$ $={\frac {2fL}{\gamma v}}$ 8 ${\frac {2fL}{v}}$ 10
Twin's calculation as to how much the other twin should have aged $T'={\frac {2L}{\gamma v}}$ 8 yr $T={\frac {2L}{v}}$ 10 yr
Length contraction
The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0, which can be combined with Equation 4 to find the relation between the lengths Δx and Δx′:
$\Delta x'={\frac {\Delta x}{\gamma }}$ for events satisfying $\Delta t'=0\ .$
This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S).
Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring time intervals between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system.
Lorentz transformation of velocities
Consider two frames S and S′ in standard configuration. A particle in S moves in the x direction with velocity vector $\mathbf {u} .$ What is its velocity $\mathbf {u'} $ in frame S′ ?
We can write
$\mathbf {|u|} =u=dx/dt\,.$
(7)
$\mathbf {|u'|} =u'=dx'/dt'\,.$
(8)
Substituting expressions for $dx'$ and $dt'$ from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed $u$ to $u'$:
$u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-vdt)}{\gamma \left(dt-{\frac {vdx}{c^{2}}}\right)}}={\frac {{\frac {dx}{dt}}-v}{1-\left({\frac {v}{c^{2}}}\right)\left({\frac {dx}{dt}}\right)}}={\frac {u-v}{1-uv/c^{2}}}.$
(9)
The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing $v$ with $-v\ .$
$u={\frac {u'+v}{1+u'v/c^{2}}}.$
(10)
For $\mathbf {u} $ not aligned along the x-axis, we write:[12]: 47–49
$\mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .$
(11)
$\mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .$
(12)
The forward and inverse transformations for this case are:
$u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .$
(13)
$u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .$
(14)
Equation 10 and Equation 14 can be interpreted as giving the resultant $\mathbf {u} $ of the two velocities $\mathbf {v} $ and $\mathbf {u'} ,$ and they replace the formula $\mathbf {u=u'+v} $ which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S′ being aligned with each other (although not necessarily in standard configuration).[12]: 47–49
We note the following points:
• If an object (e.g., a photon) were moving at the speed of light in one frame (i.e., u = ±c or u′ = ±c), then it would also be moving at the speed of light in any other frame, moving at |v| < c.
• The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c.
• If both |u| and |v| (and then also |u′| and |v′|) are small with respect to the speed of light (that is, e.g., |u/c| ≪ 1), then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
• Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.
There is nothing special about the x direction in the standard configuration. The above formalism applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See Velocity-addition formula for details.
Thomas rotation
Figure 4-5. Thomas–Wigner rotation
The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation.
Thomas rotation results from the relativity of simultaneity. In Fig. 4-5a, a rod of length $L$ in its rest frame (i.e., having a proper length of $L$) rises vertically along the y-axis in the ground frame.
In Fig. 4-5b, the same rod is observed from the frame of a rocket moving at speed $v$ to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized in the frame of the rod, relativity of simultaneity causes the observer in the rocket frame to observe (not see) the clock at the right end of the rod as being advanced in time by $Lv/c^{2},$ and the rod is correspondingly observed as tilted.[30]: 98–99
Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the spin of moving particles, where Thomas precession is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope, relating the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.[30]: 169–174
Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".[p 15][30]: 98–99
Causality and prohibition of motion faster than light
In Fig. 4-6, the time interval between the events A (the "cause") and B (the "effect") is 'time-like'; that is, there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect).
The interval AC in the diagram is 'space-like'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. But no frames are accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, paradoxes of causality would result.
For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).[44][p 16] A variety of causal paradoxes could then be constructed.
Figure 4-7. Causality violation by the use of fictitious
"instantaneous communicators"
Consider the spacetime diagrams in Fig. 4-7. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The world lines of A and B are vertical (ct), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (ct′), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.
1. Fig. 4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the $-x'$ axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives earlier than it was sent.
2. Fig. 4-7b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the $+x$ axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality.[45]
It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the $x'$ axis (and the signal from A to B slightly steeper than the $x$ axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the $ct'$ and $x'$ axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only slightly faster than the speed of light will result in causality violation.[46]
Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light in vacuum.
This is not to say that all faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light.[47] For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).[48][49]
Optical effects
Dragging effects
In 1850, Hippolyte Fizeau and Léon Foucault independently established that light travels more slowly in water than in air, thus validating a prediction of Fresnel's wave theory of light and invalidating the corresponding prediction of Newton's corpuscular theory.[50] The speed of light was measured in still water. What would be the speed of light in flowing water?
In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light.
According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed through the medium plus the speed of the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If $u'=c/n$ is the speed of light in still water, and $v$ is the speed of the water, and $u_{\pm }$ is the water-borne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then
$u_{\pm }={\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)\ .$
Fizeau's results, although consistent with Fresnel's earlier hypothesis of partial aether dragging, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since $n$ depends on wavelength, the aether must be capable of sustaining different motions at the same time.[note 8] A variety of theoretical explanations were proposed to explain Fresnel's dragging coefficient, that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies.[51]
From the point of view of special relativity, Fizeau's result is nothing but an approximation to Equation 10, the relativistic formula for composition of velocities.[29]
$u_{\pm }={\frac {u'\pm v}{1\pm u'v/c^{2}}}=$ ${\frac {c/n\pm v}{1\pm v/cn}}\approx $ $c\left({\frac {1}{n}}\pm {\frac {v}{c}}\right)\left(1\mp {\frac {v}{cn}}\right)\approx $ ${\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)$
Relativistic aberration of light
Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the aberration of light. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light.[52] (2) If the source is in motion, the displacement would be the consequence of light-time correction. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver.[53]
The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light,[54] and in 1870, George Airy tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope.[55] A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,[56] but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag.[57]
Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include[29]: 57–60
$\cos \theta '={\frac {\cos \theta +v/c}{1+(v/c)\cos \theta }}$ OR $\sin \theta '={\frac {\sin \theta }{\gamma [1+(v/c)\cos \theta ]}}$ OR $\tan {\frac {\theta '}{2}}=\left({\frac {c-v}{c+v}}\right)^{1/2}\tan {\frac {\theta }{2}}$
Relativistic longitudinal Doppler effect
The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a time dilation term, and that is the treatment described here.[58][59]
Assume the receiver and the source are moving away from each other with a relative speed $v\,$ as measured by an observer on the receiver or the source (The sign convention adopted here is that $v$ is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then
$f_{r}=\left(1-{\frac {v}{c_{s}}}\right)f_{s}$
where $c_{s}$ is the speed of sound.
For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be
$f_{r}=\gamma \left(1-\beta \right)f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.$
where
• $\beta =v/c$ and
• $\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}$ is the Lorentz factor.
An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.[60][19]
Transverse Doppler effect
The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.
Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver.
Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments.[19] In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of $\gamma $. In Fig. 5-3b, the light is redshifted by the same factor.
Measurement versus visual appearance
Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of Doppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer.
Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. But the visual appearance of an object is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.
For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be seen as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation.[p 19][p 20][61][62] A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, although the surface of the sphere and the images on it will appear distorted.[63][64]
Both Fig. 5-4 and Fig. 5-5 illustrate objects moving transversely to the line of sight. In Fig. 5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. At high speeds, the sphere in Fig. 5-5 takes on the appearance of a flattened disk tilted up to 45° from the line of sight. If the objects' motions are not strictly transverse but instead include a longitudinal component, exaggerated distortions in perspective may be seen.[65] This illusion has come to be known as Terrell rotation or the Terrell–Penrose effect.[note 9]
Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various radio galaxies, BL Lac objects, quasars, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel.[66][67][68] In Fig. 5-6, galaxy M87 streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5-4 has been stretched out.[69]
Dynamics
Section Consequences derived from the Lorentz transformation dealt strictly with kinematics, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself.
Equivalence of mass and energy
As an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference.
The energy content of an object at rest with mass m equals mc2. Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.
In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc2.
Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is (E/c, 0, 0, 0): it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes (E/c, Ev/c2, 0, 0). The momentum is equal to the energy multiplied by the velocity divided by c2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c2.
The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these do not talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations.[p 1] The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.[p 21] Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong.[70] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.[71]
Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.[p 22][note 10]
Einstein's 1905 demonstration of E = mc2
In his fourth of his 1905 Annus mirabilis papers,[p 21] Einstein presented a heuristic argument for the equivalence of mass and energy. Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time.
Einstein took as starting assumptions his recently discovered formula for relativistic Doppler shift, the laws of conservation of energy and conservation of momentum, and the relationship between the frequency of light and its energy as implied by Maxwell's equations.
Figure 6-1. Einstein's 1905 derivation of E = mc2
Fig. 6-1 (top). Consider a system of plane waves of light having frequency $f$ traveling in direction $\phi $ relative to the x-axis of reference frame S. The frequency (and hence energy) of the waves as measured in frame S′ that is moving along the x-axis at velocity $v$ is given by the relativistic Doppler shift formula which Einstein had developed in his 1905 paper on special relativity:[p 1]
${\frac {f'}{f}}={\frac {1-{\tfrac {v}{c}}\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}$
Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frame S. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle $\phi $ with respect to the x-axis. Each pulse has energy $L/2$. Because of conservation of momentum, the body remains stationary in S after emission of the two pulses. Let $E_{0}$ be the energy of the body before emission of the two pulses and $E_{1}$ after their emission.
Next, consider the same system observed from frame S′ that is moving along the x-axis at speed $v$ relative to frame S. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let $H_{0}$ be the energy of the body measured in reference frame S′ before emission of the two pulses and $H_{1}$ after their emission. We obtain the following relationships:[p 21]
$E_{0}=E_{1}+{\tfrac {1}{2}}L+{\tfrac {1}{2}}L=E_{1}+L$
${\begin{aligned}H_{0}&=H_{1}+{\tfrac {1}{2}}L{\tfrac {1-{\tfrac {v}{c}}\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}+{\tfrac {1}{2}}L{\tfrac {1+{\tfrac {v}{c}}\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}\\&=H_{1}+{\tfrac {L}{\sqrt {1-v^{2}/c^{2}}}}\end{aligned}}$
From the above equations, we obtain the following:
$\quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$
(6-1)
The two differences of form $H-E$ seen in the above equation have a straightforward physical interpretation. Since $H$ and $E$ are the energies of the arbitrary body in the moving and stationary frames, $H_{0}-E_{0}$ and $H_{1}-E_{1}$ represents the kinetic energies of the bodies before and after the emission of light (except for an additive constant that fixes the zero point of energy and is conventionally set to zero). Hence,
$\quad \quad K_{0}-K_{1}=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$
(6-2)
Taking a Taylor series expansion and neglecting higher order terms, he obtained
$\quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}$
(6-3)
Comparing the above expression with the classical expression for kinetic energy, K.E. = 1/2mv2, Einstein then noted: "If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2."
Rindler has observed that Einstein's heuristic argument suggested merely that energy contributes to mass. In 1905, Einstein's cautious expression of the mass–energy relationship allowed for the possibility that "dormant" mass might exist that would remain behind after all the energy of a body was removed. By 1907, however, Einstein was ready to assert that all inertial mass represented a reserve of energy. "To equate all mass with energy required an act of aesthetic faith, very characteristic of Einstein."[12]: 81–84 Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal.
For a variety of reasons, Einstein's original derivation is currently seldom taught. Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.[72]
Elastic collisions
Examination of the collision products generated by particle accelerators around the world provides scientists evidence of the structure of the subatomic world and the natural laws governing it. Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident particles, requires special relativity.[73]
In Newtonian mechanics, analysis of collisions involves use of the conservation laws for mass, momentum and energy. In relativistic mechanics, mass is not independently conserved, because it has been subsumed into the total relativistic energy. We illustrate the differences that arise between the Newtonian and relativistic treatments of particle collisions by examining the simple case of two perfectly elastic colliding particles of equal mass. (Inelastic collisions are discussed in Spacetime#Conservation laws. Radioactive decay may be considered a sort of time-reversed inelastic collision.[73])
Elastic scattering of charged elementary particles deviates from ideality due to the production of Bremsstrahlung radiation.[74][75]
Newtonian analysis
Fig. 6-2 provides a demonstration of the result, familiar to billiard players, that if a stationary ball is struck elastically by another one of the same mass (assuming no sidespin, or "English"), then after collision, the diverging paths of the two balls will subtend a right angle. (a) In the stationary frame, an incident sphere traveling at 2v strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±v. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ±u. Energy conservation requires that |u| = |v|. (c) Reverting to the stationary frame, the rebound velocities are v ± u. The dot product (v + u) • (v − u) = v2 − u2 = 0, indicating that the vectors are orthogonal.[12]: 26–27
Relativistic analysis
Consider the elastic collision scenario in Fig. 6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and smaller as the velocity of the incident particle approaches the speed of light:
The relativistic momentum and total relativistic energy of a particle are given by
$\quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}$
(6-4)
Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles:
$\quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}$
(6-5)
Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc2) equals the sum of the total energies of the emergent particles:
$\quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}$
(6-6)
Breaking down (6-5) into its components, replacing $v$ with the dimensionless $\beta $, and factoring out common terms from (6-5) and (6-6) yields the following:[p 23]
$\quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }$
(6-7)
$\quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }$
(6-8)
$\quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}$
(6-9)
From these we obtain the following relationships:[p 23]
$\quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}$
(6-10)
$\quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}$
(6-11)
$\quad \quad \cos {(\phi +\theta )}={\frac {(\gamma _{1}-1)\sin {\theta }\cos {\theta }}{\{(\gamma _{1}+1)^{2}\sin ^{2}\theta +4\cos ^{2}\theta \}^{1/2}}}$
(6-12)
For the symmetrical case in which $\phi =\theta $ and $\beta _{2}=\beta _{3},$ (6-12) takes on the simpler form:[p 23]
$\quad \quad \cos {\theta }={\frac {\beta _{1}}{\{2/\gamma _{1}+3\beta _{1}^{2}-2\}^{1/2}}}$
(6-13)
How far can you travel from the Earth?
Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems which exist within the limit of 100 light years from Earth. However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant 1g, it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by:
$v(t)={\frac {at}{\sqrt {1+{\frac {a^{2}t^{2}}{c^{2}}}}}}$
where v(t) is the velocity at a time t, a is the acceleration of the spaceship and t is the coordinate time as measured by people on Earth.[p 24] Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at v = 0.712c and 0.946c after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth. During their journey, people on Earth will experience more time than they do - since their clocks (all physical phenomena) would really be ticking 3.1 times faster than those of the spaceship. A 5-year round trip for the traveller will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for them (5 years accelerating, 5 decelerating, twice each) will land them back on Earth having travelled for 335 Earth years and a distance of 331 light years.[76] A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1g will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy.[76] This same time dilation is why a muon travelling close to c is observed to travel much farther than c times its half-life (when at rest).[77]
Relativity and unifying electromagnetism
Main articles: Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism
Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant form, that is, in the language of tensor calculus.[78]
Theories of relativity and quantum mechanics
Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. How general relativity and quantum mechanics can be unified is one of the unsolved problems in physics; quantum gravity and a "theory of everything", which require a unification including general relativity too, are active and ongoing areas in theoretical research.
The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics of the time.[79]
In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour,[p 25] that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only described the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),[p 25][p 26] and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic quantum mechanics.
On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles can be created and destroyed throughout space and time.
Status
Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c2 in the region of interest.[80] In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)[81] and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.
Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).
Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See classical mechanics for a more detailed discussion.
Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905,[82] and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.[14]
• The Fizeau experiment (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.
• The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.
• The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
• The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.
Particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples:
• Tests of relativistic energy and momentum – testing the limiting speed of particles
• Ives–Stilwell experiment – testing relativistic Doppler effect and time dilation
• Experimental testing of time dilation – relativistic effects on a fast-moving particle's half-life
• Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations
• Hughes–Drever experiment – testing isotropy of space and mass
• Modern searches for Lorentz violation – various modern tests
• Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
• Experiments to test the aether drag hypothesis – no "aether flow obstruction".
Technical discussion of spacetime
Main article: Minkowski space
Comparison between flat Euclidean space and Minkowski space
See also: line element
Special relativity uses a "flat" 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.
In 3D space, the differential of distance (line element) ds is defined by
$ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},$
where dx = (dx1, dx2, dx3) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X0 derived from time, such that the distance differential fulfills
$ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},$
where dX = (dX0, dX1, dX2, dX3) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1).[84] Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski spacetime.
The actual form of ds above depends on the metric and on the choices for the X0 coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: X0 = ict (this is called a Wick rotation). According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X0 = ct, rather than a "disguised" Euclidean metric using ict as the time coordinate.
Some authors use X0 = t, with factors of c elsewhere to compensate; for instance, spatial coordinates are divided by c or factors of c±2 are included in the metric tensor.[85] These numerous conventions can be superseded by using natural units where c = 1. Then space and time have equivalent units, and no factors of c appear anywhere.
3D spacetime
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space
$ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},$
we see that the null geodesics lie along a dual-cone (see Fig. 10-2) defined by the equation;
$ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}$
or simply
$dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},$
which is the equation of a circle of radius c dt.
4D spacetime
If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
$ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}$
so
$dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}.$
As illustrated in Fig. 10-3, the null geodesics can be visualized as a set of continuous concentric spheres with radii = c dt.
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance $ d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}$ away and a time d/c in the past. For this reason the null dual cone is also known as the "light cone". (The point in the lower left of the Fig. 10-2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the −t region is the information that the point is "receiving", while the cone in the +t section is the information that the point is "sending".
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought experiments in special relativity.
Transformations of physical quantities between reference frames
Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.
The Lorentz transformation in standard configuration above, that is, for a boost in the x-direction, can be recast into matrix form as follows:
${\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta x\\\gamma x-\beta \gamma ct\\y\\z\end{pmatrix}}.$
In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "four-vectors", in Minkowski spacetime. The components of vectors are written using tensor index notation, as this has numerous advantages. The notation makes it clear the equations are manifestly covariant under the Poincaré group, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other physical quantities as tensors simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used.
The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component x = (x, y, z), in a contravariant position four-vector with components:
$X^{\nu }=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z)=(ct,\mathbf {x} ).$
where we define X0 = ct so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.[86][87][88] Now the transformation of the contravariant components of the position 4-vector can be compactly written as:
$X^{\mu '}=\Lambda ^{\mu '}{}_{\nu }X^{\nu }$
where there is an implied summation on $\nu $ from 0 to 3, and $\Lambda ^{\mu '}{}_{\nu }$ is a matrix.
More generally, all contravariant components of a four-vector $T^{\nu }$ transform from one frame to another frame by a Lorentz transformation:
$T^{\mu '}=\Lambda ^{\mu '}{}_{\nu }T^{\nu }$
Examples of other 4-vectors include the four-velocity $U^{\mu },$ defined as the derivative of the position 4-vector with respect to proper time:
$U^{\mu }={\frac {dX^{\mu }}{d\tau }}=\gamma (v)(c,v_{x},v_{y},v_{z})=\gamma (v)(c,\mathbf {v} ).$
where the Lorentz factor is:
$\gamma (v)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\qquad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.$
The relativistic energy $E=\gamma (v)mc^{2}$ and relativistic momentum $\mathbf {p} =\gamma (v)m\mathbf {v} $ of an object are respectively the timelike and spacelike components of a contravariant four-momentum vector:
$P^{\mu }=mU^{\mu }=m\gamma (v)(c,v_{x},v_{y},v_{z})=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right)=\left({\frac {E}{c}},\mathbf {p} \right).$
where m is the invariant mass.
The four-acceleration is the proper time derivative of 4-velocity:
$A^{\mu }={\frac {dU^{\mu }}{d\tau }}.$
The transformation rules for three-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix.
The four-gradient of a scalar field φ transforms covariantly rather than contravariantly:
${\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial \phi }{\partial z'}}\end{pmatrix}}={\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t}}&{\dfrac {\partial \phi }{\partial x}}&{\dfrac {\partial \phi }{\partial y}}&{\dfrac {\partial \phi }{\partial z}}\end{pmatrix}}{\begin{pmatrix}\gamma &+\beta \gamma &0&0\\+\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}.$
which is the transpose of:
$(\partial _{\mu '}\phi )=\Lambda _{\mu '}{}^{\nu }(\partial _{\nu }\phi )\qquad \partial _{\mu }\equiv {\frac {\partial }{\partial x^{\mu }}}.$
only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.
More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation:
$T_{\mu '}=\Lambda _{\mu '}{}^{\nu }T_{\nu },$
where $\Lambda _{\mu '}{}^{\nu }$ is the reciprocal matrix of $\Lambda ^{\mu '}{}_{\nu }$.
The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.
More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law[89]
$T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}=\Lambda ^{\alpha '}{}_{\mu }\Lambda ^{\beta '}{}_{\nu }\cdots \Lambda ^{\zeta '}{}_{\rho }\Lambda _{\theta '}{}^{\sigma }\Lambda _{\iota '}{}^{\upsilon }\cdots \Lambda _{\kappa '}{}^{\phi }T_{\sigma \upsilon \cdots \phi }^{\mu \nu \cdots \rho }$
where $\Lambda _{\chi '}{}^{\psi }$ is the reciprocal matrix of $\Lambda ^{\chi '}{}_{\psi }$. All tensors transform by this rule.
An example of a four-dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order antisymmetric tensor.
The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor.
Metric
The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid with suitably chosen coordinates) which can be arranged in a 4 × 4 matrix:
$\eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}$
which is equal to its reciprocal, $\eta ^{\alpha \beta }$, in those frames. Throughout we use the signs as above, different authors use different conventions – see Minkowski metric alternative signs.
The Poincaré group is the most general group of transformations which preserves the Minkowski metric:
$\eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }$
and this is the physical symmetry underlying special relativity.
The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is:
$T^{\alpha }S_{\alpha }=T^{\alpha }\eta _{\alpha \beta }S^{\beta }=T_{\alpha }\eta ^{\alpha \beta }S_{\beta }={\text{invariant scalar}}$
Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself:
$|\mathbf {T} |={\sqrt {T^{\alpha }T_{\alpha }}}$
One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:
$T^{\alpha }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\gamma }T^{\gamma }{}_{\alpha }={\text{invariant scalars}},$
similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one does not need to perform Lorentz transformations to determine the invariants.
Relativistic kinematics and invariance
The coordinate differentials transform also contravariantly:
$dX^{\mu '}=\Lambda ^{\mu '}{}_{\nu }dX^{\nu }$
so the squared length of the differential of the position four-vector dXμ constructed using
$d\mathbf {X} ^{2}=dX^{\mu }\,dX_{\mu }=\eta _{\mu \nu }\,dX^{\mu }\,dX^{\nu }=-(cdt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}$
is an invariant. Notice that when the line element dX2 is negative that √−dX2 is the differential of proper time, while when dX2 is positive, √dX2 is differential of the proper distance.
The 4-velocity Uμ has an invariant form:
$\mathbf {U} ^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}\,,$
which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces:
$2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.$
So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
Relativistic dynamics and invariance
The invariant magnitude of the momentum 4-vector generates the energy–momentum relation:
$\mathbf {P} ^{2}=\eta ^{\mu \nu }P_{\mu }P_{\nu }=-\left({\frac {E}{c}}\right)^{2}+p^{2}.$
We can work out what this invariant is by first arguing that, since it is a scalar, it does not matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
$\mathbf {P} ^{2}=-\left({\frac {E_{\text{rest}}}{c}}\right)^{2}=-(mc)^{2}.$
We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated equation discussed above:
$E_{\text{rest}}=mc^{2}.$
The mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:
$F_{\nu }={\frac {dP_{\nu }}{d\tau }}=mA_{\nu }$
In the rest frame of the object, the time component of the four-force is zero unless the "invariant mass" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four-force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, dp/dt while the four-force is defined by the rate of change of momentum with respect to proper time, that is, dp/dτ.
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.
See also
• People:
• Max Planck
• Hermann Minkowski
• Max von Laue
• Arnold Sommerfeld
• Max Born
• Relativity:
• History of special relativity
• Doubly special relativity
• Bondi k-calculus
• Einstein synchronisation
• Rietdijk–Putnam argument
• Special relativity (alternative formulations)
• Relativity priority dispute
• Physics:
• Einstein's thought experiments
• physical cosmology
• Relativistic Euler equations
• Lorentz ether theory
• Moving magnet and conductor problem
• Shape waves
• Relativistic heat conduction
• Relativistic disk
• Born rigidity
• Born coordinates
• Mathematics:
• Lorentz group
• Relativity in the APS formalism
• Philosophy:
• actualism
• conventionalism
• Paradoxes:
• Ehrenfest paradox
• Bell's spaceship paradox
• Velocity composition paradox
• Lighthouse paradox
Notes
1. Einstein himself, in The Foundations of the General Theory of Relativity, Ann. Phys. 49 (1916), writes "The word 'special' is meant to intimate that the principle is restricted to the case ...". See p. 111 of The Principle of Relativity, A. Einstein, H. A. Lorentz, H. Weyl, H. Minkowski, Dover reprint of 1923 translation by Methuen and Company.]
2. Wald, General Relativity, p. 60: "... the special theory of relativity asserts that spacetime is the manifold $\mathbb {R} ^{4}$ with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..."
3. In a spacetime setting, the length of a moving rigid object is the spatial distance between the ends of the object measured at the same time. In the rest frame of the object the simultaneity is not required.
4. The results of the Michelson–Morley experiment led George Francis FitzGerald and Hendrik Lorentz independently to propose the phenomenon of length contraction. Lorentz believed that length contraction represented a physical contraction of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.[26]: 62–68
Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation[p 11] and induction of double refraction,[p 12] and the induction of torques on charged condensers moving at an angle with respect to the aether.[p 12] Lorentz was perplexed by experiments such as the Trouton–Noble experiment and the experiments of Rayleigh and Brace which failed to validate his theoretical expectations.[26]
5. For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation $t'=t-vx/c^{2}$.[p 13] Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.[27]: 51, 80
Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation described the apparent states of the field for a moving observer. True states remained those defined with respect to the ether.[28]
6. This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of $\Delta s^{2}$ is also an invariant, generated by a variant of the metric signature of spacetime.
7. The invariance of Δs2 under standard Lorentz transformation in analogous to the invariance of squared distances Δr2 under rotations in Euclidean space. Although space and time have an equal footing in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.
8. The refractive index dependence of the presumed partial aether-drag was eventually confirmed by Pieter Zeeman in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to Amsterdam's main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).[p 17][p 18]
9. Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in Einstein's Cosmos (W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length."
10. In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig, 1955.
Primary sources
1. Albert Einstein (1905) "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17: 891; English translation On the Electrodynamics of Moving Bodies by George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920).
2. "Science and Common Sense", P. W. Bridgman, The Scientific Monthly, Vol. 79, No. 1 (Jul. 1954), pp. 32–39.
3. The Electromagnetic Mass and Momentum of a Spinning Electron, G. Breit, Proceedings of the National Academy of Sciences, Vol. 12, p.451, 1926
4. Kinematics of an electron with an axis. Phil. Mag. 3:1-22. L. H. Thomas.]
5. Einstein, Autobiographical Notes, 1949.
6. Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920
7. Einstein, On the Relativity Principle and the Conclusions Drawn from It, 1907; "The Principle of Relativity and Its Consequences in Modern Physics", 1910; "The Theory of Relativity", 1911; Manuscript on the Special Theory of Relativity, 1912; Theory of Relativity, 1913; Einstein, Relativity, the Special and General Theory, 1916; The Principal Ideas of the Theory of Relativity, 1916; What Is The Theory of Relativity?, 1919; The Principle of Relativity (Princeton Lectures), 1921; Physics and Reality, 1936; The Theory of Relativity, 1949.
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22. On the Inertia of Energy Required by the Relativity Principle, A. Einstein, Annalen der Physik 23 (1907): 371–384
23. Champion, Frank Clive (1932). "On some close collisions of fast β-particles with electrons, photographed by the expansion method". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. The Royal Society Publishing. 136 (830): 630–637. Bibcode:1932RSPSA.136..630C. doi:10.1098/rspa.1932.0108. S2CID 123018629.
24. Baglio, Julien (26 May 2007). "Acceleration in special relativity: What is the meaning of "uniformly accelerated movement" ?" (PDF). Physics Department, ENS Cachan. Retrieved 22 January 2016.
25. P.A.M. Dirac (1930). "A Theory of Electrons and Protons". Proceedings of the Royal Society. A126 (801): 360–365. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013. JSTOR 95359.
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References
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81. The number of works is vast, see as example:
Sidney Coleman; Sheldon L. Glashow (1997). "Cosmic Ray and Neutrino Tests of Special Relativity". Physics Letters B. 405 (3–4): 249–252. arXiv:hep-ph/9703240. Bibcode:1997PhLB..405..249C. doi:10.1016/S0370-2693(97)00638-2. S2CID 17286330.
An overview can be found on this page
82. John D. Norton, John D. (2004). "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905". Archive for History of Exact Sciences. 59 (1): 45–105. Bibcode:2004AHES...59...45N. doi:10.1007/s00407-004-0085-6. S2CID 17459755.
83. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 978-0-7167-0344-0.
84. J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. p. 247. ISBN 978-0-470-01460-8.
85. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
86. Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, ISBN 0-07-032071-3
87. Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, pg 51, ISBN 0-7167-0344-0
88. George Sterman, An Introduction to Quantum Field Theory, pg 4, ISBN 0-521-31132-2
89. Sean M. Carroll (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley. p. 22. ISBN 978-0-8053-8732-2.
Further reading
Texts by Einstein and text about history of special relativity
• Einstein, Albert (1920). Relativity: The Special and General Theory.
• Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1-56731-136-9
• Logunov, Anatoly A. (2005). Henri Poincaré and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow.
Textbooks
• Charles Misner, Kip Thorne, and John Archibald Wheeler (1971) Gravitation. W. H. Freeman & Co. ISBN 0-7167-0334-3
• Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications.
• Wolfgang Rindler (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. ISBN 978-0-19-853952-0; ISBN 0-19-853952-5
• Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press, ISBN 0-19-927583-1; ISBN 978-0-19-927583-0
• Qadir, Asghar (1989). Relativity: An Introduction to the Special Theory. Singapore: World Scientific Publications. p. 128. Bibcode:1989rist.book.....Q. ISBN 978-9971-5-0612-4.
• French, A. P. (1968). Special Relativity (M.I.T. Introductory Physics) (1st ed.). W. W. Norton & Company. ISBN 978-0393097931.
• Silberstein, Ludwik (1914). The Theory of Relativity.
• Lawrence Sklar (1977). Space, Time and Spacetime. University of California Press. ISBN 978-0-520-03174-6.
• Lawrence Sklar (1992). Philosophy of Physics. Westview Press. ISBN 978-0-8133-0625-4.
• Sergey Stepanov (2018). Relativistic World. De Gruyter. ISBN 9783110515879.
• Taylor, Edwin, and John Archibald Wheeler (1992). Spacetime Physics (2nd ed.). W. H. Freeman & Co. ISBN 0-7167-2327-1.
• Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. ISBN 0-7167-4345-0.
Journal articles
• Alvager, T.; Farley, F. J. M.; Kjellman, J.; Wallin, L.; et al. (1964). "Test of the Second Postulate of Special Relativity in the GeV region". Physics Letters. 12 (3): 260–262. Bibcode:1964PhL....12..260A. doi:10.1016/0031-9163(64)91095-9.
• Darrigol, Olivier (2004). "The Mystery of the Poincaré–Einstein Connection". Isis. 95 (4): 614–26. doi:10.1086/430652. PMID 16011297. S2CID 26997100.
• Wolf, Peter; Petit, Gerard (1997). "Satellite test of Special Relativity using the Global Positioning System". Physical Review A. 56 (6): 4405–09. Bibcode:1997PhRvA..56.4405W. doi:10.1103/PhysRevA.56.4405.
• Special Relativity Scholarpedia
• Rindler, Wolfgang (2011). "Special relativity: Kinematics". Scholarpedia. 6 (2): 8520. Bibcode:2011SchpJ...6.8520R. doi:10.4249/scholarpedia.8520.
External links
Wikisource has original text related to this article:
Relativity: The Special and General Theory
Wikisource has original works on the topic: Relativity
Wikibooks has a book on the topic of: Special Relativity
Wikiversity has learning resources about Special Relativity
Look up special relativity in Wiktionary, the free dictionary.
Original works
• Zur Elektrodynamik bewegter Körper Einstein's original work in German, Annalen der Physik, Bern 1905
• On the Electrodynamics of Moving Bodies English Translation as published in the 1923 book The Principle of Relativity.
Special relativity for a general audience (no mathematical knowledge required)
• Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
• Einstein Online Archived 2010-02-01 at the Wayback Machine Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
• Audio: Cain/Gay (2006) – Astronomy Cast. Einstein's Theory of Special Relativity
Special relativity explained (using simple or more advanced mathematics)
• Bondi K-Calculus – A simple introduction to the special theory of relativity.
• Greg Egan's Foundations.
• The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus.
• Relativity Calculator: Special Relativity – An algebraic and integral calculus derivation for E = mc2.
• MathPages – Reflections on Relativity A complete online book on relativity with an extensive bibliography.
• Special Relativity An introduction to special relativity at the undergraduate level.
• Relativity: the Special and General Theory at Project Gutenberg, by Albert Einstein
• Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.
• Understanding Special Relativity The theory of special relativity in an easily understandable way.
• An Introduction to the Special Theory of Relativity (1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format).
• Lecture Notes on Special Relativity by J D Cresser Department of Physics Macquarie University.
• SpecialRelativity.net – An overview with visualizations and minimal mathematics.
• Relativity 4-ever? The problem of superluminal motion is discussed in an entertaining manner.
Visualization
• Raytracing Special Relativity Software visualizing several scenarios under the influence of special relativity.
• Real Time Relativity Archived 2013-05-08 at the Wayback Machine The Australian National University. Relativistic visual effects experienced through an interactive program.
• Spacetime travel A variety of visualizations of relativistic effects, from relativistic motion to black holes.
• Through Einstein's Eyes Archived 2013-05-14 at the Wayback Machine The Australian National University. Relativistic visual effects explained with movies and images.
• Warp Special Relativity Simulator A computer program to show the effects of traveling close to the speed of light.
• Animation clip on YouTube visualizing the Lorentz transformation.
• Original interactive FLASH Animations from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.
• lightspeed An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.
• Animation showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.
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| Wikipedia |
Constructible function
In complexity theory, a time-constructible function is a function f from natural numbers to natural numbers with the property that f(n) can be constructed from n by a Turing machine in the time of order f(n). The purpose of such a definition is to exclude functions that do not provide an upper bound on the runtime of some Turing machine.[1]
Time-constructible definitions
There are two different definitions of a time-constructible function. In the first definition, a function f is called time-constructible if there exists a positive integer n0 and Turing machine M which, given a string 1n consisting of n ones, stops after exactly f(n) steps for all n ≥ n0. In the second definition, a function f is called time-constructible if there exists a Turing machine M which, given a string 1n, outputs the binary representation of f(n) in O(f(n)) time (a unary representation may be used instead, since the two can be interconverted in O(f(n)) time).[1]
There is also a notion of a fully time-constructible function. A function f is called fully time-constructible if there exists a Turing machine M which, given a string 1n consisting of n ones, stops after exactly f(n) steps.[2] This definition is slightly less general than the first two but, for most applications, either definition can be used.[3]
Space-constructible definitions
Similarly, a function f is space-constructible if there exists a positive integer n0 and a Turing machine M which, given a string 1n consisting of n ones, halts after using exactly f(n) cells for all n ≥ n0. Equivalently, a function f is space-constructible if there exists a Turing machine M which, given a string 1n consisting of n ones, outputs the binary (or unary) representation of f(n), while using only O(f(n)) space.[1]
Also, a function f is fully space-constructible if there exists a Turing machine M which, given a string 1n consisting of n ones, halts after using exactly f(n) cells.[2]
Examples
All the commonly used functions f(n) (such as n, nk, 2n) are time- and space-constructible, as long as f(n) is at least cn for a constant c > 0. No function which is o(n) can be time-constructible unless it is eventually constant, since there is insufficient time to read the entire input. However, $\log(n)$ is a space-constructible function.
Applications
Time-constructible functions are used in results from complexity theory such as the time hierarchy theorem. They are important because the time hierarchy theorem relies on Turing machines that must determine in O(f(n)) time whether an algorithm has taken more than f(n) steps. This is, of course, impossible without being able to calculate f(n) in that time. Such results are typically true for all natural functions f but not necessarily true for artificially constructed f. To formulate them precisely, it is necessary to have a precise definition for a natural function f for which the theorem is true. Time-constructible functions are often used to provide such a definition.
Space-constructible functions are used similarly, for example in the space hierarchy theorem.
This article incorporates material from constructible on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
References
1. Goldreich, Oded (2008). Computational Complexity: A Conceptual Perspective. Cambridge University Press. pp. 130, 139. ISBN 978-0-521-88473-0.
2. Homer, Steven; Selman, Alan L. (2011). Computability and Complexity Theory (Second ed.). Springer. ISBN 978-1-4614-0681-5.
3. Balcázar, José Luis; Díaz, Josep; Gabarró, Joaquim (1988). Structural Complexity I. Springer-Verlag. ISBN 3-540-18622-0.
| Wikipedia |
Time-domain harmonic scaling
Time-domain harmonic scaling (TDHS) is a method for time-scale modification of speech (or other audio signals),[1] allowing the apparent rate of speech articulation to be changed without affecting the pitch-contour and the time-evolution of the formant structure.[2] TDHS differs from other time-scale modification algorithms in that time-scaling operations are performed in the time domain (not the frequency domain).[3] TDHS was proposed by D. Malah in 1979.[4]
References
1. Richard V. Cox; Ronald E. Crochiere; James D. Johnston (February 1983). "Real-time implementation of time domain harmonic scaling of speech for rate modification and coding" (PDF). IEEE Transactions on Acoustics, Speech, and Signal Processing. ASSP-31 (1): 258–272. Bibcode:1983IJSSC..18...10C. doi:10.1109/JSSC.1983.1051894.
2. Moulines, Eric & Jean Laroche. (1995). "Non-parametric techniques for pitch-scale and time-scale modification of speech" (PDF). Speech Communication. 16 (2): 175–205. doi:10.1016/0167-6393(94)00054-e.
3. D. Malah, R.E. Crochiere, and R.V. Cox. (1981). "Performance of transform and subband coding systems combined with harmonic scaling of speech" (PDF). IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (2): 273–283. doi:10.1109/tassp.1981.1163547.{{cite journal}}: CS1 maint: multiple names: authors list (link)
4. David Malah (April 1979). "Time-domain algorithms for harmonic bandwidth reduction and time scaling of speech signals" (PDF). IEEE Transactions on Acoustics, Speech, and Signal Processing. ASSP-27 (2): 121–133.
External links
• PICOLA and TDHS
| Wikipedia |
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. its energy levels and eigenstates) can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.
Approximate Hamiltonians
Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity. The Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems.
Applying perturbation theory
Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.
For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect) can be calculated. This is only approximate because the sum of a Coulomb potential with a linear potential is unstable (has no true bound states) although the tunneling time (decay rate) is very long. This instability shows up as a broadening of the energy spectrum lines, which perturbation theory fails to reproduce entirely.
The expressions produced by perturbation theory are not exact, but they can lead to accurate results as long as the expansion parameter, say α, is very small. Typically, the results are expressed in terms of finite power series in α that seem to converge to the exact values when summed to higher order. After a certain order n ~ 1/α however, the results become increasingly worse since the series are usually divergent (being asymptotic series). There exist ways to convert them into convergent series, which can be evaluated for large-expansion parameters, most efficiently by the variational method. Even convergent perturbations can converge to the wrong answer and divergent perturbations expansions can sometimes give good results at lower order.[1]
In the theory of quantum electrodynamics (QED), in which the electron–photon interaction is treated perturbatively, the calculation of the electron's magnetic moment has been found to agree with experiment to eleven decimal places.[2] In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms.
Large perturbations
Under some circumstances, perturbation theory is an invalid approach to take. This happens when the system we wish to describe cannot be described by a small perturbation imposed on some simple system. In quantum chromodynamics, for instance, the interaction of quarks with the gluon field cannot be treated perturbatively at low energies because the coupling constant (the expansion parameter) becomes too large, violating the requirement that corrections must be small.
Non-adiabatic states
Perturbation theory also fails to describe states that are not generated adiabatically from the "free model", including bound states and various collective phenomena such as solitons. Imagine, for example, that we have a system of free (i.e. non-interacting) particles, to which an attractive interaction is introduced. Depending on the form of the interaction, this may create an entirely new set of eigenstates corresponding to groups of particles bound to one another. An example of this phenomenon may be found in conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs. When faced with such systems, one usually turns to other approximation schemes, such as the variational method and the WKB approximation. This is because there is no analogue of a bound particle in the unperturbed model and the energy of a soliton typically goes as the inverse of the expansion parameter. However, if we "integrate" over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the order of exp(−1/g) or exp(−1/g2) in the perturbation parameter g. Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions for which the perturbative expansion is not valid.
Difficult computations
The problem of non-perturbative systems has been somewhat alleviated by the advent of modern computers. It has become practical to obtain numerical non-perturbative solutions for certain problems, using methods such as density functional theory. These advances have been of particular benefit to the field of quantum chemistry.[3] Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in particle physics for generating theoretical results that can be compared with experiment.
Time-independent perturbation theory
Time-independent perturbation theory is one of two categories of perturbation theory, the other being time-dependent perturbation (see next section). In time-independent perturbation theory, the perturbation Hamiltonian is static (i.e., possesses no time dependence). Time-independent perturbation theory was presented by Erwin Schrödinger in a 1926 paper,[4] shortly after he produced his theories in wave mechanics. In this paper Schrödinger referred to earlier work of Lord Rayleigh,[5] who investigated harmonic vibrations of a string perturbed by small inhomogeneities. This is why this perturbation theory is often referred to as Rayleigh–Schrödinger perturbation theory.[6]
First order corrections
The process begins with an unperturbed Hamiltonian H0, which is assumed to have no time dependence.[7] It has known energy levels and eigenstates, arising from the time-independent Schrödinger equation:
$H_{0}\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(0)}\right\rangle ,\qquad n=1,2,3,\cdots $
For simplicity, it is assumed that the energies are discrete. The (0) superscripts denote that these quantities are associated with the unperturbed system. Note the use of bra–ket notation.
A perturbation is then introduced to the Hamiltonian. Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. Thus, V is formally a Hermitian operator. Let λ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is:
$H=H_{0}+\lambda V$
The energy levels and eigenstates of the perturbed Hamiltonian are again given by the time-independent Schrödinger equation,
$\left(H_{0}+\lambda V\right)|n\rangle =E_{n}|n\rangle .$
The objective is to express En and $|n\rangle $ in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, they can be written as a (Maclaurin) power series in λ,
${\begin{aligned}E_{n}&=E_{n}^{(0)}+\lambda E_{n}^{(1)}+\lambda ^{2}E_{n}^{(2)}+\cdots \\[1ex]|n\rangle &=\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\lambda ^{2}\left|n^{(2)}\right\rangle +\cdots \end{aligned}}$
where
${\begin{aligned}E_{n}^{(k)}&={\frac {1}{k!}}{\frac {d^{k}E_{n}}{d\lambda ^{k}}}{\bigg |}_{\lambda =0}\\[1ex]\left|n^{(k)}\right\rangle &=\left.{\frac {1}{k!}}{\frac {d^{k}|n\rangle }{d\lambda ^{k}}}\right|_{\lambda =0.}\end{aligned}}$
When k = 0, these reduce to the unperturbed values, which are the first term in each series. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as the order is increased.
Substituting the power series expansion into the Schrödinger equation produces:
$\left(H_{0}+\lambda V\right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\cdots \right)=\left(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\cdots \right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\cdots \right).$
Expanding this equation and comparing coefficients of each power of λ results in an infinite series of simultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed system,
$H_{0}\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(0)}\right\rangle .$
The first-order equation is
$H_{0}\left|n^{(1)}\right\rangle +V\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(1)}\right\rangle +E_{n}^{(1)}\left|n^{(0)}\right\rangle .$
Operating through by $\langle n^{(0)}|$, the first term on the left-hand side cancels the first term on the right-hand side. (Recall, the unperturbed Hamiltonian is Hermitian). This leads to the first-order energy shift,
$E_{n}^{(1)}=\left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle .$
This is simply the expectation value of the perturbation Hamiltonian while the system is in the unperturbed eigenstate.
This result can be interpreted in the following way: supposing that the perturbation is applied, but the system is kept in the quantum state $|n^{(0)}\rangle $, which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by $\langle n^{(0)}|V|n^{(0)}\rangle $. However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as $|n^{(0)}\rangle $. These further shifts are given by the second and higher order corrections to the energy.
Before corrections to the energy eigenstate are computed, the issue of normalization must be addressed. Supposing that
$\left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle =1,$
but perturbation theory also assumes that $\langle n|n\rangle =1$.
Then at first order in λ, the following must be true:
$\left(\left\langle n^{(0)}\right|+\lambda \left\langle n^{(1)}\right|\right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle \right)=1$
$\left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle +\lambda \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle +\lambda \left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle +{\cancel {\lambda ^{2}\left\langle n^{(1)}\right|\left.n^{(1)}\right\rangle }}=1$
$\left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle +\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle =0.$
Since the overall phase is not determined in quantum mechanics, without loss of generality, in time-independent theory it can be assumed that $\langle n^{(0)}|n^{(1)}\rangle $ is purely real. Therefore,
$\left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle =\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle =-\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle ,$
leading to
$\left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle =0.$
To obtain the first-order correction to the energy eigenstate, the expression for the first-order energy correction is inserted back into the result shown above, equating the first-order coefficients of λ. Then by using the resolution of the identity:
${\begin{aligned}V\left|n^{(0)}\right\rangle &=\left(\sum _{k\neq n}\left|k^{(0)}\right\rangle \left\langle k^{(0)}\right|\right)V\left|n^{(0)}\right\rangle +\left(\left|n^{(0)}\right\rangle \left\langle n^{(0)}\right|\right)V\left|n^{(0)}\right\rangle \\&=\sum _{k\neq n}\left|k^{(0)}\right\rangle \left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle +E_{n}^{(1)}\left|n^{(0)}\right\rangle ,\end{aligned}}$
where the $|k^{(0)}\rangle $ are in the orthogonal complement of $|n^{(0)}\rangle $, i.e., the other eigenvectors.
The first-order equation may thus be expressed as
$\left(E_{n}^{(0)}-H_{0}\right)\left|n^{(1)}\right\rangle =\sum _{k\neq n}\left|k^{(0)}\right\rangle \left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle .$
Supposing that the zeroth-order energy level is not degenerate, i.e. that there is no eigenstate of H0 in the orthogonal complement of $|n^{(0)}\rangle $ with the energy $E_{n}^{(0)}$. After renaming the summation dummy index above as $k'$, any $k\neq n$ can be chosen and multiplying the first-order equation through by $\langle k^{(0)}|$ gives
$\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left\langle k^{(0)}\right.\left|n^{(1)}\right\rangle =\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle .$
The above $\langle k^{(0)}|n^{(1)}\rangle $ also gives us the component of the first-order correction along $|k^{(0)}\rangle $.
Thus, in total, the result is,
$\left|n^{(1)}\right\rangle =\sum _{k\neq n}{\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}\left|k^{(0)}\right\rangle .$
The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k ≠ n. Each term is proportional to the matrix element $\langle k^{(0)}|V|n^{(0)}\rangle $, which is a measure of how much the perturbation mixes eigenstate n with eigenstate k; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies. The expression is singular if any of these states have the same energy as state n, which is why it was assumed that there is no degeneracy. The above formula for the perturbed eigenstates also implies that the perturbation theory can be legitimately used only when the absolute magnitude of the matrix elements of the perturbation is small compared with the corresponding differences in the unperturbed energy levels, i.e., $|\langle k^{(0)}|\lambda V|n^{(0)}\rangle |\ll |E_{n}^{(0)}-E_{k}^{(0)}|.$
Second-order and higher-order corrections
We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. Our normalization prescription gives that
$2\left\langle n^{(0)}\right|\left.n^{(2)}\right\rangle +\left\langle n^{(1)}\right|\left.n^{(1)}\right\rangle =0.$
Up to second order, the expressions for the energies and (normalized) eigenstates are:
$E_{n}(\lambda )=E_{n}^{(0)}+\lambda \left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle +\lambda ^{2}\sum _{k\neq n}{\frac {\left|\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle \right|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}}+O(\lambda ^{3})$
${\begin{aligned}|n(\lambda )\rangle =\left|n^{(0)}\right\rangle &+\lambda \sum _{k\neq n}\left|k^{(0)}\right\rangle {\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}+\lambda ^{2}\sum _{k\neq n}\sum _{\ell \neq n}\left|k^{(0)}\right\rangle {\frac {\left\langle k^{(0)}\right|V\left|\ell ^{(0)}\right\rangle \left\langle \ell ^{(0)}\right|V\left|n^{(0)}\right\rangle }{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left(E_{n}^{(0)}-E_{\ell }^{(0)}\right)}}\\[1ex]&-\lambda ^{2}\sum _{k\neq n}\left|k^{(0)}\right\rangle {\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle \left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle }{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}}-{\frac {1}{2}}\lambda ^{2}\left|n^{(0)}\right\rangle \sum _{k\neq n}{\frac {|\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle |^{2}}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}}+O(\lambda ^{3}).\end{aligned}}$
If an intermediate normalization is taken (it means, if we require that $\langle n^{(0)}|n(\lambda )\rangle =1$), we obtain the same expression for the second-order correction to the wave function, except for the last term.
Extending the process further, the third-order energy correction can be shown to be [8]
$E_{n}^{(3)}=\sum _{k\neq n}\sum _{m\neq n}{\frac {\langle n^{(0)}|V|m^{(0)}\rangle \langle m^{(0)}|V|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle }{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)\left(E_{n}^{(0)}-E_{k}^{(0)}\right)}}-\langle n^{(0)}|V|n^{(0)}\rangle \sum _{m\neq n}{\frac {|\langle n^{(0)}|V|m^{(0)}\rangle |^{2}}{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)^{2}}}.$
Corrections to fifth order (energies) and fourth order (states) in compact notation
If we introduce the notation,
$V_{nm}\equiv \langle n^{(0)}|V|m^{(0)}\rangle ,$
$E_{nm}\equiv E_{n}^{(0)}-E_{m}^{(0)},$
then the energy corrections to fifth order can be written
${\begin{aligned}E_{n}^{(1)}&=V_{nn}\\E_{n}^{(2)}&={\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}\\E_{n}^{(3)}&={\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}-V_{nn}{\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}\\E_{n}^{(4)}&={\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}}}-{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{2}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}-V_{nn}{\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}}}-V_{nn}{\frac {V_{nk_{4}}V_{k_{4}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{4}}^{2}}}+V_{nn}^{2}{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{3}}}\\&={\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}}}-E_{n}^{(2)}{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{2}}}-2V_{nn}{\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}}}+V_{nn}^{2}{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{3}}}\\E_{n}^{(5)}&={\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}E_{nk_{5}}}}-{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{2}E_{nk_{5}}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}-{\frac {V_{nk_{5}}V_{k_{5}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{5}}^{2}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}-{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}\\&\quad -V_{nn}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}E_{nk_{5}}}}-V_{nn}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{4}}^{2}E_{nk_{5}}}}-V_{nn}{\frac {V_{nk_{5}}V_{k_{5}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{5}}^{2}}}+V_{nn}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}+2V_{nn}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{3}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}\\&\quad +V_{nn}^{2}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{3}E_{nk_{5}}}}+V_{nn}^{2}{\frac {V_{nk_{5}}V_{k_{5}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{5}}^{2}}}+V_{nn}^{2}{\frac {V_{nk_{5}}V_{k_{5}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{5}}^{3}}}-V_{nn}^{3}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{4}}}\\&={\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}E_{nk_{5}}}}-2E_{n}^{(2)}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{2}E_{nk_{5}}}}-{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}\\&\quad +V_{nn}\left(-2{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}E_{nk_{5}}}}-{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{4}}^{2}E_{nk_{5}}}}+{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}+2E_{n}^{(2)}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{3}}}\right)\\&\quad +V_{nn}^{2}\left(2{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{3}E_{nk_{5}}}}+{\frac {V_{nk_{5}}V_{k_{5}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{5}}^{2}}}\right)-V_{nn}^{3}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{4}}}\end{aligned}}$
and the states to fourth order can be written
${\begin{aligned}|n^{(1)}\rangle &={\frac {V_{k_{1}n}}{E_{nk_{1}}}}|k_{1}^{(0)}\rangle \\|n^{(2)}\rangle &=\left({\frac {V_{k_{1}k_{2}}V_{k_{2}n}}{E_{nk_{1}}E_{nk_{2}}}}-{\frac {V_{nn}V_{k_{1}n}}{E_{nk_{1}}^{2}}}\right)|k_{1}^{(0)}\rangle -{\frac {1}{2}}{\frac {V_{nk_{1}}V_{k_{1}n}}{E_{k_{1}n}^{2}}}|n^{(0)}\rangle \\|n^{(3)}\rangle &={\Bigg [}-{\frac {V_{k_{1}k_{2}}V_{k_{2}k_{3}}V_{k_{3}n}}{E_{k_{1}n}E_{nk_{2}}E_{nk_{3}}}}+{\frac {V_{nn}V_{k_{1}k_{2}}V_{k_{2}n}}{E_{k_{1}n}E_{nk_{2}}}}\left({\frac {1}{E_{nk_{1}}}}+{\frac {1}{E_{nk_{2}}}}\right)-{\frac {|V_{nn}|^{2}V_{k_{1}n}}{E_{k_{1}n}^{3}}}+{\frac {|V_{nk_{2}}|^{2}V_{k_{1}n}}{E_{k_{1}n}E_{nk_{2}}}}\left({\frac {1}{E_{nk_{1}}}}+{\frac {1}{2E_{nk_{2}}}}\right){\Bigg ]}|k_{1}^{(0)}\rangle \\&\quad +{\Bigg [}-{\frac {V_{nk_{2}}V_{k_{2}k_{1}}V_{k_{1}n}+V_{k_{2}n}V_{k_{1}k_{2}}V_{nk_{1}}}{2E_{nk_{2}}^{2}E_{nk_{1}}}}+{\frac {|V_{nk_{1}}|^{2}V_{nn}}{E_{nk_{1}}^{3}}}{\Bigg ]}|n^{(0)}\rangle \\|n^{(4)}\rangle &={\Bigg [}{\frac {V_{k_{1}k_{2}}V_{k_{2}k_{3}}V_{k_{3}k_{4}}V_{k_{4}k_{2}}+V_{k_{3}k_{2}}V_{k_{1}k_{2}}V_{k_{4}k_{3}}V_{k_{2}k_{4}}}{2E_{k_{1}n}E_{k_{2}k_{3}}^{2}E_{k_{2}k_{4}}}}-{\frac {V_{k_{2}k_{3}}V_{k_{3}k_{4}}V_{k_{4}n}V_{k_{1}k_{2}}}{E_{k_{1}n}E_{k_{2}n}E_{nk_{3}}E_{nk_{4}}}}+{\frac {V_{k_{1}k_{2}}}{E_{k_{1}n}}}\left({\frac {|V_{k_{2}k_{3}}|^{2}V_{k_{2}k_{2}}}{E_{k_{2}k_{3}}^{3}}}-{\frac {|V_{nk_{3}}|^{2}V_{k_{2}n}}{E_{k_{3}n}^{2}E_{k_{2}n}}}\right)\\&\quad +{\frac {V_{nn}V_{k_{1}k_{2}}V_{k_{3}n}V_{k_{2}k_{3}}}{E_{k_{1}n}E_{nk_{3}}E_{k_{2}n}}}\left({\frac {1}{E_{nk_{3}}}}+{\frac {1}{E_{k_{2}n}}}+{\frac {1}{E_{k_{1}n}}}\right)+{\frac {|V_{k_{2}n}|^{2}V_{k_{1}k_{3}}}{E_{nk_{2}}E_{k_{1}n}}}\left({\frac {V_{k_{3}n}}{E_{nk_{1}}E_{nk_{3}}}}-{\frac {V_{k_{3}k_{1}}}{E_{k_{3}k_{1}}^{2}}}\right)-{\frac {V_{nn}\left(V_{k_{3}k_{2}}V_{k_{1}k_{3}}V_{k_{2}k_{1}}+V_{k_{3}k_{1}}V_{k_{2}k_{3}}V_{k_{1}k_{2}}\right)}{2E_{k_{1}n}E_{k_{1}k_{3}}^{2}E_{k_{1}k_{2}}}}\\&\quad +{\frac {|V_{nn}|^{2}}{E_{k_{1}n}}}\left({\frac {V_{k_{1}n}V_{nn}}{E_{k_{1}n}^{3}}}+{\frac {V_{k_{1}k_{2}}V_{k_{2}n}}{E_{k_{2}n}^{3}}}\right)-{\frac {|V_{k_{1}k_{2}}|^{2}V_{nn}V_{k_{1}n}}{E_{k_{1}n}E_{k_{1}k_{2}}^{3}}}{\Bigg ]}|k_{1}^{(0)}\rangle +{\frac {1}{2}}\left[{\frac {V_{nk_{1}}V_{k_{1}k_{2}}}{E_{nk_{1}}E_{k_{2}n}^{2}}}\left({\frac {V_{k_{2}n}V_{nn}}{E_{k_{2}n}}}-{\frac {V_{k_{2}k_{3}}V_{k_{3}n}}{E_{nk_{3}}}}\right)\right.\\&\quad \left.-{\frac {V_{k_{1}n}V_{k_{2}k_{1}}}{E_{k_{1}n}^{2}E_{nk_{2}}}}\left({\frac {V_{k_{3}k_{2}}V_{nk_{3}}}{E_{nk_{3}}}}+{\frac {V_{nn}V_{nk_{2}}}{E_{nk_{2}}}}\right)+{\frac {|V_{nk_{1}}|^{2}}{E_{k_{1}n}^{2}}}\left({\frac {3|V_{nk_{2}}|^{2}}{4E_{k_{2}n}^{2}}}-{\frac {2|V_{nn}|^{2}}{E_{k_{1}n}^{2}}}\right)-{\frac {V_{k_{2}k_{3}}V_{k_{3}k_{1}}|V_{nk_{1}}|^{2}}{E_{nk_{3}}^{2}E_{nk_{1}}E_{nk_{2}}}}\right]|n^{(0)}\rangle \end{aligned}}$
All terms involved kj should be summed over kj such that the denominator does not vanish.
It is possible to relate the k-th order correction to the energy En to the k-point connected correlation function of the perturbation V in the state $|n^{(0)}\rangle $. For $k=2$, one has to consider the inverse Laplace transform $\rho _{n,2}(s)$ of the two-point correlator:
$\langle n^{(0)}|V(\tau )V(0)|n^{(0)}\rangle -\langle n^{(0)}|V|n^{(0)}\rangle ^{2}=\mathrel {\mathop {:} } \int _{\mathbb {R} }\!ds\;\rho _{n,2}(s)\,e^{-(s-E_{n}^{(0)})\tau }$
where $V(\tau )=e^{H_{0}\tau }Ve^{-H_{0}\tau }$ is the perturbing operator V in the interaction picture, evolving in Euclidean time. Then
$E_{n}^{(2)}=-\int _{\mathbb {R} }\!{\frac {ds}{s-E_{n}^{(0)}}}\,\rho _{n,2}(s).$
Similar formulas exist to all orders in perturbation theory, allowing one to express $E_{n}^{(k)}$ in terms of the inverse Laplace transform $\rho _{n,k}$ of the connected correlation function
$\langle n^{(0)}|V(\tau _{1}+\ldots +\tau _{k-1})\dotsm V(\tau _{1}+\tau _{2})V(\tau _{1})V(0)|n^{(0)}\rangle _{\text{conn}}=\langle n^{(0)}|V(\tau _{1}+\ldots +\tau _{k-1})\dotsm V(\tau _{1}+\tau _{2})V(\tau _{1})V(0)|n^{(0)}\rangle -{\text{subtractions}}.$
To be precise, if we write
$\langle n^{(0)}|V(\tau _{1}+\ldots +\tau _{k-1})\dotsm V(\tau _{1}+\tau _{2})V(\tau _{1})V(0)|n^{(0)}\rangle _{\text{conn}}=\int _{\mathbb {R} }\,\prod _{i=1}^{k-1}ds_{i}\,e^{-(s_{i}-E_{n}^{(0)})\tau _{i}}\,\rho _{n,k}(s_{1},\ldots ,s_{k-1})\,$
then the k-th order energy shift is given by [9]
$E_{n}^{(k)}=(-1)^{k-1}\int _{\mathbb {R} }\,\prod _{i=1}^{k-1}{\frac {ds_{i}}{s_{i}-E_{n}^{(0)}}}\,\rho _{n,k}(s_{1},\ldots ,s_{k-1}).$
Effects of degeneracy
Suppose that two or more energy eigenstates of the unperturbed Hamiltonian are degenerate. The first-order energy shift is not well defined, since there is no unique way to choose a basis of eigenstates for the unperturbed system. The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all.
This is manifested in the calculation of the perturbed eigenstate via the fact that the operator
$E_{n}^{(0)}-H_{0}$
does not have a well-defined inverse.
Let D denote the subspace spanned by these degenerate eigenstates. No matter how small the perturbation is, in the degenerate subspace D the energy differences between the eigenstates of H are non-zero, so complete mixing of at least some of these states is assured. Typically, the eigenvalues will split, and the eigenspaces will become simple (one-dimensional), or at least of smaller dimension than D.
The successful perturbations will not be "small" relative to a poorly chosen basis of D. Instead, we consider the perturbation "small" if the new eigenstate is close to the subspace D. The new Hamiltonian must be diagonalized in D, or a slight variation of D, so to speak. These perturbed eigenstates in D are now the basis for the perturbation expansion,
$|n\rangle =\sum _{k\in D}\alpha _{nk}|k^{(0)}\rangle +\lambda |n^{(1)}\rangle .$
For the first-order perturbation, we need solve the perturbed Hamiltonian restricted to the degenerate subspace D,
$V|k^{(0)}\rangle =\epsilon _{k}|k^{(0)}\rangle +{\text{small}}\qquad \forall |k^{(0)}\rangle \in D,$
simultaneously for all the degenerate eigenstates, where $\epsilon _{k}$ are first-order corrections to the degenerate energy levels, and "small" is a vector of $O(\lambda )$ orthogonal to D. This amounts to diagonalizing the matrix
$\langle k^{(0)}|V|l^{(0)}\rangle =V_{kl}\qquad \forall \;|k^{(0)}\rangle ,|l^{(0)}\rangle \in D.$
This procedure is approximate, since we neglected states outside the D subspace ("small"). The splitting of degenerate energies $\epsilon _{k}$ is generally observed. Although the splitting may be small, $O(\lambda )$, compared to the range of energies found in the system, it is crucial in understanding certain details, such as spectral lines in Electron Spin Resonance experiments.
Higher-order corrections due to other eigenstates outside D can be found in the same way as for the non-degenerate case,
$\left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle =\sum _{k\not \in D}\left(\langle k^{(0)}|V|n^{(0)}\rangle \right)|k^{(0)}\rangle .$
The operator on the left-hand side is not singular when applied to eigenstates outside D, so we can write
$|n^{(1)}\rangle =\sum _{k\not \in D}{\frac {\langle k^{(0)}|V|n^{(0)}\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}|k^{(0)}\rangle ,$
but the effect on the degenerate states is of $O(\lambda )$.
Near-degenerate states should also be treated similarly, when the original Hamiltonian splits aren't larger than the perturbation in the near-degenerate subspace. An application is found in the nearly free electron model, where near-degeneracy, treated properly, gives rise to an energy gap even for small perturbations. Other eigenstates will only shift the absolute energy of all near-degenerate states simultaneously.
Degeneracy lifted to first order
Let us consider degenerate energy eigenstates and a perturbation that completely lifts the degeneracy to first order of correction.
The perturbed Hamiltonian is denoted as
${\hat {H}}={\hat {H}}_{0}+\lambda {\hat {V}}\,,$
where ${\hat {H}}_{0}$ is the unperturbed Hamiltonian, ${\hat {V}}$ is the perturbation operator, and $0<\lambda <1$ is the parameter of the perturbation.
Let us focus on the degeneracy of the $n$-th unperturbed energy $E_{n}^{(0)}$. We will denote the unperturbed states in this degenerate subspace as $\left|\psi _{nk}^{(0)}\right\rangle $ and the other unperturbed states as $\left|\psi _{m}^{(0)}\right\rangle $, where $k$ is the index of the unperturbed state in the degenerate subspace and $m\neq n$ represents all other energy eigenstates with energies different from $E_{n}^{(0)}$. The eventual degeneracy among the other states with $\forall m\neq n$ does not change our arguments. All states $\left|\psi _{nk}^{(0)}\right\rangle $ with various values of $k$ share the same energy $E_{n}^{(0)}$ when there is no perturbation, i.e., when $\lambda =0$. The energies $E_{m}^{(0)}$ of the other states $\left|\psi _{m}^{(0)}\right\rangle $ with $m\neq n$ are all different from $E_{n}^{(0)}$, but not necessarily unique, i.e. not necessarily always different among themselves.
By $V_{nl,nk}$ and $V_{m,nk}$, we denote the matrix elements of the perturbation operator ${\hat {V}}$ in the basis of the unperturbed eigenstates. We assume that the basis vectors $\left|\psi _{nk}^{(0)}\right\rangle $ in the degenerate subspace are chosen such that the matrix elements $V_{nl,nk}\equiv \left\langle \psi _{nl}^{(0)}\right|{\hat {V}}\left|\psi _{nk}^{(0)}\right\rangle $ are diagonal. Assuming also that the degeneracy is completely lifted to the first order, i.e. that $E_{nl}^{(1)}\neq E_{nk}^{(1)}$ if $l\neq k$, we have the following formulae for the energy correction to the second order in $\lambda $
$E_{nk}=E_{n}^{0}+\lambda V_{nk,nk}+\lambda ^{2}\sum \limits _{m\neq n}{\frac {\left|V_{m,nk}\right|^{2}}{E_{n}^{(0)}-E_{m}^{(0)}}}+{\mathcal {O}}(\lambda ^{3})\,,$
and for the state correction to the first order in $\lambda $
$\left|\psi _{nk}\right\rangle =\left|\psi _{nk}^{(0)}\right\rangle +\lambda \sum \limits _{m\neq n}{\frac {V_{m,nk}}{E_{m}^{(0)}-E_{n}^{(0)}}}\left(-\left|\psi _{m}^{(0)}\right\rangle +\sum \limits _{l\neq k}{\frac {V_{nl,m}}{E_{nl}^{(1)}-E_{nk}^{(1)}}}\left|\psi _{nl}^{(0)}\right\rangle \right)+{\mathcal {O}}(\lambda ^{2})\,.$
Notice that here the first order correction to the state is orthogonal to the unperturbed state,
$\left\langle \psi _{nk}^{(0)}|\psi _{nk}^{(1)}\right\rangle =0\,.$
Generalization to multi-parameter case
The generalization of time-independent perturbation theory to the case where there are multiple small parameters $x^{\mu }=(x^{1},x^{2},\cdots )$ in place of λ can be formulated more systematically using the language of differential geometry, which basically defines the derivatives of the quantum states and calculates the perturbative corrections by taking derivatives iteratively at the unperturbed point.
Hamiltonian and force operator
From the differential geometric point of view, a parameterized Hamiltonian is considered as a function defined on the parameter manifold that maps each particular set of parameters $(x^{1},x^{2},\cdots )$ to an Hermitian operator H(x μ) that acts on the Hilbert space. The parameters here can be external field, interaction strength, or driving parameters in the quantum phase transition. Let En(x μ) and $|n(x^{\mu })\rangle $ be the n-th eigenenergy and eigenstate of H(x μ) respectively. In the language of differential geometry, the states $|n(x^{\mu })\rangle $ form a vector bundle over the parameter manifold, on which derivatives of these states can be defined. The perturbation theory is to answer the following question: given $E_{n}(x_{0}^{\mu })$ and $|n(x_{0}^{\mu })\rangle $ at an unperturbed reference point $x_{0}^{\mu }$, how to estimate the En(x μ) and $|n(x^{\mu })\rangle $ at x μ close to that reference point.
Without loss of generality, the coordinate system can be shifted, such that the reference point $x_{0}^{\mu }=0$ is set to be the origin. The following linearly parameterized Hamiltonian is frequently used
$H(x^{\mu })=H(0)+x^{\mu }F_{\mu }.$
If the parameters x μ are considered as generalized coordinates, then Fμ should be identified as the generalized force operators related to those coordinates. Different indices μ label the different forces along different directions in the parameter manifold. For example, if x μ denotes the external magnetic field in the μ-direction, then Fμ should be the magnetization in the same direction.
Perturbation theory as power series expansion
The validity of perturbation theory lies on the adiabatic assumption, which assumes the eigenenergies and eigenstates of the Hamiltonian are smooth functions of parameters such that their values in the vicinity region can be calculated in power series (like Taylor expansion) of the parameters:
${\begin{aligned}E_{n}(x^{\mu })&=E_{n}+x^{\mu }\partial _{\mu }E_{n}+{\frac {1}{2!}}x^{\mu }x^{\nu }\partial _{\mu }\partial _{\nu }E_{n}+\cdots \\[1ex]\left|n(x^{\mu })\right\rangle &=|n\rangle +x^{\mu }|\partial _{\mu }n\rangle +{\frac {1}{2!}}x^{\mu }x^{\nu }|\partial _{\mu }\partial _{\nu }n\rangle +\cdots \end{aligned}}$
Here ∂μ denotes the derivative with respect to x μ. When applying to the state $|\partial _{\mu }n\rangle $, it should be understood as the covariant derivative if the vector bundle is equipped with non-vanishing connection. All the terms on the right-hand-side of the series are evaluated at x μ = 0, e.g. En ≡ En(0) and $|n\rangle \equiv |n(0)\rangle $. This convention will be adopted throughout this subsection, that all functions without the parameter dependence explicitly stated are assumed to be evaluated at the origin. The power series may converge slowly or even not converge when the energy levels are close to each other. The adiabatic assumption breaks down when there is energy level degeneracy, and hence the perturbation theory is not applicable in that case.
Hellmann–Feynman theorems
The above power series expansion can be readily evaluated if there is a systematic approach to calculate the derivates to any order. Using the chain rule, the derivatives can be broken down to the single derivative on either the energy or the state. The Hellmann–Feynman theorems are used to calculate these single derivatives. The first Hellmann–Feynman theorem gives the derivative of the energy,
$\partial _{\mu }E_{n}=\langle n|\partial _{\mu }H|n\rangle $
The second Hellmann–Feynman theorem gives the derivative of the state (resolved by the complete basis with m ≠ n),
$\langle m|\partial _{\mu }n\rangle ={\frac {\langle m|\partial _{\mu }H|n\rangle }{E_{n}-E_{m}}},\qquad \langle \partial _{\mu }m|n\rangle ={\frac {\langle m|\partial _{\mu }H|n\rangle }{E_{m}-E_{n}}}.$
For the linearly parameterized Hamiltonian, ∂μH simply stands for the generalized force operator Fμ.
The theorems can be simply derived by applying the differential operator ∂μ to both sides of the Schrödinger equation $H|n\rangle =E_{n}|n\rangle ,$ which reads
$\partial _{\mu }H|n\rangle +H|\partial _{\mu }n\rangle =\partial _{\mu }E_{n}|n\rangle +E_{n}|\partial _{\mu }n\rangle .$
Then overlap with the state $\langle m|$ from left and make use of the Schrödinger equation $\langle m|H=\langle m|E_{m}$ again,
$\langle m|\partial _{\mu }H|n\rangle +E_{m}\langle m|\partial _{\mu }n\rangle =\partial _{\mu }E_{n}\langle m|n\rangle +E_{n}\langle m|\partial _{\mu }n\rangle .$
Given that the eigenstates of the Hamiltonian always form an orthonormal basis $\langle m|n\rangle =\delta _{mn}$, the cases of m = n and m ≠ n can be discussed separately. The first case will lead to the first theorem and the second case to the second theorem, which can be shown immediately by rearranging the terms. With the differential rules given by the Hellmann–Feynman theorems, the perturbative correction to the energies and states can be calculated systematically.
Correction of energy and state
To the second order, the energy correction reads
$E_{n}(x^{\mu })=\langle n|H|n\rangle +\langle n|\partial _{\mu }H|n\rangle x^{\mu }+\Re \sum _{m\neq n}{\frac {\langle n|\partial _{\nu }H|m\rangle \langle m|\partial _{\mu }H|n\rangle }{E_{n}-E_{m}}}x^{\mu }x^{\nu }+\cdots ,$
where $\Re $ denotes the real part function. The first order derivative ∂μEn is given by the first Hellmann–Feynman theorem directly. To obtain the second order derivative ∂μ∂νEn, simply applying the differential operator ∂μ to the result of the first order derivative $\langle n|\partial _{\nu }H|n\rangle $, which reads
$\partial _{\mu }\partial _{\nu }E_{n}=\langle \partial _{\mu }n|\partial _{\nu }H|n\rangle +\langle n|\partial _{\mu }\partial _{\nu }H|n\rangle +\langle n|\partial _{\nu }H|\partial _{\mu }n\rangle .$
Note that for linearly parameterized Hamiltonian, there is no second derivative ∂μ∂νH = 0 on the operator level. Resolve the derivative of state by inserting the complete set of basis,
$\partial _{\mu }\partial _{\nu }E_{n}=\sum _{m}\left(\langle \partial _{\mu }n|m\rangle \langle m|\partial _{\nu }H|n\rangle +\langle n|\partial _{\nu }H|m\rangle \langle m|\partial _{\mu }n\rangle \right),$
then all parts can be calculated using the Hellmann–Feynman theorems. In terms of Lie derivatives, $\langle \partial _{\mu }n|n\rangle =\langle n|\partial _{\mu }n\rangle =0$ according to the definition of the connection for the vector bundle. Therefore, the case m = n can be excluded from the summation, which avoids the singularity of the energy denominator. The same procedure can be carried on for higher order derivatives, from which higher order corrections are obtained.
The same computational scheme is applicable for the correction of states. The result to the second order is as follows
${\begin{aligned}\left|n\left(x^{\mu }\right)\right\rangle =|n\rangle &+\sum _{m\neq n}{\frac {\langle m|\partial _{\mu }H|n\rangle }{E_{n}-E_{m}}}|m\rangle x^{\mu }\\&+\left(\sum _{m\neq n}\sum _{l\neq n}{\frac {\langle m|\partial _{\mu }H|l\rangle \langle l|\partial _{\nu }H|n\rangle }{(E_{n}-E_{m})(E_{n}-E_{l})}}|m\rangle -\sum _{m\neq n}{\frac {\langle m|\partial _{\mu }H|n\rangle \langle n|\partial _{\nu }H|n\rangle }{(E_{n}-E_{m})^{2}}}|m\rangle -{\frac {1}{2}}\sum _{m\neq n}{\frac {\langle n|\partial _{\mu }H|m\rangle \langle m|\partial _{\nu }H|n\rangle }{(E_{n}-E_{m})^{2}}}|n\rangle \right)x^{\mu }x^{\nu }+\cdots .\end{aligned}}$
Both energy derivatives and state derivatives will be involved in deduction. Whenever a state derivative is encountered, resolve it by inserting the complete set of basis, then the Hellmann-Feynman theorem is applicable. Because differentiation can be calculated systematically, the series expansion approach to the perturbative corrections can be coded on computers with symbolic processing software like Mathematica.
Effective Hamiltonian
Let H(0) be the Hamiltonian completely restricted either in the low-energy subspace ${\mathcal {H}}_{L}$ or in the high-energy subspace ${\mathcal {H}}_{H}$, such that there is no matrix element in H(0) connecting the low- and the high-energy subspaces, i.e. $\langle m|H(0)|l\rangle =0$ if $m\in {\mathcal {H}}_{L},l\in {\mathcal {H}}_{H}$. Let Fμ = ∂μH be the coupling terms connecting the subspaces. Then when the high energy degrees of freedoms are integrated out, the effective Hamiltonian in the low energy subspace reads[10]
$H_{mn}^{\text{eff}}\left(x^{\mu }\right)=\langle m|H|n\rangle +\delta _{nm}\langle m|\partial _{\mu }H|n\rangle x^{\mu }+{\frac {1}{2!}}\sum _{l\in {\mathcal {H}}_{H}}\left({\frac {\langle m|\partial _{\mu }H|l\rangle \langle l|\partial _{\nu }H|n\rangle }{E_{m}-E_{l}}}+{\frac {\langle m|\partial _{\nu }H|l\rangle \langle l|\partial _{\mu }H|n\rangle }{E_{n}-E_{l}}}\right)x^{\mu }x^{\nu }+\cdots .$
Here $m,n\in {\mathcal {H}}_{L}$ are restricted in the low energy subspace. The above result can be derived by power series expansion of $\langle m|H(x^{\mu })|n\rangle $.
In a formal way it is possible to define an effective Hamiltonian that gives exactly the low-lying energy states and wavefunctions.[11] In practice, some kind of approximation (perturbation theory) is generally required.
Time-dependent perturbation theory
Method of variation of constants
Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H0.[12]
Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Thus, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. One is interested in the following quantities:
• The time-dependent expectation value of some observable A, for a given initial state.
• The time-dependent expansion coefficients (w.r.t. a given time-dependent state) of those basis states that are energy eigenkets (eigenvectors) in the unperturbed system.
The first quantity is important because it gives rise to the classical result of an A measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take A to be the displacement in the x-direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent dielectric polarization of a hydrogen gas. With an appropriate choice of perturbation (i.e. an oscillating electric potential), this allows one to calculate the AC permittivity of the gas.
The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly useful in laser physics, where one is interested in the populations of different atomic states in a gas when a time-dependent electric field is applied. These probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening) and particle decay in particle physics and nuclear physics.
We will briefly examine the method behind Dirac's formulation of time-dependent perturbation theory. Choose an energy basis ${|n\rangle }$ for the unperturbed system. (We drop the (0) superscripts for the eigenstates, because it is not useful to speak of energy levels and eigenstates for the perturbed system.)
If the unperturbed system is an eigenstate (of the Hamiltonian) $|j\rangle $ at time t = 0, its state at subsequent times varies only by a phase (in the Schrödinger picture, where state vectors evolve in time and operators are constant),
$|j(t)\rangle =e^{-iE_{j}t/\hbar }|j\rangle ~.$
Now, introduce a time-dependent perturbing Hamiltonian V(t). The Hamiltonian of the perturbed system is
$H=H_{0}+V(t)~.$
Let $|\psi (t)\rangle $ denote the quantum state of the perturbed system at time t. It obeys the time-dependent Schrödinger equation,
$H|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ~.$
The quantum state at each instant can be expressed as a linear combination of the complete eigenbasis of $|n\rangle $:
$|\psi (t)\rangle =\sum _{n}c_{n}(t)e^{-iE_{n}t/\hbar }|n\rangle ~,$
(1)
where the cn(t)s are to be determined complex functions of t which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the Dirac picture).
We have explicitly extracted the exponential phase factors $\exp(-iE_{n}t/\hbar )$ on the right hand side. This is only a matter of convention, and may be done without loss of generality. The reason we go to this trouble is that when the system starts in the state $|j\rangle $ and no perturbation is present, the amplitudes have the convenient property that, for all t, cj(t) = 1 and cn(t) = 0 if n ≠ j.
The square of the absolute amplitude cn(t) is the probability that the system is in state n at time t, since
$\left|c_{n}(t)\right|^{2}=\left|\langle n|\psi (t)\rangle \right|^{2}~.$
Plugging into the Schrödinger equation and using the fact that ∂/∂t acts by a product rule, one obtains
$\sum _{n}\left(i\hbar {\frac {dc_{n}}{dt}}-c_{n}(t)V(t)\right)e^{-iE_{n}t/\hbar }|n\rangle =0~.$
By resolving the identity in front of V and multiplying through by the bra $\langle n|$ on the left, this can be reduced to a set of coupled differential equations for the amplitudes,
${\frac {dc_{n}}{dt}}={\frac {-i}{\hbar }}\sum _{k}\langle n|V(t)|k\rangle \,c_{k}(t)\,e^{-i(E_{k}-E_{n})t/\hbar }~.$
where we have used equation (1) to evaluate the sum on n in the second term, then used the fact that $\langle k|\Psi (t)\rangle =c_{k}(t)e^{-iE_{k}t/\hbar }$.
The matrix elements of V play a similar role as in time-independent perturbation theory, being proportional to the rate at which amplitudes are shifted between states. Note, however, that the direction of the shift is modified by the exponential phase factor. Over times much longer than the energy difference Ek − En, the phase winds around 0 several times. If the time-dependence of V is sufficiently slow, this may cause the state amplitudes to oscillate. (For example, such oscillations are useful for managing radiative transitions in a laser.)
Up to this point, we have made no approximations, so this set of differential equations is exact. By supplying appropriate initial values cn(t), we could in principle find an exact (i.e., non-perturbative) solution. This is easily done when there are only two energy levels (n = 1, 2), and this solution is useful for modelling systems like the ammonia molecule.
However, exact solutions are difficult to find when there are many energy levels, and one instead looks for perturbative solutions. These may be obtained by expressing the equations in an integral form,
$c_{n}(t)=c_{n}(0)-{\frac {i}{\hbar }}\sum _{k}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle \,c_{k}(t')\,e^{-i(E_{k}-E_{n})t'/\hbar }~.$
Repeatedly substituting this expression for cn back into right hand side, yields an iterative solution,
$c_{n}(t)=c_{n}^{(0)}+c_{n}^{(1)}+c_{n}^{(2)}+\cdots $
where, for example, the first-order term is
$c_{n}^{(1)}(t)={\frac {-i}{\hbar }}\sum _{k}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle \,c_{k}^{(0)}\,e^{-i(E_{k}-E_{n})t'/\hbar }~.$
To the same approximation, the summation in the above expression can be removed since in the unperturbed state $c_{k}^{(0)}=\delta _{kn}$ so that we have
$c_{n}^{(1)}(t)={\frac {-i}{\hbar }}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle \,e^{-i(E_{k}-E_{n})t'/\hbar }~.$
Several further results follow from this, such as Fermi's golden rule, which relates the rate of transitions between quantum states to the density of states at particular energies; or the Dyson series, obtained by applying the iterative method to the time evolution operator, which is one of the starting points for the method of Feynman diagrams.
Method of Dyson series
Time-dependent perturbations can be reorganized through the technique of the Dyson series. The Schrödinger equation
$H(t)|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}$
has the formal solution
$|\psi (t)\rangle =T\exp {\left[-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'H(t')\right]}|\psi (t_{0})\rangle ~,$
where T is the time ordering operator,
$TA(t_{1})A(t_{2})={\begin{cases}A(t_{1})A(t_{2})&t_{1}>t_{2}\\A(t_{2})A(t_{1})&t_{2}>t_{1}\end{cases}}~.$
Thus, the exponential represents the following Dyson series,
$|\psi (t)\rangle =\left[1-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt_{1}H(t_{1})-{\frac {1}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}H(t_{1})H(t_{2})+\ldots \right]|\psi (t_{0})\rangle ~.$
Note that in the second term, the 1/2! factor exactly cancels the double contribution due to the time-ordering operator, etc.
Consider the following perturbation problem
$[H_{0}+\lambda V(t)]|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}~,$
assuming that the parameter λ is small and that the problem $H_{0}|n\rangle =E_{n}|n\rangle $ has been solved.
Perform the following unitary transformation to the interaction picture (or Dirac picture),
$|\psi (t)\rangle =e^{-{\frac {i}{\hbar }}H_{0}(t-t_{0})}|\psi _{I}(t)\rangle ~.$
Consequently, the Schrödinger equation simplifies to
$\lambda e^{{\frac {i}{\hbar }}H_{0}(t-t_{0})}V(t)e^{-{\frac {i}{\hbar }}H_{0}(t-t_{0})}|\psi _{I}(t)\rangle =i\hbar {\frac {\partial |\psi _{I}(t)\rangle }{\partial t}}~,$
so it is solved through the above Dyson series,
$|\psi _{I}(t)\rangle =\left[1-{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}e^{{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}-{\frac {\lambda ^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}e^{{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}e^{{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}+\ldots \right]|\psi (t_{0})\rangle ~,$
as a perturbation series with small λ.
Using the solution of the unperturbed problem $H_{0}|n\rangle =E_{n}|n\rangle $ and $\sum _{n}|n\rangle \langle n|=1$ (for the sake of simplicity assume a pure discrete spectrum), yields, to first order,
$|\psi _{I}(t)\rangle =\left[1-{\frac {i\lambda }{\hbar }}\sum _{m}\sum _{n}\int _{t_{0}}^{t}dt_{1}\langle m|V(t_{1})|n\rangle e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle \langle n|+\ldots \right]|\psi (t_{0})\rangle ~.$
Thus, the system, initially in the unperturbed state $|\alpha \rangle =|\psi (t_{0})\rangle $, by dint of the perturbation can go into the state $|\beta \rangle $. The corresponding transition probability amplitude to first order is
$A_{\alpha \beta }=-{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}\langle \beta |V(t_{1})|\alpha \rangle e^{-{\frac {i}{\hbar }}(E_{\alpha }-E_{\beta })(t_{1}-t_{0})}~,$
as detailed in the previous section——while the corresponding transition probability to a continuum is furnished by Fermi's golden rule.
As an aside, note that time-independent perturbation theory is also organized inside this time-dependent perturbation theory Dyson series. To see this, write the unitary evolution operator, obtained from the above Dyson series, as
$U(t)=1-{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}e^{{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}-{\frac {\lambda ^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}e^{{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}e^{{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}+\cdots $
and take the perturbation V to be time-independent.
Using the identity resolution
$\sum _{n}|n\rangle \langle n|=1$
with $H_{0}|n\rangle =E_{n}|n\rangle $ for a pure discrete spectrum, write
${\begin{aligned}U(t)=1&-\left[{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}\sum _{m}\sum _{n}\langle m|V|n\rangle e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle \langle n|\right]\\[5mu]&-\left[{\frac {\lambda ^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\sum _{m}\sum _{n}\sum _{q}e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}\langle m|V|n\rangle \langle n|V|q\rangle e^{-{\frac {i}{\hbar }}(E_{q}-E_{n})(t_{2}-t_{0})}|m\rangle \langle q|\right]+\cdots \end{aligned}}$
It is evident that, at second order, one must sum on all the intermediate states. Assume $t_{0}=0$ and the asymptotic limit of larger times. This means that, at each contribution of the perturbation series, one has to add a multiplicative factor $e^{-\epsilon t}$ in the integrands for ε arbitrarily small. Thus the limit t → ∞ gives back the final state of the system by eliminating all oscillating terms, but keeping the secular ones. The integrals are thus computable, and, separating the diagonal terms from the others yields
${\begin{aligned}U(t)=1&-{\frac {i\lambda }{\hbar }}\sum _{n}\langle n|V|n\rangle t-{\frac {i\lambda ^{2}}{\hbar }}\sum _{m\neq n}{\frac {\langle n|V|m\rangle \langle m|V|n\rangle }{E_{n}-E_{m}}}t-{\frac {1}{2}}{\frac {\lambda ^{2}}{\hbar ^{2}}}\sum _{m,n}\langle n|V|m\rangle \langle m|V|n\rangle t^{2}+\cdots \\&+\lambda \sum _{m\neq n}{\frac {\langle m|V|n\rangle }{E_{n}-E_{m}}}|m\rangle \langle n|+\lambda ^{2}\sum _{m\neq n}\sum _{q\neq n}\sum _{n}{\frac {\langle m|V|n\rangle \langle n|V|q\rangle }{(E_{n}-E_{m})(E_{q}-E_{n})}}|m\rangle \langle q|+\cdots \end{aligned}}$
where the time secular series yields the eigenvalues of the perturbed problem specified above, recursively; whereas the remaining time-constant part yields the corrections to the stationary eigenfunctions also given above ($|n(\lambda )\rangle =U(0;\lambda )|n\rangle )$.)
The unitary evolution operator is applicable to arbitrary eigenstates of the unperturbed problem and, in this case, yields a secular series that holds at small times.
Strong perturbation theory
In a similar way as for small perturbations, it is possible to develop a strong perturbation theory. Consider as usual the Schrödinger equation
$H(t)|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}$
and we consider the question if a dual Dyson series exists that applies in the limit of a perturbation increasingly large. This question can be answered in an affirmative way [13] and the series is the well-known adiabatic series.[14] This approach is quite general and can be shown in the following way. Consider the perturbation problem
$[H_{0}+\lambda V(t)]|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}$
being λ→ ∞. Our aim is to find a solution in the form
$|\psi \rangle =|\psi _{0}\rangle +{\frac {1}{\lambda }}|\psi _{1}\rangle +{\frac {1}{\lambda ^{2}}}|\psi _{2}\rangle +\ldots $
but a direct substitution into the above equation fails to produce useful results. This situation can be adjusted making a rescaling of the time variable as $\tau =\lambda t$ producing the following meaningful equations
${\begin{aligned}V(t)|\psi _{0}\rangle &=i\hbar {\frac {\partial |\psi _{0}\rangle }{\partial \tau }}\\[1ex]V(t)|\psi _{1}\rangle +H_{0}|\psi _{0}\rangle &=i\hbar {\frac {\partial |\psi _{1}\rangle }{\partial \tau }}\\[1ex]&\;\,\vdots \end{aligned}}$
that can be solved once we know the solution of the leading order equation. But we know that in this case we can use the adiabatic approximation. When $V(t)$ does not depend on time one gets the Wigner-Kirkwood series that is often used in statistical mechanics. Indeed, in this case we introduce the unitary transformation
$|\psi (t)\rangle =e^{-{\frac {i}{\hbar }}\lambda V(t-t_{0})}|\psi _{F}(t)\rangle $
that defines a free picture as we are trying to eliminate the interaction term. Now, in dual way with respect to the small perturbations, we have to solve the Schrödinger equation
$e^{{\frac {i}{\hbar }}\lambda V(t-t_{0})}H_{0}e^{-{\frac {i}{\hbar }}\lambda V(t-t_{0})}|\psi _{F}(t)\rangle =i\hbar {\frac {\partial |\psi _{F}(t)\rangle }{\partial t}}$
and we see that the expansion parameter λ appears only into the exponential and so, the corresponding Dyson series, a dual Dyson series, is meaningful at large λs and is
$|\psi _{F}(t)\rangle =\left[1-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt_{1}e^{{\frac {i}{\hbar }}\lambda V(t_{1}-t_{0})}H_{0}e^{-{\frac {i}{\hbar }}\lambda V(t_{1}-t_{0})}-{\frac {1}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}e^{{\frac {i}{\hbar }}\lambda V(t_{1}-t_{0})}H_{0}e^{-{\frac {i}{\hbar }}\lambda V(t_{1}-t_{0})}e^{{\frac {i}{\hbar }}\lambda V(t_{2}-t_{0})}H_{0}e^{-{\frac {i}{\hbar }}\lambda V(t_{2}-t_{0})}+\cdots \right]|\psi (t_{0})\rangle .$
After the rescaling in time $\tau =\lambda t$ we can see that this is indeed a series in $1/\lambda $ justifying in this way the name of dual Dyson series. The reason is that we have obtained this series simply interchanging H0 and V and we can go from one to another applying this exchange. This is called duality principle in perturbation theory. The choice $H_{0}=p^{2}/2m$ yields, as already said, a Wigner-Kirkwood series that is a gradient expansion. The Wigner-Kirkwood series is a semiclassical series with eigenvalues given exactly as for WKB approximation.[15]
Examples
Example of first-order perturbation theory – ground-state energy of the quartic oscillator
Consider the quantum harmonic oscillator with the quartic potential perturbation and the Hamiltonian
$H=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}x^{2}}{2}}+\lambda x^{4}.$
The ground state of the harmonic oscillator is
$\psi _{0}=\left({\frac {\alpha }{\pi }}\right)^{\frac {1}{4}}e^{-\alpha x^{2}/2}$
($\alpha =m\omega /\hbar $), and the energy of unperturbed ground state is
$E_{0}^{(0)}={\tfrac {1}{2}}\hbar \omega $
Using the first-order correction formula, we get
$E_{0}^{(1)}=\lambda \left({\frac {\alpha }{\pi }}\right)^{\frac {1}{2}}\int e^{-\alpha x^{2}/2}x^{4}e^{-\alpha x^{2}/2}dx=\lambda \left({\frac {\alpha }{\pi }}\right)^{\frac {1}{2}}{\frac {\partial ^{2}}{\partial \alpha ^{2}}}\int e^{-\alpha x^{2}}dx,$
or
$E_{0}^{(1)}=\lambda \left({\frac {\alpha }{\pi }}\right)^{\frac {1}{2}}{\frac {\partial ^{2}}{\partial \alpha ^{2}}}\left({\frac {\pi }{\alpha }}\right)^{\frac {1}{2}}=\lambda {\frac {3}{4}}{\frac {1}{\alpha ^{2}}}={\frac {3}{4}}{\frac {\hbar ^{2}\lambda }{m^{2}\omega ^{2}}}.$
Example of first- and second-order perturbation theory – quantum pendulum
Consider the quantum-mathematical pendulum with the Hamiltonian
$H=-{\frac {\hbar ^{2}}{2ma^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}-\lambda \cos \phi $
with the potential energy $-\lambda \cos \phi $ taken as the perturbation i.e.
$V=-\cos \phi .$
The unperturbed normalized quantum wave functions are those of the rigid rotor and are given by
$\psi _{n}(\phi )={\frac {e^{in\phi }}{\sqrt {2\pi }}},$
and the energies
$E_{n}^{(0)}={\frac {\hbar ^{2}n^{2}}{2ma^{2}}}.$
The first-order energy correction to the rotor due to the potential energy is
$E_{n}^{(1)}=-{\frac {1}{2\pi }}\int e^{-in\phi }\cos \phi e^{in\phi }=-{\frac {1}{2\pi }}\int \cos \phi =0.$
Using the formula for the second-order correction, one gets
$E_{n}^{(2)}={\frac {ma^{2}}{2\pi ^{2}\hbar ^{2}}}\sum _{k}{\frac {\left|\int e^{-ik\phi }\cos \phi e^{in\phi }\,d\phi \right|^{2}}{n^{2}-k^{2}}},$
or
$E_{n}^{(2)}={\frac {ma^{2}}{2\hbar ^{2}}}\sum _{k}{\frac {\left|\left(\delta _{n,1-k}+\delta _{n,-1-k}\right)\right|^{2}}{n^{2}-k^{2}}},$
or
$E_{n}^{(2)}={\frac {ma^{2}}{2\hbar ^{2}}}\left({\frac {1}{2n-1}}+{\frac {1}{-2n-1}}\right)={\frac {ma^{2}}{\hbar ^{2}}}{\frac {1}{4n^{2}-1}}.$
Potential energy as a perturbation
When the unperturbed state is a free motion of a particle with kinetic energy $E$, the solution of the Schrödinger equation
$\nabla ^{2}\psi ^{(0)}+k^{2}\psi ^{(0)}=0$
corresponds to plane waves with wavenumber $ k={\sqrt {2mE/\hbar ^{2}}}$. If there is a weak potential energy $U(x,y,z)$ present in the space, in the first approximation, the perturbed state is described by the equation
$\nabla ^{2}\psi ^{(1)}+k^{2}\psi ^{(1)}={\frac {2mU}{\hbar ^{2}}}\psi ^{(0)},$
whose particular integral is[16]
$\psi ^{(1)}(x,y,z)=-{\frac {m}{2\pi \hbar ^{2}}}\int \psi ^{(0)}U(x',y',z'){\frac {e^{ikr}}{r}}\,dx'dy'dz',$
where $r^{2}=(x-x')^{2}+(y-y')^{2}+(z-z')^{2}$. In the two-dimensional case, the solution is
$\psi ^{(1)}(x,y)=-{\frac {im}{2\hbar ^{2}}}\int \psi ^{(0)}U(x',y')H_{0}^{(1)}(kr)\,dx'dy',$
where $r^{2}=(x-x')^{2}+(y-y')^{2}$ and $H_{0}^{(1)}$ is the Hankel function of the first kind. In the one-dimensional case, the solution is
$\psi ^{(1)}(x)=-{\frac {im}{\hbar ^{2}}}\int \psi ^{(0)}U(x'){\frac {e^{ikr}}{k}}\,dx',$
where $r=|x-x'|$.
Applications
• Rabi cycle
• Fermi's golden rule
• Muon spin spectroscopy
• Perturbed angular correlation
References
1. Simon, Barry (1982). "Large orders and summability of eigenvalue perturbation theory: A mathematical overview". International Journal of Quantum Chemistry. 21: 3–25. doi:10.1002/qua.560210103.
2. Aoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, Toichiro; Nio, Makiko (2012). "Tenth-order QED lepton anomalous magnetic moment: Eighth-order vertices containing a second-order vacuum polarization". Physical Review D. 85 (3): 033007. arXiv:1110.2826. Bibcode:2012PhRvD..85c3007A. doi:10.1103/PhysRevD.85.033007. S2CID 119279420.
3. van Mourik, T.; Buhl, M.; Gaigeot, M.-P. (10 February 2014). "Density functional theory across chemistry, physics and biology". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 372 (2011): 20120488. Bibcode:2014RSPTA.37220488V. doi:10.1098/rsta.2012.0488. PMC 3928866. PMID 24516181.
4. Schrödinger, E. (1926). "Quantisierung als Eigenwertproblem" [Quantization as an eigenvalue problem]. Annalen der Physik (in German). 80 (13): 437–490. Bibcode:1926AnP...385..437S. doi:10.1002/andp.19263851302.
5. Rayleigh, J. W. S. (1894). Theory of Sound. Vol. I (2nd ed.). London: Macmillan. pp. 115–118. ISBN 978-1-152-06023-4.
6. Sulejmanpasic, Tin; Ünsal, Mithat (2018-07-01). "Aspects of perturbation theory in quantum mechanics: The BenderWuMathematica® package". Computer Physics Communications. 228: 273–289. Bibcode:2018CoPhC.228..273S. doi:10.1016/j.cpc.2017.11.018. ISSN 0010-4655. S2CID 46923647.
7. Sakurai, J.J., and Napolitano, J. (1964,2011). Modern Quantum Mechanics (2nd ed.), Addison Wesley ISBN 978-0-8053-8291-4. Chapter 5
8. Landau, L. D.; Lifschitz, E. M. (1977). Quantum Mechanics: Non-relativistic Theory (3rd ed.). Pergamon Press. ISBN 978-0-08-019012-9.
9. Hogervorst M, Meineri M, Penedones J, Salehi Vaziri K (2021). "Hamiltonian truncation in Anti-de Sitter spacetime". Journal of High Energy Physics. 2021 (8): 63. arXiv:2104.10689. Bibcode:2021JHEP...08..063H. doi:10.1007/JHEP08(2021)063. S2CID 233346724.
10. Bir, Gennadiĭ Levikovich; Pikus, Grigoriĭ Ezekielevich (1974). "Chapter 15: Perturbation theory for the degenerate case". Symmetry and Strain-induced Effects in Semiconductors. Wiley. ISBN 978-0-470-07321-6.
11. Soliverez, Carlos E. (1981). "General Theory of Effective Hamiltonians". Physical Review A. 24 (1): 4–9. Bibcode:1981PhRvA..24....4S. doi:10.1103/PhysRevA.24.4 – via Academia.Edu.
12. Albert Messiah (1966). Quantum Mechanics, North Holland, John Wiley & Sons. ISBN 0486409244; J. J. Sakurai (1994). Modern Quantum Mechanics (Addison-Wesley) ISBN 9780201539295.
13. Frasca, M. (1998). "Duality in Perturbation Theory and the Quantum Adiabatic Approximation". Physical Review A. 58 (5): 3439–3442. arXiv:hep-th/9801069. Bibcode:1998PhRvA..58.3439F. doi:10.1103/PhysRevA.58.3439. S2CID 2699775.
14. Mostafazadeh, A. (1997). "Quantum adiabatic approximation and the geometric phase". Physical Review A. 55 (3): 1653–1664. arXiv:hep-th/9606053. Bibcode:1997PhRvA..55.1653M. doi:10.1103/PhysRevA.55.1653. S2CID 17059815.
15. Frasca, Marco (2007). "A strongly perturbed quantum system is a semiclassical system". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 463 (2085): 2195–2200. arXiv:hep-th/0603182. Bibcode:2007RSPSA.463.2195F. doi:10.1098/rspa.2007.1879. S2CID 19783654.
16. Lifshitz, E. M., & LD and Sykes Landau (JB). (1965). Quantum Mechanics; Non-relativistic Theory. Pergamon Press.
External links
Wikiquote has quotations related to Perturbation theory (quantum mechanics).
• "L1.1 General problem. Non-degenerate perturbation theory". YouTube. MIT OpenCourseWare. 14 February 2019. Archived from the original on 2021-12-12. (lecture by Barton Zwiebach)
• "L1.2 Setting up the perturbative equations". YouTube. MIT OpenCourseWare. 14 February 2019. Archived from the original on 2021-12-12.
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| Wikipedia |
Time-inhomogeneous hidden Bernoulli model
Time-inhomogeneous hidden Bernoulli model (TI-HBM) is an alternative to hidden Markov model (HMM) for automatic speech recognition. Contrary to HMM, the state transition process in TI-HBM is not a Markov-dependent process, rather it is a generalized Bernoulli (an independent) process. This difference leads to elimination of dynamic programming at state-level in TI-HBM decoding process. Thus, the computational complexity of TI-HBM for probability evaluation and state estimation is $O(NL)$ (instead of $O(N^{2}L)$ in the HMM case, where $N$ and $L$ are number of states and observation sequence length respectively). The TI-HBM is able to model acoustic-unit duration (e.g. phone/word duration) by using a built-in parameter named survival probability. The TI-HBM is simpler and faster than HMM in a phoneme recognition task, but its performance is comparable to HMM.
For details, see or .
References
• Jahanshah Kabudian, M. Mehdi Homayounpour, S. Mohammad Ahadi, "Bernoulli versus Markov: Investigation of state transition regime in switching-state acoustic models," Signal Processing, vol. 89, no. 4, pp. 662–668, April 2009.
• Jahanshah Kabudian, M. Mehdi Homayounpour, S. Mohammad Ahadi, "Time-inhomogeneous hidden Bernoulli model: An alternative to hidden Markov model for automatic speech recognition," Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4101–4104, Las Vegas, Nevada, USA, March 2008.
| Wikipedia |
Time reversibility
A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed.
A deterministic process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the sign of time. A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process.
Mathematics
In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one, so that for every state there exists a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution of any one state and the forward-time evolution of another corresponding state, given by the operator equation:
$U_{-t}=\pi \,U_{t}\,\pi $
Any time-independent structures (e.g. critical points or attractors) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.
Physics
In physics, the laws of motion of classical mechanics exhibit time reversibility, as long as the operator π reverses the conjugate momenta of all the particles of the system, i.e. $\mathbf {p} \rightarrow \mathbf {-p} $ (T-symmetry).
In quantum mechanical systems, however, the weak nuclear force is not invariant under T-symmetry alone; if weak interactions are present, reversible dynamics are still possible, but only if the operator π also reverses the signs of all the charges and the parity of the spatial co-ordinates (C-symmetry and P-symmetry). This reversibility of several linked properties is known as CPT symmetry.
Thermodynamic processes can be reversible or irreversible, depending on the change in entropy during the process. Note, however, that the fundamental laws that underlie the thermodynamic processes are all time-reversible (classical laws of motion and laws of electrodynamics),[1] which means that on the microscopic level, if one were to keep track of all the particles and all the degrees of freedom, the many-body system processes are all reversible; However, such analysis is beyond the capability of any human being (or artificial intelligence), and the macroscopic properties (like entropy and temperature) of many-body system are only defined from the statistics of the ensembles. When we talk about such macroscopic properties in thermodynamics, in certain cases, we can see irreversibility in the time evolution of these quantities on a statistical level. Indeed, the second law of thermodynamics predicates that the entropy of the entire universe must not decrease, not because the probability of that is zero, but because it is so unlikely that it is a statistical impossibility for all practical considerations (see Crooks fluctuation theorem).
Stochastic processes
A stochastic process is time-reversible if the joint probabilities of the forward and reverse state sequences are the same for all sets of time increments { τs }, for s = 1, ..., k for any k:[2]
$p(x_{t},x_{t+\tau _{1}},x_{t+\tau _{2}},\ldots ,x_{t+\tau _{k}})=p(x_{t'},x_{t'-\tau _{1}},x_{t'-\tau _{2}},\ldots ,x_{t'-\tau _{k}})$.
A univariate stationary Gaussian process is time-reversible. Markov processes can only be reversible if their stationary distributions have the property of detailed balance:
$p(x_{t}=i,x_{t+1}=j)=\,p(x_{t}=j,x_{t+1}=i)$.
Kolmogorov's criterion defines the condition for a Markov chain or continuous-time Markov chain to be time-reversible.
Time reversal of numerous classes of stochastic processes has been studied, including Lévy processes,[3] stochastic networks (Kelly's lemma),[4] birth and death processes,[5] Markov chains,[6] and piecewise deterministic Markov processes.[7]
Waves and optics
Time reversal method works based on the linear reciprocity of the wave equation, which states that the time reversed solution of a wave equation is also a solution to the wave equation since standard wave equations only contain even derivatives of the unknown variables.[8] Thus, the wave equation is symmetrical under time reversal, so the time reversal of any valid solution is also a solution. This means that a wave's path through space is valid when travelled in either direction.
Time reversal signal processing[9] is a process in which this property is used to reverse a received signal; this signal is then re-emitted and a temporal compression occurs, resulting in a reversal of the initial excitation waveform being played at the initial source.
See also
• T-symmetry
• Memorylessness
• Markov property
• Reversible computing
Notes
1. David Albert on Time and Chance
2. Tong (1990), Section 4.4
3. Jacod, J.; Protter, P. (1988). "Time Reversal on Levy Processes". The Annals of Probability. 16 (2): 620. doi:10.1214/aop/1176991776. JSTOR 2243828.
4. Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability. 8 (2): 416–432. doi:10.2307/1425912. JSTOR 1425912. S2CID 204177645.
5. Tanaka, H. (1989). "Time Reversal of Random Walks in One-Dimension". Tokyo Journal of Mathematics. 12: 159–174. doi:10.3836/tjm/1270133555.
6. Norris, J. R. (1998). Markov Chains. Cambridge University Press. ISBN 978-0521633963.
7. Löpker, A.; Palmowski, Z. (2013). "On time reversal of piecewise deterministic Markov processes". Electronic Journal of Probability. 18. arXiv:1110.3813. doi:10.1214/EJP.v18-1958. S2CID 1453859.
8. Parvasi, Seyed Mohammad; Ho, Siu Chun Michael; Kong, Qingzhao; Mousavi, Reza; Song, Gangbing (19 July 2016). "Real time bolt preload monitoring using piezoceramic transducers and time reversal technique—a numerical study with experimental verification". Smart Materials and Structures. 25 (8): 085015. Bibcode:2016SMaS...25h5015P. doi:10.1088/0964-1726/25/8/085015. ISSN 0964-1726. S2CID 113510522.
9. Anderson, B. E., M. Griffa, C. Larmat, T.J. Ulrich, and P.A. Johnson, "Time reversal", Acoust. Today, 4 (1), 5-16 (2008). https://acousticstoday.org/time-reversal-brian-e-anderson/
References
• Isham, V. (1991) "Modelling stochastic phenomena". In: Stochastic Theory and Modelling, Hinkley, DV., Reid, N., Snell, E.J. (Eds). Chapman and Hall. ISBN 978-0-412-30590-0.
• Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford UP. ISBN 0-19-852300-9
| Wikipedia |
Time-scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.
History
Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger.[1] However, similar ideas have been used before and go back at least to the introduction of the Riemann–Stieltjes integral, which unifies sums and integrals.
Dynamic equations
Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts.[2] The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.
The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus. Dynamic equations on a time scale have a potential for applications such as in population dynamics. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population.
Formal definitions
A time scale (or measure chain) is a closed subset of the real line $\mathbb {R} $. The common notation for a general time scale is $\mathbb {T} $.
The two most commonly encountered examples of time scales are the real numbers $\mathbb {R} $ and the discrete time scale $h\mathbb {Z} $.
A single point in a time scale is defined as:
$t:t\in \mathbb {T} $
Operations on time scales
The forward jump and backward jump operators represent the closest point in the time scale on the right and left of a given point $t$, respectively. Formally:
$\sigma (t)=\inf\{s\in \mathbb {T} :s>t\}$ (forward shift/jump operator)
$\rho (t)=\sup\{s\in \mathbb {T} :s<t\}$ (backward shift/jump operator)
The graininess $\mu $ is the distance from a point to the closest point on the right and is given by:
$\mu (t)=\sigma (t)-t.$
For a right-dense $t$, $\sigma (t)=t$ and $\mu (t)=0$.
For a left-dense $t$, $\rho (t)=t.$
Classification of points
For any $t\in \mathbb {T} $, $t$ is:
• left dense if $\rho (t)=t$
• right dense if $\sigma (t)=t$
• left scattered if $\rho (t)<t$
• right scattered if $\sigma (t)>t$
• dense if both left dense and right dense
• isolated if both left scattered and right scattered
As illustrated by the figure at right:
• Point $t_{1}$ is dense
• Point $t_{2}$ is left dense and right scattered
• Point $t_{3}$ is isolated
• Point $t_{4}$ is left scattered and right dense
Continuity
Continuity of a time scale is redefined as equivalent to density. A time scale is said to be right-continuous at point $t$ if it is right dense at point $t$. Similarly, a time scale is said to be left-continuous at point $t$ if it is left dense at point $t$.
Derivative
Take a function:
$f:\mathbb {T} \to \mathbb {R} ,$
(where R could be any Banach space, but is set to the real line for simplicity).
Definition: The delta derivative (also Hilger derivative) $f^{\Delta }(t)$ exists if and only if:
For every $\varepsilon >0$ there exists a neighborhood $U$ of $t$ such that:
$\left|f(\sigma (t))-f(s)-f^{\Delta }(t)(\sigma (t)-s)\right|\leq \varepsilon \left|\sigma (t)-s\right|$
for all $s$ in $U$.
Take $\mathbb {T} =\mathbb {R} .$ Then $\sigma (t)=t$, $\mu (t)=0$, $f^{\Delta }=f'$; is the derivative used in standard calculus. If $\mathbb {T} =\mathbb {Z} $ (the integers), $\sigma (t)=t+1$, $\mu (t)=1$, $f^{\Delta }=\Delta f$ is the forward difference operator used in difference equations.
Integration
The delta integral is defined as the antiderivative with respect to the delta derivative. If $F(t)$ has a continuous derivative $f(t)=F^{\Delta }(t)$ one sets
$\int _{r}^{s}f(t)\Delta (t)=F(s)-F(r).$
Laplace transform and z-transform
A Laplace transform can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal[2] to a modified Z-transform:
${\mathcal {Z}}'\{x[z]\}={\frac {{\mathcal {Z}}\{x[z+1]\}}{z+1}}$
Partial differentiation
Partial differential equations and partial difference equations are unified as partial dynamic equations on time scales.[3][4][5]
Multiple integration
Multiple integration on time scales is treated in Bohner (2005).[6]
Stochastic dynamic equations on time scales
Stochastic differential equations and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.[7]
Measure theory on time scales
Associated with every time scale is a natural measure[8][9] defined via
$\mu ^{\Delta }(A)=\lambda (\rho ^{-1}(A)),$
where $\lambda $ denotes Lebesgue measure and $\rho $ is the backward shift operator defined on $\mathbb {R} $. The delta integral turns out to be the usual Lebesgue–Stieltjes integral with respect to this measure
$\int _{r}^{s}f(t)\Delta t=\int _{[r,s)}f(t)d\mu ^{\Delta }(t)$
and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure[10]
$f^{\Delta }(t)={\frac {df}{d\mu ^{\Delta }}}(t).$
Distributions on time scales
The Dirac delta and Kronecker delta are unified on time scales as the Hilger delta:[11][12]
$\delta _{a}^{\mathbb {H} }(t)={\begin{cases}{\dfrac {1}{\mu (a)}},&t=a\\0,&t\neq a\end{cases}}$
Integral equations on time scales
Integral equations and summation equations are unified as integral equations on time scales.
Fractional calculus on time scales
Fractional calculus on time scales is treated in Bastos, Mozyrska, and Torres.[13]
See also
• Analysis on fractals for dynamic equations on a Cantor set.
• Multiple-scale analysis
• Method of averaging
• Krylov–Bogoliubov averaging method
References
1. Hilger, Stefan (1989). Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (PhD thesis). Universität Würzburg. OCLC 246538565.
2. Martin Bohner & Allan Peterson (2001). Dynamic Equations on Time Scales. Birkhäuser. ISBN 978-0-8176-4225-9.
3. Ahlbrandt, Calvin D.; Morian, Christina (2002). "Partial differential equations on time scales". Journal of Computational and Applied Mathematics. 141 (1–2): 35–55. Bibcode:2002JCoAM.141...35A. doi:10.1016/S0377-0427(01)00434-4.
4. Jackson, B. (2006). "Partial dynamic equations on time scales". Journal of Computational and Applied Mathematics. 186 (2): 391–415. Bibcode:2006JCoAM.186..391J. doi:10.1016/j.cam.2005.02.011.
5. Bohner, M.; Guseinov, G. S. (2004). "Partial differentiation on time scales" (PDF). Dynamic Systems and Applications. 13: 351–379.
6. Bohner, M; Guseinov, GS (2005). "Multiple integration on time scales". Dynamic Systems and Applications. CiteSeerX 10.1.1.79.8824.
7. Sanyal, Suman (2008). Stochastic Dynamic Equations (PhD thesis). Missouri University of Science and Technology. ProQuest 304364901.
8. Guseinov, G. S. (2003). "Integration on time scales". J. Math. Anal. Appl. 285: 107–127. doi:10.1016/S0022-247X(03)00361-5.
9. Deniz, A. (2007). Measure theory on time scales (PDF) (Master's thesis). İzmir Institute of Technology.
10. Eckhardt, J.; Teschl, G. (2012). "On the connection between the Hilger and Radon–Nikodym derivatives". J. Math. Anal. Appl. 385 (2): 1184–1189. arXiv:1102.2511. doi:10.1016/j.jmaa.2011.07.041. S2CID 17178288.
11. Davis, John M.; Gravagne, Ian A.; Jackson, Billy J.; Marks, Robert J. II; Ramos, Alice A. (2007). "The Laplace transform on time scales revisited". J. Math. Anal. Appl. 332 (2): 1291–1307. Bibcode:2007JMAA..332.1291D. doi:10.1016/j.jmaa.2006.10.089.
12. Davis, John M.; Gravagne, Ian A.; Marks, Robert J. II (2010). "Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series". Circuits, Systems and Signal Processing. 29 (6): 1141–1165. doi:10.1007/s00034-010-9196-2. S2CID 16404013.
13. Bastos, Nuno R. O.; Mozyrska, Dorota; Torres, Delfim F. M. (2011). "Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform". International Journal of Mathematics & Computation. 11 (J11): 1–9. arXiv:1012.1555. Bibcode:2010arXiv1012.1555B.
Further reading
• Agarwal, Ravi; Bohner, Martin; O’Regan, Donal; Peterson, Allan (2002). "Dynamic equations on time scales: a survey". Journal of Computational and Applied Mathematics. 141 (1–2): 1–26. Bibcode:2002JCoAM.141....1A. doi:10.1016/S0377-0427(01)00432-0.
• Dynamic Equations on Time Scales Special issue of Journal of Computational and Applied Mathematics (2002)
• Dynamic Equations And Applications Special Issue of Advances in Difference Equations (2006)
• Dynamic Equations on Time Scales: Qualitative Analysis and Applications Special issue of Nonlinear Dynamics And Systems Theory (2009)
External links
• The Baylor University Time Scales Group
• Timescalewiki.org
| Wikipedia |
Time Warp Edit Distance
Time Warp Edit Distance (TWED) is a measure of similarity (or dissimilarity) for discrete time series matching with time 'elasticity'. In comparison to other distance measures, (e.g. DTW (dynamic time warping) or LCS (longest common subsequence problem)), TWED is a metric. Its computational time complexity is $O(n^{2})$, but can be drastically reduced in some specific situations by using a corridor to reduce the search space. Its memory space complexity can be reduced to $O(n)$. It was first proposed in 2009 by P.-F. Marteau.
Definition
$\delta _{\lambda ,\nu }(A_{1}^{p},B_{1}^{q})=Min{\begin{cases}\delta _{\lambda ,\nu }(A_{1}^{p-1},B_{1}^{q})+\Gamma (a_{p}^{'}\to \Lambda )&{\rm {delete\ in\ A}}\\\delta _{\lambda ,\nu }(A_{1}^{p-1},B_{1}^{q-1})+\Gamma (a_{p}^{'}\to b_{q}^{'})&{\rm {match\ or\ substitution}}\\\delta _{\lambda ,\nu }(A_{1}^{p},B_{1}^{q-1})+\Gamma (\Lambda \to b_{q}^{'})&{\rm {delete\ in\ B}}\end{cases}}$
whereas
$\Gamma (\alpha _{p}^{'}\to \Lambda )=d_{LP}(a_{p}^{'},a_{p-1}^{'})+\nu \cdot (t_{a_{p}}-t_{a_{p-1}})+\lambda $
$\Gamma (\alpha _{p}^{'}\to b_{q}^{'})=d_{LP}(a_{p}^{'},b_{q}^{'})+d_{LP}(a_{p-1}^{'},b_{q-1}^{'})+\nu \cdot (|t_{a_{p}}-t_{b_{q}}|+|t_{a_{p-1}}-t_{b_{q-1}}|)$
$\Gamma (\Lambda \to b_{q}^{'})=d_{LP}(b_{p}^{'},b_{p-1}^{'})+\nu \cdot (t_{b_{q}}-t_{b_{q-1}})+\lambda $
Whereas the recursion $\delta _{\lambda ,\nu }$ is initialized as:
$\delta _{\lambda ,\nu }(A_{1}^{0},B_{1}^{0})=0,$
$\delta _{\lambda ,\nu }(A_{1}^{0},B_{1}^{j})=\infty \ {\rm {{for\ }j\geq 1}}$
$\delta _{\lambda ,\nu }(A_{1}^{i},B_{1}^{0})=\infty \ {\rm {{for\ }i\geq 1}}$
with $a'_{0}=b'_{0}=0$
Implementations
An implementation of the TWED algorithm in C with a Python wrapper is available at [1]
TWED is also implemented into the Time Series Subsequence Search Python package (TSSEARCH for short) available at .
An R implementation of TWED has been integrated into the TraMineR, a R package for mining, describing and visualizing sequences of states or events, and more generally discrete sequence data.[2]
Additionally, cuTWED is a CUDA- accelerated implementation of TWED which uses an improved algorithm due to G. Wright (2020). This method is linear in memory and massively parallelized. cuTWED is written in CUDA C/C++, comes with Python bindings, and also includes Python bindings for Marteau's reference C implementation.
Python
import numpy as np
def dlp(A, B, p=2):
cost = np.sum(np.power(np.abs(A - B), p))
return np.power(cost, 1 / p)
def twed(A, timeSA, B, timeSB, nu, _lambda):
# [distance, DP] = TWED( A, timeSA, B, timeSB, lambda, nu )
# Compute Time Warp Edit Distance (TWED) for given time series A and B
#
# A := Time series A (e.g. [ 10 2 30 4])
# timeSA := Time stamp of time series A (e.g. 1:4)
# B := Time series B
# timeSB := Time stamp of time series B
# lambda := Penalty for deletion operation
# nu := Elasticity parameter - nu >=0 needed for distance measure
# Reference :
# Marteau, P.; F. (2009). "Time Warp Edit Distance with Stiffness Adjustment for Time Series Matching".
# IEEE Transactions on Pattern Analysis and Machine Intelligence. 31 (2): 306–318. arXiv:cs/0703033
# http://people.irisa.fr/Pierre-Francois.Marteau/
# Check if input arguments
if len(A) != len(timeSA):
print("The length of A is not equal length of timeSA")
return None, None
if len(B) != len(timeSB):
print("The length of B is not equal length of timeSB")
return None, None
if nu < 0:
print("nu is negative")
return None, None
# Add padding
A = np.array([0] + list(A))
timeSA = np.array([0] + list(timeSA))
B = np.array([0] + list(B))
timeSB = np.array([0] + list(timeSB))
n = len(A)
m = len(B)
# Dynamical programming
DP = np.zeros((n, m))
# Initialize DP Matrix and set first row and column to infinity
DP[0, :] = np.inf
DP[:, 0] = np.inf
DP[0, 0] = 0
# Compute minimal cost
for i in range(1, n):
for j in range(1, m):
# Calculate and save cost of various operations
C = np.ones((3, 1)) * np.inf
# Deletion in A
C[0] = (
DP[i - 1, j]
+ dlp(A[i - 1], A[i])
+ nu * (timeSA[i] - timeSA[i - 1])
+ _lambda
)
# Deletion in B
C[1] = (
DP[i, j - 1]
+ dlp(B[j - 1], B[j])
+ nu * (timeSB[j] - timeSB[j - 1])
+ _lambda
)
# Keep data points in both time series
C[2] = (
DP[i - 1, j - 1]
+ dlp(A[i], B[j])
+ dlp(A[i - 1], B[j - 1])
+ nu * (abs(timeSA[i] - timeSB[j]) + abs(timeSA[i - 1] - timeSB[j - 1]))
)
# Choose the operation with the minimal cost and update DP Matrix
DP[i, j] = np.min(C)
distance = DP[n - 1, m - 1]
return distance, DP
Backtracking, to find the most cost-efficient path:
def backtracking(DP):
# [ best_path ] = BACKTRACKING ( DP )
# Compute the most cost-efficient path
# DP := DP matrix of the TWED function
x = np.shape(DP)
i = x[0] - 1
j = x[1] - 1
# The indices of the paths are save in opposite direction
# path = np.ones((i + j, 2 )) * np.inf;
best_path = []
steps = 0
while i != 0 or j != 0:
best_path.append((i - 1, j - 1))
C = np.ones((3, 1)) * np.inf
# Keep data points in both time series
C[0] = DP[i - 1, j - 1]
# Deletion in A
C[1] = DP[i - 1, j]
# Deletion in B
C[2] = DP[i, j - 1]
# Find the index for the lowest cost
idx = np.argmin(C)
if idx == 0:
# Keep data points in both time series
i = i - 1
j = j - 1
elif idx == 1:
# Deletion in A
i = i - 1
j = j
else:
# Deletion in B
i = i
j = j - 1
steps = steps + 1
best_path.append((i - 1, j - 1))
best_path.reverse()
return best_path[1:]
MATLAB
function [distance, DP] = twed(A, timeSA, B, timeSB, lambda, nu)
% [distance, DP] = TWED( A, timeSA, B, timeSB, lambda, nu )
% Compute Time Warp Edit Distance (TWED) for given time series A and B
%
% A := Time series A (e.g. [ 10 2 30 4])
% timeSA := Time stamp of time series A (e.g. 1:4)
% B := Time series B
% timeSB := Time stamp of time series B
% lambda := Penalty for deletion operation
% nu := Elasticity parameter - nu >=0 needed for distance measure
%
% Code by: P.-F. Marteau - http://people.irisa.fr/Pierre-Francois.Marteau/
% Check if input arguments
if length(A) ~= length(timeSA)
warning('The length of A is not equal length of timeSA')
return
end
if length(B) ~= length(timeSB)
warning('The length of B is not equal length of timeSB')
return
end
if nu < 0
warning('nu is negative')
return
end
% Add padding
A = [0 A];
timeSA = [0 timeSA];
B = [0 B];
timeSB = [0 timeSB];
% Dynamical programming
DP = zeros(length(A), length(B));
% Initialize DP Matrix and set first row and column to infinity
DP(1, :) = inf;
DP(:, 1) = inf;
DP(1, 1) = 0;
n = length(timeSA);
m = length(timeSB);
% Compute minimal cost
for i = 2:n
for j = 2:m
cost = Dlp(A(i), B(j));
% Calculate and save cost of various operations
C = ones(3, 1) * inf;
% Deletion in A
C(1) = DP(i - 1, j) + Dlp(A(i - 1), A(i)) + nu * (timeSA(i) - timeSA(i - 1)) + lambda;
% Deletion in B
C(2) = DP(i, j - 1) + Dlp(B(j - 1), B(j)) + nu * (timeSB(j) - timeSB(j - 1)) + lambda;
% Keep data points in both time series
C(3) = DP(i - 1, j - 1) + Dlp(A(i), B(j)) + Dlp(A(i - 1), B(j - 1)) + ...
nu * (abs(timeSA(i) - timeSB(j)) + abs(timeSA(i - 1) - timeSB(j - 1)));
% Choose the operation with the minimal cost and update DP Matrix
DP(i, j) = min(C);
end
end
distance = DP(n, m);
% Function to calculate euclidean distance
function [cost] = Dlp(A, B)
cost = sqrt(sum((A - B) .^ 2, 2));
end
end
Backtracking, to find the most cost-efficient path:
function [path] = backtracking(DP)
% [ path ] = BACKTRACKING ( DP )
% Compute the most cost-efficient path
% DP := DP matrix of the TWED function
x = size(DP);
i = x(1);
j = x(2);
% The indices of the paths are save in opposite direction
path = ones(i + j, 2) * Inf;
steps = 1;
while (i ~= 1 || j ~= 1)
path(steps, :) = [i; j];
C = ones(3, 1) * inf;
% Keep data points in both time series
C(1) = DP(i - 1, j - 1);
% Deletion in A
C(2) = DP(i - 1, j);
% Deletion in B
C(3) = DP(i, j - 1);
% Find the index for the lowest cost
[~, idx] = min(C);
switch idx
case 1
% Keep data points in both time series
i = i - 1;
j = j - 1;
case 2
% Deletion in A
i = i - 1;
j = j;
case 3
% Deletion in B
i = i;
j = j - 1;
end
steps = steps + 1;
end
path(steps, :) = [i j];
% Path was calculated in reversed direction.
path = path(1:steps, :);
path = path(end: - 1:1, :);
end
References
1. Marcus-Voß and Jeremie Zumer, pytwed. "Github repository". GitHub. Retrieved 2020-09-11.
2. TraMineR. "Website on the servers of the Geneva University, CH". Retrieved 2016-09-11.
• Marteau, P.; F. (2009). "Time Warp Edit Distance with Stiffness Adjustment for Time Series Matching". IEEE Transactions on Pattern Analysis and Machine Intelligence. 31 (2): 306–318. arXiv:cs/0703033. doi:10.1109/TPAMI.2008.76. PMID 19110495. S2CID 10049446.
| Wikipedia |
Time dependent vector field
In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.
Part of a series of articles about
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Definition
A time dependent vector field on a manifold M is a map from an open subset $\Omega \subset \mathbb {R} \times M$ on $TM$
${\begin{aligned}X:\Omega \subset \mathbb {R} \times M&\longrightarrow TM\\(t,x)&\longmapsto X(t,x)=X_{t}(x)\in T_{x}M\end{aligned}}$
such that for every $(t,x)\in \Omega $, $X_{t}(x)$ is an element of $T_{x}M$.
For every $t\in \mathbb {R} $ such that the set
$\Omega _{t}=\{x\in M\mid (t,x)\in \Omega \}\subset M$
is nonempty, $X_{t}$ is a vector field in the usual sense defined on the open set $\Omega _{t}\subset M$.
Associated differential equation
Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:
${\frac {dx}{dt}}=X(t,x)$
which is called nonautonomous by definition.
Integral curve
An integral curve of the equation above (also called an integral curve of X) is a map
$\alpha :I\subset \mathbb {R} \longrightarrow M$
such that $\forall t_{0}\in I$, $(t_{0},\alpha (t_{0}))$ is an element of the domain of definition of X and
${\frac {d\alpha }{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{0}}=X(t_{0},\alpha (t_{0}))$.
Equivalence with time-independent vector fields
A time dependent vector field $X$ on $M$ can be thought of as a vector field ${\tilde {X}}$ on $\mathbb {R} \times M,$ where ${\tilde {X}}(t,p)\in T_{(t,p)}(\mathbb {R} \times M)$ does not depend on $t.$
Conversely, associated with a time-dependent vector field $X$ on $M$ is a time-independent one ${\tilde {X}}$
$\mathbb {R} \times M\ni (t,p)\mapsto {\dfrac {\partial }{\partial t}}{\Biggl |}_{t}+X(p)\in T_{(t,p)}(\mathbb {R} \times M)$
on $\mathbb {R} \times M.$ In coordinates,
${\tilde {X}}(t,x)=(1,X(t,x)).$
The system of autonomous differential equations for ${\tilde {X}}$ is equivalent to that of non-autonomous ones for $X,$ and $x_{t}\leftrightarrow (t,x_{t})$ is a bijection between the sets of integral curves of $X$ and ${\tilde {X}},$ respectively.
Flow
The flow of a time dependent vector field X, is the unique differentiable map
$F:D(X)\subset \mathbb {R} \times \Omega \longrightarrow M$
such that for every $(t_{0},x)\in \Omega $,
$t\longrightarrow F(t,t_{0},x)$
is the integral curve $\alpha $ of X that satisfies $\alpha (t_{0})=x$.
Properties
We define $F_{t,s}$ as $F_{t,s}(p)=F(t,s,p)$
1. If $(t_{1},t_{0},p)\in D(X)$ and $(t_{2},t_{1},F_{t_{1},t_{0}}(p))\in D(X)$ then $F_{t_{2},t_{1}}\circ F_{t_{1},t_{0}}(p)=F_{t_{2},t_{0}}(p)$
2. $\forall t,s$, $F_{t,s}$ is a diffeomorphism with inverse $F_{s,t}$.
Applications
Let X and Y be smooth time dependent vector fields and $F$ the flow of X. The following identity can be proved:
${\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}(F_{t,t_{0}}^{*}Y_{t})_{p}=\left(F_{t_{1},t_{0}}^{*}\left([X_{t_{1}},Y_{t_{1}}]+{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}Y_{t}\right)\right)_{p}$
Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that $\eta $ is a smooth time dependent tensor field:
${\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}(F_{t,t_{0}}^{*}\eta _{t})_{p}=\left(F_{t_{1},t_{0}}^{*}\left({\mathcal {L}}_{X_{t_{1}}}\eta _{t_{1}}+{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}\eta _{t}\right)\right)_{p}$
This last identity is useful to prove the Darboux theorem.
References
• Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.
| Wikipedia |
Time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function.[1] The variable denoting time is usually written as $t$.
Notation
A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \frac {dx} {dt}
A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E.
${\dot {x}}$
(This is called Newton's notation)
Higher time derivatives are also used: the second derivative with respect to time is written as
${\frac {d^{2}x}{dt^{2}}}$
with the corresponding shorthand of ${\ddot {x}}$.
As a generalization, the time derivative of a vector, say:
$\mathbf {v} =\left[v_{1},\ v_{2},\ v_{3},\ldots \right]$
is defined as the vector whose components are the derivatives of the components of the original vector. That is,
${\frac {d\mathbf {v} }{dt}}=\left[{\frac {dv_{1}}{dt}},{\frac {dv_{2}}{dt}},{\frac {dv_{3}}{dt}},\ldots \right].$
Use in physics
Time derivatives are a key concept in physics. For example, for a changing position $x$, its time derivative ${\dot {x}}$ is its velocity, and its second derivative with respect to time, ${\ddot {x}}$, is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk. See motion graphs and derivatives.
A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another:
• force is the time derivative of momentum
• power is the time derivative of energy
• electric current is the time derivative of electric charge
and so on.
A common occurrence in physics is the time derivative of a vector, such as velocity or displacement. In dealing with such a derivative, both magnitude and orientation may depend upon time.
Example: circular motion
See also: Uniform circular motion and Centripetal force
For example, consider a particle moving in a circular path. Its position is given by the displacement vector $r=x{\hat {\imath }}+y{\hat {\jmath }}$, related to the angle, θ, and radial distance, r, as defined in the figure:
${\begin{aligned}x&=r\cos(\theta )\\y&=r\sin(\theta )\end{aligned}}$
For this example, we assume that θ = t. Hence, the displacement (position) at any time t is given by
$\mathbf {r} (t)=r\cos(t){\hat {\imath }}+r\sin(t){\hat {\jmath }}$
This form shows the motion described by r(t) is in a circle of radius r because the magnitude of r(t) is given by
$|\mathbf {r} (t)|={\sqrt {\mathbf {r} (t)\cdot \mathbf {r} (t)}}={\sqrt {x(t)^{2}+y(t)^{2}}}=r\,{\sqrt {\cos ^{2}(t)+\sin ^{2}(t)}}=r$
using the trigonometric identity sin2(t) + cos2(t) = 1 and where $\cdot $ is the usual Euclidean dot product.
With this form for the displacement, the velocity now is found. The time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a vector made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in this case, the velocity vector is:
${\begin{aligned}\mathbf {v} (t)={\frac {d\,\mathbf {r} (t)}{dt}}&=r\left[{\frac {d\,\cos(t)}{dt}},{\frac {d\,\sin(t)}{dt}}\right]\\&=r\ [-\sin(t),\ \cos(t)]\\&=[-y(t),x(t)].\end{aligned}}$
Thus the velocity of the particle is nonzero even though the magnitude of the position (that is, the radius of the path) is constant. The velocity is directed perpendicular to the displacement, as can be established using the dot product:
$\mathbf {v} \cdot \mathbf {r} =[-y,x]\cdot [x,y]=-yx+xy=0\,.$
Acceleration is then the time-derivative of velocity:
$\mathbf {a} (t)={\frac {d\,\mathbf {v} (t)}{dt}}=[-x(t),-y(t)]=-\mathbf {r} (t)\,.$
The acceleration is directed inward, toward the axis of rotation. It points opposite to the position vector and perpendicular to the velocity vector. This inward-directed acceleration is called centripetal acceleration.
In differential geometry
In differential geometry, quantities are often expressed with respect to the local covariant basis, $\mathbf {e} _{i}$, where i ranges over the number of dimensions. The components of a vector $\mathbf {U} $ expressed this way transform as a contravariant tensor, as shown in the expression $\mathbf {U} =U^{i}\mathbf {e} _{i}$, invoking Einstein summation convention. If we want to calculate the time derivatives of these components along a trajectory, so that we have $\mathbf {U} (t)=U^{i}(t)\mathbf {e} _{i}(t)$, we can define a new operator, the invariant derivative $\delta $, which will continue to return contravariant tensors:[2]
${\begin{aligned}{\frac {\delta U^{i}}{\delta t}}={\frac {dU^{i}}{dt}}+V^{j}\Gamma _{jk}^{i}U^{k}\\\end{aligned}}$
where $V^{j}={\frac {dx^{j}}{dt}}$ (with $x^{j}$ being the jth coordinate) captures the components of the velocity in the local covariant basis, and $\Gamma _{jk}^{i}$ are the Christoffel symbols for the coordinate system. Note that explicit dependence on t has been repressed in the notation. We can then write:
${\begin{aligned}{\frac {d\mathbf {U} }{dt}}={\frac {\delta U^{i}}{\delta t}}\mathbf {e} _{i}\\\end{aligned}}$
as well as:
${\begin{aligned}{\frac {d^{2}\mathbf {U} }{dt^{2}}}={\frac {\delta ^{2}U^{i}}{\delta t^{2}}}\mathbf {e} _{i}\\\end{aligned}}$
In terms of the covariant derivative, $\nabla _{j}$, we have:
${\begin{aligned}{\frac {\delta U^{i}}{\delta t}}=V^{j}\nabla _{j}U^{i}\\\end{aligned}}$
Use in economics
In economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives.[3]: ch. 1-3 One situation involves a stock variable and its time derivative, a flow variable. Examples include:
• The flow of net fixed investment is the time derivative of the capital stock.
• The flow of inventory investment is the time derivative of the stock of inventories.
• The growth rate of the money supply is the time derivative of the money supply divided by the money supply itself.
Sometimes the time derivative of a flow variable can appear in a model:
• The growth rate of output is the time derivative of the flow of output divided by output itself.
• The growth rate of the labor force is the time derivative of the labor force divided by the labor force itself.
And sometimes there appears a time derivative of a variable which, unlike the examples above, is not measured in units of currency:
• The time derivative of a key interest rate can appear.
• The inflation rate is the growth rate of the price level—that is, the time derivative of the price level divided by the price level itself.
See also
• Differential calculus
• Notation for differentiation
• Circular motion
• Centripetal force
• Spatial derivative
• Temporal rate
References
1. Chiang, Alpha C., Fundamental Methods of Mathematical Economics, McGraw-Hill, third edition, 1984, ch. 14, 15, 18.
2. Grinfeld, Pavel. "Tensor Calculus 6d: Velocity, Acceleration, Jolt and the New δ/δt-derivative". YouTube. Archived from the original on 2021-12-13.
3. See for example Romer, David (1996). Advanced Macroeconomics. McGraw-Hill. ISBN 0-07-053667-8.
| Wikipedia |
Time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various separate instants in the case of discrete time. An oscilloscope is a tool commonly used to visualize real-world signals in the time domain. A time-domain graph shows how a signal changes with time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.
Though most precisely referring to time in physics, the term time domain may occasionally informally refer to position in space when dealing with spatial frequencies, as a substitute for the more precise term spatial domain.
Origin of term
The use of the contrasting terms time domain and frequency domain developed in U.S. communication engineering in the late 1940s, with the terms appearing together without definition by 1950.[1] When an analysis uses the second or one of its multiples as a unit of measurement, then it is in the time domain. When analysis concerns the reciprocal units such as Hertz, then it is in the frequency domain.
See also
• Frequency domain
• Fourier transform
• Laplace transform
• Blackman–Tukey transform
References
1. Lee, Y. W.; Cheatham, T. P. Jr.; Wiesner, J. B. (1950). "Application of Correlation Analysis to the Detection of Periodic Signals in Noise". Proceedings of the IRE. 38 (10): 1165–1171. doi:10.1109/JRPROC.1950.233423. S2CID 51671133.
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| Wikipedia |
Time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future.[1][2][3] It is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience.[4][5][6][7] Time is often referred to as a fourth dimension, along with three spatial dimensions.[8]
Time
Current time
02:53, 25 August 2023 (UTC)
Major concepts
• Past
• Present
• Future
• Eternity
• of the world
Fields of study
• Archaeology
• Chronology
• History
• Horology
• Metrology
• Paleontology
• Futurology
Philosophy
• Presentism
• Eternalism
• Event
• Fatalism
• Religion
• Mythology
• Creation
• End time
• Day of Judgement
• Immortality
• Afterlife
• Reincarnation
• Kalachakra
• Measurement
• Standards
• ISO 8601
• Metric
• Hexadecimal
• Science
• Naturalism
• Chronobiology
• Cosmogony
• Evolution
• Radiometric dating
• Ultimate fate of the universe
• Time in physics
Related topics
• Motion
• Space
• Spacetime
• Time travel
Time is one of the seven fundamental physical quantities in both the International System of Units (SI) and International System of Quantities. The SI base unit of time is the second, which is defined by measuring the electronic transition frequency of caesium atoms. General relativity is the primary framework for understanding how spacetime works.[9] Through advances in both theoretical and experimental investigations of spacetime, it has been shown that time can be distorted and dilated, particularly at the edges of black holes.
Throughout history, time has been an important subject of study in religion, philosophy, and science. Temporal measurement has occupied scientists and technologists and was a prime motivation in navigation and astronomy. Time is also of significant social importance, having economic value ("time is money") as well as personal value, due to an awareness of the limited time in each day and in human life spans.
Definition
Defining time in a manner applicable to all fields without circularity has consistently eluded scholars.[7][10] Nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems.[11][12][13] In physics, time is used to define other quantities, such as velocity, so defining time in terms of such quantities would result in circularity of definition.[14]
Time in physics is operationally defined as "what a clock reads".[6][15][16] This operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event constitutes one standard unit such as the second, is highly useful in the conduct of both advanced experiments and everyday affairs of life. There are many systems for determining what time it is. Periodic events and periodic motion have long served as standards for units of time. Examples include the apparent motion of the sun across the sky, the phases of the moon, and the passage of a free-swinging pendulum. More modern systems include the Global Positioning System, other satellite systems, Coordinated Universal Time and mean solar time. In general, the numbers obtained from different time systems differ from one another, but with careful measurements they can be synchronized.
The operational definition of time does not address what the fundamental nature of time is. Investigations into the relationship between space and time led physicists to define the spacetime continuum, where every event is assigned four numbers representing its time and position (the event's coordinates). Examples of events are the collision of two particles, the explosion of a supernova, or the arrival of a rocket ship. General relativity explains why the observed time of an event may be different for different observers. In general relativity, the question of what time it is now only has meaning relative to a particular observer. Distance and time are intimately related, and the time required for light to travel a specific distance is the same for all observers, as first publicly demonstrated by Michelson and Morley. Events can be separated in many directions in space, but if two events are separated by time, then one event must precede the other, and all observers will agree on this. General relativity does not address the nature of time for extremely small intervals where quantum mechanics holds. In quantum mechanics, time is treated as a universal and absolute parameter, differing from general relativity's notion of independent clocks. Reconciling these two theories is known as the problem of time. As of 2023, there is no generally accepted theory of quantum general relativity.[17]
Measurement
Generally speaking, methods of temporal measurement, or chronometry, take two distinct forms: the calendar, a mathematical tool for organising intervals of time,[18] and the clock, a physical mechanism that counts the passage of time. In day-to-day life, the clock is consulted for periods less than a day, whereas the calendar is consulted for periods longer than a day. Increasingly, personal electronic devices display both calendars and clocks simultaneously. The number (as on a clock dial or calendar) that marks the occurrence of a specified event as to hour or date is obtained by counting from a fiducial epoch – a central reference point.
History of the calendar
Artifacts from the Paleolithic suggest that the moon was used to reckon time as early as 6,000 years ago.[19] Lunar calendars were among the first to appear, with years of either 12 or 13 lunar months (either 354 or 384 days). Without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months. Lunisolar calendars have a thirteenth month added to some years to make up for the difference between a full year (now known to be about 365.24 days) and a year of just twelve lunar months. The numbers twelve and thirteen came to feature prominently in many cultures, at least partly due to this relationship of months to years. Other early forms of calendars originated in Mesoamerica, particularly in ancient Mayan civilization. These calendars were religiously and astronomically based, with 18 months in a year and 20 days in a month, plus five epagomenal days at the end of the year.[20]
The reforms of Julius Caesar in 45 BC put the Roman world on a solar calendar. This Julian calendar was faulty in that its intercalation still allowed the astronomical solstices and equinoxes to advance against it by about 11 minutes per year. Pope Gregory XIII introduced a correction in 1582; the Gregorian calendar was only slowly adopted by different nations over a period of centuries, but it is now by far the most commonly used calendar around the world.
During the French Revolution, a new clock and calendar were invented as part of the dechristianization of France, and to create a more rational system in order to replace the Gregorian calendar. The French Republican Calendar's days consisted of ten hours of a hundred minutes of a hundred seconds, which marked a deviation from the base 12 (duodecimal) system used in many other devices by many cultures. The system was abolished in 1806.[21]
History of other devices
A large variety of devices have been invented to measure time. The study of these devices is called horology.[22]
An Egyptian device that dates to c. 1500 BC, similar in shape to a bent T-square, measured the passage of time from the shadow cast by its crossbar on a nonlinear rule. The T was oriented eastward in the mornings. At noon, the device was turned around so that it could cast its shadow in the evening direction.[23]
A sundial uses a gnomon to cast a shadow on a set of markings calibrated to the hour. The position of the shadow marks the hour in local time. The idea to separate the day into smaller parts is credited to Egyptians because of their sundials, which operated on a duodecimal system. The importance of the number 12 is due to the number of lunar cycles in a year and the number of stars used to count the passage of night.[24]
The most precise timekeeping device of the ancient world was the water clock, or clepsydra, one of which was found in the tomb of Egyptian pharaoh Amenhotep I. They could be used to measure the hours even at night but required manual upkeep to replenish the flow of water. The ancient Greeks and the people from Chaldea (southeastern Mesopotamia) regularly maintained timekeeping records as an essential part of their astronomical observations. Arab inventors and engineers, in particular, made improvements on the use of water clocks up to the Middle Ages.[25] In the 11th century, Chinese inventors and engineers invented the first mechanical clocks driven by an escapement mechanism.
The hourglass uses the flow of sand to measure the flow of time. They were used in navigation. Ferdinand Magellan used 18 glasses on each ship for his circumnavigation of the globe (1522).[26]
Incense sticks and candles were, and are, commonly used to measure time in temples and churches across the globe. Water clocks, and, later, mechanical clocks, were used to mark the events of the abbeys and monasteries of the Middle Ages. Richard of Wallingford (1292–1336), abbot of St. Alban's abbey, famously built a mechanical clock as an astronomical orrery about 1330.[27][28]
Great advances in accurate time-keeping were made by Galileo Galilei and especially Christiaan Huygens with the invention of pendulum-driven clocks along with the invention of the minute hand by Jost Burgi.[29]
The English word clock probably comes from the Middle Dutch word klocke which, in turn, derives from the medieval Latin word clocca, which ultimately derives from Celtic and is cognate with French, Latin, and German words that mean bell. The passage of the hours at sea was marked by bells and denoted the time (see ship's bell). The hours were marked by bells in abbeys as well as at sea.
Clocks can range from watches to more exotic varieties such as the Clock of the Long Now. They can be driven by a variety of means, including gravity, springs, and various forms of electrical power, and regulated by a variety of means such as a pendulum.
Alarm clocks first appeared in ancient Greece around 250 BC with a water clock that would set off a whistle. This idea was later mechanized by Levi Hutchins and Seth E. Thomas.[29]
A chronometer is a portable timekeeper that meets certain precision standards. Initially, the term was used to refer to the marine chronometer, a timepiece used to determine longitude by means of celestial navigation, a precision firstly achieved by John Harrison. More recently, the term has also been applied to the chronometer watch, a watch that meets precision standards set by the Swiss agency COSC.
The most accurate timekeeping devices are atomic clocks, which are accurate to seconds in many millions of years,[31] and are used to calibrate other clocks and timekeeping instruments.
Atomic clocks use the frequency of electronic transitions in certain atoms to measure the second. One of the atoms used is caesium, most modern atomic clocks probe caesium with microwaves to determine the frequency of these electron vibrations.[32] Since 1967, the International System of Measurements bases its unit of time, the second, on the properties of caesium atoms. SI defines the second as 9,192,631,770 cycles of the radiation that corresponds to the transition between two electron spin energy levels of the ground state of the 133Cs atom.
Today, the Global Positioning System in coordination with the Network Time Protocol can be used to synchronize timekeeping systems across the globe.
In medieval philosophical writings, the atom was a unit of time referred to as the smallest possible division of time. The earliest known occurrence in English is in Byrhtferth's Enchiridion (a science text) of 1010–1012,[33] where it was defined as 1/564 of a momentum (11⁄2 minutes),[34] and thus equal to 15/94 of a second. It was used in the computus, the process of calculating the date of Easter.
As of May 2010, the smallest time interval uncertainty in direct measurements is on the order of 12 attoseconds (1.2 × 10−17 seconds), about 3.7 × 1026 Planck times.[35]
Units
The second (s) is the SI base unit. A minute (min) is 60 seconds in length (or, rarely, 59 or 61 seconds when leap seconds are employed), and an hour is 60 minutes or 3600 seconds in length. A day is usually 24 hours or 86,400 seconds in length; however, the duration of a calendar day can vary due to Daylight saving time and Leap seconds.
Time standards
A time standard is a specification for measuring time: assigning a number or calendar date to an instant (point in time), quantifying the duration of a time interval, and establishing a chronology (ordering of events). In modern times, several time specifications have been officially recognized as standards, where formerly they were matters of custom and practice. The invention in 1955 of the caesium atomic clock has led to the replacement of older and purely astronomical time standards such as sidereal time and ephemeris time, for most practical purposes, by newer time standards based wholly or partly on atomic time using the SI second.
International Atomic Time (TAI) is the primary international time standard from which other time standards are calculated. Universal Time (UT1) is mean solar time at 0° longitude, computed from astronomical observations. It varies from TAI because of the irregularities in Earth's rotation. Coordinated Universal Time (UTC) is an atomic time scale designed to approximate Universal Time. UTC differs from TAI by an integral number of seconds. UTC is kept within 0.9 second of UT1 by the introduction of one-second steps to UTC, the "leap second". The Global Positioning System broadcasts a very precise time signal based on UTC time.
The surface of the Earth is split up into a number of time zones. Standard time or civil time in a time zone deviates a fixed, round amount, usually a whole number of hours, from some form of Universal Time, usually UTC. Most time zones are exactly one hour apart, and by convention compute their local time as an offset from UTC. For example, time zones at sea are based on UTC. In many locations (but not at sea) these offsets vary twice yearly due to daylight saving time transitions.
Some other time standards are used mainly for scientific work. Terrestrial Time is a theoretical ideal scale realized by TAI. Geocentric Coordinate Time and Barycentric Coordinate Time are scales defined as coordinate times in the context of the general theory of relativity. Barycentric Dynamical Time is an older relativistic scale that is still in use.
Philosophy
Religion
Religions which view time as cyclical
Ancient cultures such as Incan, Mayan, Hopi, and other Native American Tribes – plus the Babylonians, ancient Greeks, Hinduism, Buddhism, Jainism, and others – have a concept of a wheel of time: they regard time as cyclical and quantic, consisting of repeating ages that happen to every being of the Universe between birth and extinction.[36]
Time as Linear for Abrahamic Religions
In general, the Islamic and Judeo-Christian world-view regards time as linear[37] and directional,[38] beginning with the act of creation by God. The traditional Christian view sees time ending, teleologically,[39] with the eschatological end of the present order of things, the "end time".
In the Old Testament book Ecclesiastes, traditionally ascribed to Solomon (970–928 BC), time (as the Hebrew word עידן, זמן iddan (age, as in "Ice age") zĕman(time) is often translated) was traditionally regarded as a medium for the passage of predestined events. (Another word, زمان" זמן" zamān, meant time fit for an event, and is used as the modern Arabic, Persian, and Hebrew equivalent to the English word "time".)
Time in Greek mythology
The Greek language denotes two distinct principles, Chronos and Kairos. The former refers to numeric, or chronological, time. The latter, literally "the right or opportune moment", relates specifically to metaphysical or Divine time. In theology, Kairos is qualitative, as opposed to quantitative.[40]
In Greek mythology, Chronos (ancient Greek: Χρόνος) is identified as the Personification of Time. His name in Greek means "time" and is alternatively spelled Chronus (Latin spelling) or Khronos. Chronos is usually portrayed as an old, wise man with a long, gray beard, such as "Father Time". Some English words whose etymological root is khronos/chronos include chronology, chronometer, chronic, anachronism, synchronise, and chronicle.
Time in Kabbalah
According to Kabbalists, "time" is a paradox[41] and an illusion.[42] Both the future and the past are recognised to be combined and simultaneously present.
In Western philosophy
Two contrasting viewpoints on time divide prominent philosophers. One view is that time is part of the fundamental structure of the universe – a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is sometimes referred to as Newtonian time.[43][44] The opposing view is that time does not refer to any kind of "container" that events and objects "move through", nor to any entity that "flows", but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events. This second view, in the tradition of Gottfried Leibniz[15] and Immanuel Kant,[45][46] holds that time is neither an event nor a thing, and thus is not itself measurable nor can it be travelled.
Furthermore, it may be that there is a subjective component to time, but whether or not time itself is "felt", as a sensation, or is a judgment, is a matter of debate.[2][6][7][47][48]
In Philosophy, time was questioned throughout the centuries; what time is and if it is real or not. Ancient Greek philosophers asked if time was linear or cyclical and if time was endless or finite.[49] These philosophers had different ways of explaining time; for instance, ancient Indian philosophers had something called the Wheel of Time. It is believed that there was repeating ages over the lifespan of the universe.[50] This led to beliefs like cycles of rebirth and reincarnation.[50] The Greek philosophers believe that the universe was infinite, and was an illusion to humans.[50] Plato believed that time was made by the Creator at the same instant as the heavens.[50] He also says that time is a period of motion of the heavenly bodies.[50] Aristotle believed that time correlated to movement, that time did not exist on its own but was relative to motion of objects.[50] He also believed that time was related to the motion of celestial bodies; the reason that humans can tell time was because of orbital periods and therefore there was a duration on time.[51]
The Vedas, the earliest texts on Indian philosophy and Hindu philosophy dating back to the late 2nd millennium BC, describe ancient Hindu cosmology, in which the universe goes through repeated cycles of creation, destruction and rebirth, with each cycle lasting 4,320 million years.[52] Ancient Greek philosophers, including Parmenides and Heraclitus, wrote essays on the nature of time.[53] Plato, in the Timaeus, identified time with the period of motion of the heavenly bodies. Aristotle, in Book IV of his Physica defined time as 'number of movement in respect of the before and after'.[54]
In Book 11 of his Confessions, St. Augustine of Hippo ruminates on the nature of time, asking, "What then is time? If no one asks me, I know: if I wish to explain it to one that asketh, I know not." He begins to define time by what it is not rather than what it is,[55] an approach similar to that taken in other negative definitions. However, Augustine ends up calling time a "distention" of the mind (Confessions 11.26) by which we simultaneously grasp the past in memory, the present by attention, and the future by expectation.
Isaac Newton believed in absolute space and absolute time; Leibniz believed that time and space are relational.[56] The differences between Leibniz's and Newton's interpretations came to a head in the famous Leibniz–Clarke correspondence.
Philosophers in the 17th and 18th century questioned if time was real and absolute, or if it was an intellectual concept that humans use to understand and sequence events.[49] These questions lead to realism vs anti-realism; the realists believed that time is a fundamental part of the universe, and be perceived by events happening in a sequence, in a dimension.[57] Isaac Newton said that we are merely occupying time, he also says that humans can only understand relative time.[57] Relative time is a measurement of objects in motion.[57] The anti-realists believed that time is merely a convenient intellectual concept for humans to understand events.[57] This means that time was useless unless there were objects that it could interact with, this was called relational time.[57] René Descartes, John Locke, and David Hume said that one's mind needs to acknowledge time, in order to understand what time is.[51] Immanuel Kant believed that we can not know what something is unless we experience it first hand.[58]
Time is not an empirical concept. For neither co-existence nor succession would be perceived by us, if the representation of time did not exist as a foundation a priori. Without this presupposition, we could not represent to ourselves that things exist together at one and the same time, or at different times, that is, contemporaneously, or in succession.
Immanuel Kant, Critique of Pure Reason (1781), trans. Vasilis Politis (London: Dent., 1991), p. 54.
Immanuel Kant, in the Critique of Pure Reason, described time as an a priori intuition that allows us (together with the other a priori intuition, space) to comprehend sense experience.[59] With Kant, neither space nor time are conceived as substances, but rather both are elements of a systematic mental framework that necessarily structures the experiences of any rational agent, or observing subject. Kant thought of time as a fundamental part of an abstract conceptual framework, together with space and number, within which we sequence events, quantify their duration, and compare the motions of objects. In this view, time does not refer to any kind of entity that "flows," that objects "move through," or that is a "container" for events. Spatial measurements are used to quantify the extent of and distances between objects, and temporal measurements are used to quantify the durations of and between events. Time was designated by Kant as the purest possible schema of a pure concept or category.
Henri Bergson believed that time was neither a real homogeneous medium nor a mental construct, but possesses what he referred to as Duration. Duration, in Bergson's view, was creativity and memory as an essential component of reality.[60]
According to Martin Heidegger we do not exist inside time, we are time. Hence, the relationship to the past is a present awareness of having been, which allows the past to exist in the present. The relationship to the future is the state of anticipating a potential possibility, task, or engagement. It is related to the human propensity for caring and being concerned, which causes "being ahead of oneself" when thinking of a pending occurrence. Therefore, this concern for a potential occurrence also allows the future to exist in the present. The present becomes an experience, which is qualitative instead of quantitative. Heidegger seems to think this is the way that a linear relationship with time, or temporal existence, is broken or transcended.[61] We are not stuck in sequential time. We are able to remember the past and project into the future – we have a kind of random access to our representation of temporal existence; we can, in our thoughts, step out of (ecstasis) sequential time.[62]
Modern era philosophers asked: is time real or unreal, is time happening all at once or a duration, is time tensed or tenseless, and is there a future to be?[49] There is a theory called the tenseless or B-theory; this theory says that any tensed terminology can be replaced with tenseless terminology.[63] For example, "we will win the game" can be replaced with "we do win the game", taking out the future tense. On the other hand, there is a theory called the tense or A-theory; this theory says that our language has tense verbs for a reason and that the future can not be determined.[63] There is also something called imaginary time, this was from Stephen Hawking, he says that space and imaginary time are finite but have no boundaries.[63] Imaginary time is not real or unreal, it is something that is hard to visualize.[63] Philosophers can agree that physical time exists outside of the human mind and is objective, and psychological time is mind-dependent and subjective.[51]
Unreality
In 5th century BC Greece, Antiphon the Sophist, in a fragment preserved from his chief work On Truth, held that: "Time is not a reality (hypostasis), but a concept (noêma) or a measure (metron)." Parmenides went further, maintaining that time, motion, and change were illusions, leading to the paradoxes of his follower Zeno.[64] Time as an illusion is also a common theme in Buddhist thought.[65][66]
J. M. E. McTaggart's 1908 The Unreality of Time argues that, since every event has the characteristic of being both present and not present (i.e., future or past), that time is a self-contradictory idea (see also The flow of time).
These arguments often center on what it means for something to be unreal. Modern physicists generally believe that time is as real as space – though others, such as Julian Barbour in his book The End of Time, argue that quantum equations of the universe take their true form when expressed in the timeless realm containing every possible now or momentary configuration of the universe, called "platonia" by Barbour.
A modern philosophical theory called presentism views the past and the future as human-mind interpretations of movement instead of real parts of time (or "dimensions") which coexist with the present. This theory rejects the existence of all direct interaction with the past or the future, holding only the present as tangible. This is one of the philosophical arguments against time travel. This contrasts with eternalism (all time: present, past and future, is real) and the growing block theory (the present and the past are real, but the future is not).
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Until Einstein's reinterpretation of the physical concepts associated with time and space in 1907, time was considered to be the same everywhere in the universe, with all observers measuring the same time interval for any event.[67] Non-relativistic classical mechanics is based on this Newtonian idea of time.
Einstein, in his special theory of relativity,[68] postulated the constancy and finiteness of the speed of light for all observers. He showed that this postulate, together with a reasonable definition for what it means for two events to be simultaneous, requires that distances appear compressed and time intervals appear lengthened for events associated with objects in motion relative to an inertial observer.
The theory of special relativity finds a convenient formulation in Minkowski spacetime, a mathematical structure that combines three dimensions of space with a single dimension of time. In this formalism, distances in space can be measured by how long light takes to travel that distance, e.g., a light-year is a measure of distance, and a meter is now defined in terms of how far light travels in a certain amount of time. Two events in Minkowski spacetime are separated by an invariant interval, which can be either space-like, light-like, or time-like. Events that have a time-like separation cannot be simultaneous in any frame of reference, there must be a temporal component (and possibly a spatial one) to their separation. Events that have a space-like separation will be simultaneous in some frame of reference, and there is no frame of reference in which they do not have a spatial separation. Different observers may calculate different distances and different time intervals between two events, but the invariant interval between the events is independent of the observer (and his or her velocity).
Arrow of time
Unlike space, where an object can travel in the opposite directions (and in 3 dimensions), time appears to have only one dimension and only one direction – the past lies behind, fixed and immutable, while the future lies ahead and is not necessarily fixed. Yet most laws of physics allow any process to proceed both forward and in reverse. There are only a few physical phenomena, that violate the reversibility of time. This time directionality is known as the arrow of time. Acknowledged examples of the arrow of time are:[69][70][71][72][73][74][75][76]
1. Radiative arrow of time, manifested in waves (e.g. light and sound) travelling only expanding (rather than focusing) in time (see light cone);
2. Entropic arrow of time: according to the second law of thermodynamics an isolated system evolves toward a larger disorder rather than orders spontaneously;
3. Quantum arrow time, which is related to irreversibility of measurement in quantum mechanics according to the Copenhagen interpretation of quantum mechanics;
4. Weak arrow of time: prefence for a certain time direction of weak force in particle physics (see violation of CP symmetry);
5. Cosmological arrow of time, which follows the accelerated expansion of the Universe after the Big Bang.
The relationship(s) between these different Arrows of Time is a hotly debated topic in theoretical physics.[77]
Classical mechanics
In non-relativistic classical mechanics, Newton's concept of "relative, apparent, and common time" can be used in the formulation of a prescription for the synchronization of clocks. Events seen by two different observers in motion relative to each other produce a mathematical concept of time that works sufficiently well for describing the everyday phenomena of most people's experience. In the late nineteenth century, physicists encountered problems with the classical understanding of time, in connection with the behavior of electricity and magnetism. Einstein resolved these problems by invoking a method of synchronizing clocks using the constant, finite speed of light as the maximum signal velocity. This led directly to the conclusion that observers in motion relative to one another measure different elapsed times for the same event.
Spacetime
Main article: Spacetime
Time has historically been closely related with space, the two together merging into spacetime in Einstein's special relativity and general relativity. According to these theories, the concept of time depends on the spatial reference frame of the observer, and the human perception, as well as the measurement by instruments such as clocks, are different for observers in relative motion. For example, if a spaceship carrying a clock flies through space at (very nearly) the speed of light, its crew does not notice a change in the speed of time on board their vessel because everything traveling at the same speed slows down at the same rate (including the clock, the crew's thought processes, and the functions of their bodies). However, to a stationary observer watching the spaceship fly by, the spaceship appears flattened in the direction it is traveling and the clock on board the spaceship appears to move very slowly.
On the other hand, the crew on board the spaceship also perceives the observer as slowed down and flattened along the spaceship's direction of travel, because both are moving at very nearly the speed of light relative to each other. Because the outside universe appears flattened to the spaceship, the crew perceives themselves as quickly traveling between regions of space that (to the stationary observer) are many light years apart. This is reconciled by the fact that the crew's perception of time is different from the stationary observer's; what seems like seconds to the crew might be hundreds of years to the stationary observer. In either case, however, causality remains unchanged: the past is the set of events that can send light signals to an entity and the future is the set of events to which an entity can send light signals.[78][79]
Dilation
Einstein showed in his thought experiments that people travelling at different speeds, while agreeing on cause and effect, measure different time separations between events, and can even observe different chronological orderings between non-causally related events. Though these effects are typically minute in the human experience, the effect becomes much more pronounced for objects moving at speeds approaching the speed of light. Subatomic particles exist for a well-known average fraction of a second in a lab relatively at rest, but when travelling close to the speed of light they are measured to travel farther and exist for much longer than when at rest. According to the special theory of relativity, in the high-speed particle's frame of reference, it exists, on the average, for a standard amount of time known as its mean lifetime, and the distance it travels in that time is zero, because its velocity is zero. Relative to a frame of reference at rest, time seems to "slow down" for the particle. Relative to the high-speed particle, distances seem to shorten. Einstein showed how both temporal and spatial dimensions can be altered (or "warped") by high-speed motion.
Einstein (The Meaning of Relativity): "Two events taking place at the points A and B of a system K are simultaneous if they appear at the same instant when observed from the middle point, M, of the interval AB. Time is then defined as the ensemble of the indications of similar clocks, at rest relative to K, which register the same simultaneously."
Einstein wrote in his book, Relativity, that simultaneity is also relative, i.e., two events that appear simultaneous to an observer in a particular inertial reference frame need not be judged as simultaneous by a second observer in a different inertial frame of reference.
Relativistic versus Newtonian
The animations visualise the different treatments of time in the Newtonian and the relativistic descriptions. At the heart of these differences are the Galilean and Lorentz transformations applicable in the Newtonian and relativistic theories, respectively.
In the figures, the vertical direction indicates time. The horizontal direction indicates distance (only one spatial dimension is taken into account), and the thick dashed curve is the spacetime trajectory ("world line") of the observer. The small dots indicate specific (past and future) events in spacetime.
The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. In both pictures the view of spacetime changes when the observer accelerates.
In the Newtonian description these changes are such that time is absolute:[80] the movements of the observer do not influence whether an event occurs in the 'now' (i.e., whether an event passes the horizontal line through the observer).
However, in the relativistic description the observability of events is absolute: the movements of the observer do not influence whether an event passes the "light cone" of the observer. Notice that with the change from a Newtonian to a relativistic description, the concept of absolute time is no longer applicable: events move up and down in the figure depending on the acceleration of the observer.
Quantization
Time quantization is a hypothetical concept. In the modern established physical theories (the Standard Model of Particles and Interactions and General Relativity) time is not quantized.
Planck time (~ 5.4 × 10−44 seconds) is the unit of time in the system of natural units known as Planck units. Current established physical theories are believed to fail at this time scale, and many physicists expect that the Planck time might be the smallest unit of time that could ever be measured, even in principle. Tentative physical theories that describe this time scale exist; see for instance loop quantum gravity.
Thermodynamics
The second law of thermodynamics states that entropy must increase over time (see Entropy). This can be in either direction – Brian Greene theorizes that, according to the equations, the change in entropy occurs symmetrically whether going forward or backward in time. So entropy tends to increase in either direction, and our current low-entropy universe is a statistical aberration, in a similar manner as tossing a coin often enough that eventually heads will result ten times in a row. However, this theory is not supported empirically in local experiment.[81]
Travel
Time travel is the concept of moving backwards or forwards to different points in time, in a manner analogous to moving through space, and different from the normal "flow" of time to an earthbound observer. In this view, all points in time (including future times) "persist" in some way. Time travel has been a plot device in fiction since the 19th century. Travelling backwards or forwards in time has never been verified as a process, and doing so presents many theoretical problems and contradictive logic which to date have not been overcome. Any technological device, whether fictional or hypothetical, that is used to achieve time travel is known as a time machine.
A central problem with time travel to the past is the violation of causality; should an effect precede its cause, it would give rise to the possibility of a temporal paradox. Some interpretations of time travel resolve this by accepting the possibility of travel between branch points, parallel realities, or universes.
Another solution to the problem of causality-based temporal paradoxes is that such paradoxes cannot arise simply because they have not arisen. As illustrated in numerous works of fiction, free will either ceases to exist in the past or the outcomes of such decisions are predetermined. As such, it would not be possible to enact the grandfather paradox because it is a historical fact that one's grandfather was not killed before his child (one's parent) was conceived. This view does not simply hold that history is an unchangeable constant, but that any change made by a hypothetical future time traveller would already have happened in his or her past, resulting in the reality that the traveller moves from. More elaboration on this view can be found in the Novikov self-consistency principle.
Perception
The specious present refers to the time duration wherein one's perceptions are considered to be in the present. The experienced present is said to be 'specious' in that, unlike the objective present, it is an interval and not a durationless instant. The term specious present was first introduced by the psychologist E.R. Clay, and later developed by William James.[82]
Biopsychology
The brain's judgment of time is known to be a highly distributed system, including at least the cerebral cortex, cerebellum and basal ganglia as its components. One particular component, the suprachiasmatic nuclei, is responsible for the circadian (or daily) rhythm, while other cell clusters appear capable of shorter-range (ultradian) timekeeping.
Psychoactive drugs can impair the judgment of time. Stimulants can lead both humans and rats to overestimate time intervals,[83][84] while depressants can have the opposite effect.[85] The level of activity in the brain of neurotransmitters such as dopamine and norepinephrine may be the reason for this.[86] Such chemicals will either excite or inhibit the firing of neurons in the brain, with a greater firing rate allowing the brain to register the occurrence of more events within a given interval (speed up time) and a decreased firing rate reducing the brain's capacity to distinguish events occurring within a given interval (slow down time).[87]
Mental chronometry is the use of response time in perceptual-motor tasks to infer the content, duration, and temporal sequencing of cognitive operations.
Early childhood education
Children's expanding cognitive abilities allow them to understand time more clearly. Two- and three-year-olds' understanding of time is mainly limited to "now and not now". Five- and six-year-olds can grasp the ideas of past, present, and future. Seven- to ten-year-olds can use clocks and calendars.[88]
Alterations
In addition to psychoactive drugs, judgments of time can be altered by temporal illusions (like the kappa effect),[89] age,[90] and hypnosis.[91] The sense of time is impaired in some people with neurological diseases such as Parkinson's disease and attention deficit disorder.
Psychologists assert that time seems to go faster with age, but the literature on this age-related perception of time remains controversial.[92] Those who support this notion argue that young people, having more excitatory neurotransmitters, are able to cope with faster external events.[87]
Spatial conceptualization
Although time is regarded as an abstract concept, there is increasing evidence that time is conceptualized in the mind in terms of space.[93] That is, instead of thinking about time in a general, abstract way, humans think about time in a spatial way and mentally organize it as such. Using space to think about time allows humans to mentally organize temporal events in a specific way.
This spatial representation of time is often represented in the mind as a Mental Time Line (MTL).[94] Using space to think about time allows humans to mentally organize temporal order. These origins are shaped by many environmental factors[93]––for example, literacy appears to play a large role in the different types of MTLs, as reading/writing direction provides an everyday temporal orientation that differs from culture to culture.[94] In western cultures, the MTL may unfold rightward (with the past on the left and the future on the right) since people read and write from left to right.[94] Western calendars also continue this trend by placing the past on the left with the future progressing toward the right. Conversely, Arabic, Farsi, Urdu and Israeli-Hebrew speakers read from right to left, and their MTLs unfold leftward (past on the right with future on the left), and evidence suggests these speakers organize time events in their minds like this as well.[94]
This linguistic evidence that abstract concepts are based in spatial concepts also reveals that the way humans mentally organize time events varies across cultures––that is, a certain specific mental organization system is not universal. So, although Western cultures typically associate past events with the left and future events with the right according to a certain MTL, this kind of horizontal, egocentric MTL is not the spatial organization of all cultures. Although most developed nations use an egocentric spatial system, there is recent evidence that some cultures use an allocentric spatialization, often based on environmental features.[93]
A study of the indigenous Yupno people of Papua New Guinea focused on the directional gestures used when individuals used time-related words.[93] When speaking of the past (such as "last year" or "past times"), individuals gestured downhill, where the river of the valley flowed into the ocean. When speaking of the future, they gestured uphill, toward the source of the river. This was common regardless of which direction the person faced, revealing that the Yupno people may use an allocentric MTL, in which time flows uphill.[93]
A similar study of the Pormpuraawans, an aboriginal group in Australia, revealed a similar distinction in which when asked to organize photos of a man aging "in order," individuals consistently placed the youngest photos to the east and the oldest photos to the west, regardless of which direction they faced.[95] This directly clashed with an American group that consistently organized the photos from left to right. Therefore, this group also appears to have an allocentric MTL, but based on the cardinal directions instead of geographical features.[95]
The wide array of distinctions in the way different groups think about time leads to the broader question that different groups may also think about other abstract concepts in different ways as well, such as causality and number.[93]
Use
In sociology and anthropology, time discipline is the general name given to social and economic rules, conventions, customs, and expectations governing the measurement of time, the social currency and awareness of time measurements, and people's expectations concerning the observance of these customs by others. Arlie Russell Hochschild[96][97] and Norbert Elias[98] have written on the use of time from a sociological perspective.
The use of time is an important issue in understanding human behavior, education, and travel behavior. Time-use research is a developing field of study. The question concerns how time is allocated across a number of activities (such as time spent at home, at work, shopping, etc.). Time use changes with technology, as the television or the Internet created new opportunities to use time in different ways. However, some aspects of time use are relatively stable over long periods of time, such as the amount of time spent traveling to work, which despite major changes in transport, has been observed to be about 20–30 minutes one-way for a large number of cities over a long period.
Time management is the organization of tasks or events by first estimating how much time a task requires and when it must be completed, and adjusting events that would interfere with its completion so it is done in the appropriate amount of time. Calendars and day planners are common examples of time management tools.
Sequence of events
A sequence of events, or series of events, is a sequence of items, facts, events, actions, changes, or procedural steps, arranged in time order (chronological order), often with causality relationships among the items.[99][100][101] Because of causality, cause precedes effect, or cause and effect may appear together in a single item, but effect never precedes cause. A sequence of events can be presented in text, tables, charts, or timelines. The description of the items or events may include a timestamp. A sequence of events that includes the time along with place or location information to describe a sequential path may be referred to as a world line.
Uses of a sequence of events include stories,[102] historical events (chronology), directions and steps in procedures,[103] and timetables for scheduling activities. A sequence of events may also be used to help describe processes in science, technology, and medicine. A sequence of events may be focused on past events (e.g., stories, history, chronology), on future events that must be in a predetermined order (e.g., plans, schedules, procedures, timetables), or focused on the observation of past events with the expectation that the events will occur in the future (e.g., processes, projections). The use of a sequence of events occurs in fields as diverse as machines (cam timer), documentaries (Seconds From Disaster), law (choice of law), finance (directional-change intrinsic time), computer simulation (discrete event simulation), and electric power transmission[104] (sequence of events recorder). A specific example of a sequence of events is the timeline of the Fukushima Daiichi nuclear disaster.
See also
• List of UTC timing centers
• Loschmidt's paradox
• Time metrology
Organizations
• Antiquarian Horological Society – AHS (United Kingdom)
• Chronometrophilia (Switzerland)
• Deutsche Gesellschaft für Chronometrie – DGC (Germany)
• National Association of Watch and Clock Collectors – NAWCC (United States)
Miscellaneous arts and sciences
• Date and time representation by country
• List of cycles
• Nonlinear narrative
• Philosophy of physics
• Rate (mathematics)
Miscellaneous units
• Fiscal year
• Half-life
• Hexadecimal time
• Tithi
• Unix epoch
References
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80. Andersen, Holly; Rick Grush (2009). "A brief history of time-consciousness: historical precursors to James and Husserl" (PDF). Journal of the History of Philosophy. 47 (2): 277–307. doi:10.1353/hph.0.0118. S2CID 16379171. Archived from the original (PDF) on 16 February 2008. Retrieved 9 April 2011.
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82. Cheng, Ruey-Kuang; Macdonald, Christopher J.; Meck, Warren H. (2006). "Differential effects of cocaine and ketamine on time estimation: Implications for neurobiological models of interval timing" (online abstract). Pharmacology Biochemistry and Behavior. 85 (1): 114–122. doi:10.1016/j.pbb.2006.07.019. PMID 16920182. S2CID 42295255. Archived from the original on 10 August 2011. Retrieved 9 April 2011.
83. Tinklenberg, Jared R.; Walton T. Roth1; Bert S. Kopell (January 1976). "Marijuana and ethanol: Differential effects on time perception, heart rate, and subjective response". Psychopharmacology. 49 (3): 275–279. doi:10.1007/BF00426830. PMID 826945. S2CID 25928542.
84. Arzy, Shahar; Istvan Molnar-Szakacs; Olaf Blanke (18 June 2008). "Self in Time: Imagined Self-Location Influences Neural Activity Related to Mental Time Travel". The Journal of Neuroscience. 28 (25): 6502–6507. doi:10.1523/JNEUROSCI.5712-07.2008. PMC 6670885. PMID 18562621.
85. Carter, Rita (2009). The Human Brain Book. Dorling Kindersley Publishing. pp. 186–187. ISBN 978-0-7566-5441-2. Archived from the original on 13 May 2016. Retrieved 27 February 2016.
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Further reading
• Barbour, Julian (1999). The End of Time: The Next Revolution in Our Understanding of the Universe. Oxford University Press. ISBN 978-0-19-514592-2.
• Craig Callendar, Introducing Time, Icon Books, 2010, ISBN 978-1-84831-120-6
• Das, Tushar Kanti (1990). The Time Dimension: An Interdisciplinary Guide. New York: Praeger. ISBN 978-0-275-92681-6. – Research bibliography
• Davies, Paul (1996). About Time: Einstein's Unfinished Revolution. New York: Simon & Schuster Paperbacks. ISBN 978-0-684-81822-1.
• Feynman, Richard (1994) [1965]. The Character of Physical Law. Cambridge (Mass): The MIT Press. pp. 108–126. ISBN 978-0-262-56003-0.
• Galison, Peter (1992). Einstein's Clocks and Poincaré's Maps: Empires of Time. New York: W.W. Norton. ISBN 978-0-393-02001-4.
• Benjamin Gal-Or, Cosmology, Physics and Philosophy, Springer Verlag, 1981, 1983, 1987, ISBN 0-387-90581-2, 0-387-96526-2.
• Charlie Gere, (2005) Art, Time and Technology: Histories of the Disappearing Body, Berg
• Highfield, Roger (1992). Arrow of Time: A Voyage through Science to Solve Time's Greatest Mystery. Random House. ISBN 978-0-449-90723-8.
• Landes, David (2000). Revolution in Time. Harvard University Press. ISBN 978-0-674-00282-1.
• Lebowitz, Joel L. (2008). "Time's arrow and Boltzmann's entropy". Scholarpedia. 3 (4): 3448. Bibcode:2008SchpJ...3.3448L. doi:10.4249/scholarpedia.3448.
• Mermin, N. David (2005). It's About Time: Understanding Einstein's Relativity. Princeton University Press. ISBN 978-0-691-12201-4.
• Morris, Richard (1985). Time's Arrows: Scientific Attitudes Toward Time. New York: Simon and Schuster. ISBN 978-0-671-61766-0.
• Penrose, Roger (1999) [1989]. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. New York: Oxford University Press. pp. 391–417. ISBN 978-0-19-286198-6. Archived from the original on 26 December 2010. Retrieved 9 April 2011.
• Price, Huw (1996). Time's Arrow and Archimedes' Point. Oxford University Press. ISBN 978-0-19-511798-1. Retrieved 9 April 2011.
• Reichenbach, Hans (1999) [1956]. The Direction of Time. New York: Dover. ISBN 978-0-486-40926-9.
• Rovelli, Carlo (2006). What is time? What is space?. Rome: Di Renzo Editore. ISBN 978-88-8323-146-9. Archived from the original on 27 January 2007.
• Rovelli, Carlo (2018). The Order of Time. New York: Riverhead. ISBN 978-0735216105.
• Stiegler, Bernard, Technics and Time, 1: The Fault of Epimetheus
• Roberto Mangabeira Unger and Lee Smolin, The Singular Universe and the Reality of Time, Cambridge University Press, 2014, ISBN 978-1-107-07406-4.
• Whitrow, Gerald J. (1973). The Nature of Time. Holt, Rinehart and Wilson (New York).
• Whitrow, Gerald J. (1980). The Natural Philosophy of Time. Clarendon Press (Oxford).
• Whitrow, Gerald J. (1988). Time in History. The evolution of our general awareness of time and temporal perspective. Oxford University Press. ISBN 978-0-19-285211-3.
External links
• Different systems of measuring time
• Time on In Our Time at the BBC
• Time in the Internet Encyclopedia of Philosophy, by Bradley Dowden.
• Le Poidevin, Robin (Winter 2004). "The Experience and Perception of Time". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy. Retrieved 9 April 2011.
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| Wikipedia |
Time–frequency representation
A time–frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency.[1] Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved by using a formulation often called "Time–Frequency Distribution", abbreviated as TFD.
See also: Time–frequency analysis
TFRs are often complex-valued fields over time and frequency, where the modulus of the field represents either amplitude or "energy density" (the concentration of the root mean square over time and frequency), and the argument of the field represents phase.
Background and motivation
A signal, as a function of time, may be considered as a representation with perfect time resolution. In contrast, the magnitude of the Fourier transform (FT) of the signal may be considered as a representation with perfect spectral resolution but with no time information because the magnitude of the FT conveys frequency content but it fails to convey when, in time, different events occur in the signal.
TFRs provide a bridge between these two representations in that they provide some temporal information and some spectral information simultaneously. Thus, TFRs are useful for the representation and analysis of signals containing multiple time-varying frequencies.
Formulation of TFRs and TFDs
One form of TFR (or TFD) can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such representations and formulations are known as quadratic or "bilinear" TFRs or TFDs (QTFRs or QTFDs) because the representation is quadratic in the signal (see Bilinear time–frequency distribution). This formulation was first described by Eugene Wigner in 1932 in the context of quantum mechanics and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the Wigner–Ville distribution, as it was shown in [2] that Wigner's formula needed to use the analytic signal defined in Ville's paper to be useful as a representation and for a practical analysis. Today, QTFRs include the spectrogram (squared magnitude of short-time Fourier transform), the scaleogram (squared magnitude of Wavelet transform) and the smoothed pseudo-Wigner distribution.
Although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms, also called "interferences". The cross-terms caused by the bilinear structure of TFDs and TFRs may be useful in some applications such as classification as the cross-terms provide extra detail for the recognition algorithm. However, in some other applications, these cross-terms may plague certain quadratic TFRs and they would need to be reduced. One way to do this is obtained by comparing the signal with a different function. Such resulting representations are known as linear TFRs because the representation is linear in the signal. An example of such a representation is the windowed Fourier transform (also known as the short-time Fourier transform) which localises the signal by modulating it with a window function, before performing the Fourier transform to obtain the frequency content of the signal in the region of the window.
Wavelet transforms
Wavelet transforms, in particular the continuous wavelet transform, expand the signal in terms of wavelet functions which are localised in both time and frequency. Thus the wavelet transform of a signal may be represented in terms of both time and frequency.
The notions of time, frequency, and amplitude used to generate a TFR from a wavelet transform were originally developed intuitively. In 1992, a quantitative derivation of these relationships was published, based upon a stationary phase approximation.[3]
Linear canonical transformation
Main article: Linear canonical transformation
Linear canonical transformations are the linear transforms of the time–frequency representation that preserve the symplectic form. These include and generalize the Fourier transform, fractional Fourier transform, and others, thus providing a unified view of these transforms in terms of their action on the time–frequency domain.
See also
• Newland transform
• Reassignment method
• Time–frequency analysis for music signals
References
1. E. Sejdić, I. Djurović, J. Jiang, "Time-frequency feature representation using energy concentration: An overview of recent advances," Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009.
2. B. Boashash, "Note on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech. and Signal Processing, vol. 36, issue 9, pp 1518–1521, Sept. 1988. doi:10.1109/29.90380
3. Delprat, N., Escudii, B., Guillemain, P., Kronland-Martinet, R., Tchamitchian, P., and Torrksani, B. (1992). "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies". IEEE Transactions on Information Theory. 38 (2): 644–664. doi:10.1109/18.119728.{{cite journal}}: CS1 maint: multiple names: authors list (link)
External links
• DiscreteTFDs — software for computing time–frequency distributions
• TFTB — Time–Frequency ToolBox
• Time stretched short time Fourier transform for time-frequency analysis of ultra wideband signals
| Wikipedia |
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