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| {{Infobox scientist
| | My name is Weldon Pinckney. I life in Bloomington (United States).<br><br>Feel free to surf to my page :: [http://speedtax.com.au/bbs/index.php?mid=Speedtax_qna&page=1&listStyle=webzine&document_srl=2911919 Bookbyte Coupon November 2014] |
| | name = Eduard Study
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| | image = EduardStudy.jpg
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| | image_size =
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| | caption =
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| | birth_date = {{Birth date|1862|03|23|df=y}}
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| | birth_place = [[Coburg]]
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| | death_date = {{Death date and age|1930|01|06|1862|03|23|df=y}}
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| | death_place = [[Bonn]]
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| | nationality = German
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| | fields = [[Mathematics]]
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| | workplaces =
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| | alma_mater = [[Munich]]
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| | doctoral_advisor = [[Philipp Ludwig Seidel]]<br/>[[Gustav Conrad Bauer]]
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| | doctoral_students =
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| | known_for = ''Geometrie der Dynamen''<br/>[[Invariant theory]]<br/>[[Spherical trigonometry]]
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| | awards =
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| }}
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| '''Eduard Study''' (March 23, 1862 – January 6, 1930) was a [[Germany|German]] [[mathematician]] known for work on [[invariant theory]] of ternary forms (1889) and for the study of [[spherical trigonometry]]. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.
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| Study was born in [[Coburg]] in the Duchy of [[Saxe-Coburg-Gotha]]. He died in [[Bonn]].
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| ==Career==
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| Eduard Study began his university career in Jena, Strasbourg, Leipzig, and Munich. He loved to study biology, especially entomology. He was awarded the doctorate in mathematics at the [[University of Munich]] in 1884. [[Paul Gordan]], an expert in [[invariant theory]] was at Leipzig, and Study returned there as Privatdozent. In 1888 he moved to Marburg and in 1893 embarked on a speaking tour in the U.S.A. He appeared at the primordial International Congress of Mathematicians in Chicago as part of the [[World's Columbian Exposition]] and took part in mathematics at [[Johns Hopkins University]]. Back in Germany, in 1894, he was appointed extraordinary professor at Göttingen. Then he gained the rank of full professor in 1897 at Greifswald. In 1904 he was called to the [[University of Bonn]] as the position held by [[Rudolf Lipschitz]] was vacant. There he settled until retirement in 1927.
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| ==Euclidean space group and dual quaternions==
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| In 1891 Eduard Study published "Of Motions and Translations, in two parts". It treats the [[Euclidean group]] E(3). The second part of his article
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| introduces the [[associative algebra]] of [[dual quaternion]]s, that is numbers
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| :<math>q = a + bi + cj + dk \!</math>
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| where ''a'', ''b'', ''c'', and ''d'' are [[dual numbers]] and {1, ''i'', ''j'', ''k''} multiply as in the [[quaternion group]]. Actually Study uses notation such that
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| :<math>e_0 = 1,\ e_1 = i,\ e_2 = j,\ e_3 = k, \!</math>
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| :<math>\varepsilon _0 = \varepsilon ,\ \varepsilon _1 = \varepsilon i,\ \varepsilon _2 = \varepsilon j,\ \varepsilon _3 = \varepsilon k. \!</math>
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| The multiplication table is found on page 520 of volume 39 (1891) in [[Mathematische Annalen]] under the title "Von Bewegungen und Umlegungen, I. und II. Abhandlungen".
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| Eduard Study cites [[William Kingdon Clifford]] as an earlier source on these [[biquaternion]]s. In 1901 Study published ''Geometrie der Dynamen''<ref>E. Study (1903) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03150002 Geometrie der Dynamen], from ''Historical Math Monographs'' at [[Cornell University]]</ref> also using dual quaternions. In 1913 he wrote a review article treating both E(3) and [[elliptic geometry]]. This article, "Foundations and goals of analytical kinematics"<ref>E. Study (1913), Delphinich translator, [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study-analytical_kinematics.pdf "Foundations and goals of analytical kinematics"] from Neo-classical physics</ref> develops the field of [[kinematics]], in particular exhibiting an element of E(3) as a [[screw theory#Homography|homography of dual quaternions]].
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| Study's use of [[abstract algebra]] was noted in ''A History of Algebra'' (1985) by [[B. L. van der Waerden]]. On the other hand, Joe Rooney recounts these developments in relation to kinematics.<ref>Joe Rooney [http://oro.open.ac.uk/8455/01/chapter4(020507).pdf William Kingdon Clifford], Department of Design and Innovation, the Open University, London.</ref>
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| ==Hypercomplex numbers==
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| {{Main|Hypercomplex number}}
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| Study showed an early interest in systems of complex numbers and their application to transformation groups with his article in 1890.<ref>E. Study (1890) D.H. Delphenich translator, [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study_-_complex_numbers_and_transformation_groups.pdf "On systems of complex numbers and their applications to the theory of transformation groups"]</ref> He addressed this popular subject again in 1898 in [[Klein's encyclopedia]]. The essay explored [[quaternion]]s and other hypercomplex number systems.<ref>E. Study (1898) "Theorie der gemeinen und höhern komplexen Grössen", [[Klein's encyclopedia|''Encyclopädie der mathematischen Wissenschaften]] I A '''4''' 147–83 </ref> This 34 page article was expanded to 138 pages in 1908 by [[Élie Cartan]], who surveyed the hypercomplex systems in ''Encyclopédie des sciences mathématiques pures et appliqueés''. Cartan acknowledged Eduard Study's guidance, in his title, with the words "after Eduard Study".
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| In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads:<ref>M.A. Akivis & B.A. Rosenfeld (1993) ''Elie Cartan (1869 — 1951)'', [[American Mathematical Society]], pp. 68–9</ref>
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| : [Study] defined the algebra °'''H''' of 'semiquaternions' with the units 1, ''i'', ''ε'', ''η'' having the properties <math>i^2 = -1, \ \varepsilon ^2 = 0, \ i \varepsilon = - \varepsilon i = \eta. \! </math> | |
| : Semiquaternions are often called 'Study's quaternions'.
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| In 1985 Helmut Karzel and Günter Kist developed "Study's quaternions" as the kinematic algebra corresponding to the [[group of motions]] of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ring of [[2 × 2 real matrices]] which Karzel and Kist cast as the kinematic algebras of the elliptic plane and hyperbolic plane respectively. See the "Motivation and Historical Review" at page 437 of ''Rings and Geometry'', R. Kaya editor.
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| Some of the other hypercomplex systems that Study worked with are [[dual numbers]], [[dual quaternion]]s, and [[split-biquaternion]]s, all being
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| [[associative algebra]]s over '''R'''.
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| ==Ruled surfaces==
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| Study's work with [[dual number]]s and [[line coordinates]] was noted by [[Heinrich Guggenheimer]] in 1963 in his book ''Differential Geometry'' (see pages 162–5). He cites and proves the following theorem of Study: The oriented lines in '''R'''<sup>3</sup> are in one-to-one correspondence with the points of the dual unit sphere in '''D'''<sup>3</sup>. Later he says "A differentiable curve '''A'''(''u'') on the dual unit sphere, depending on a ''real'' parameter ''u'', represents a differentiable family of straight lines in '''R'''<sup>3</sup>: a [[ruled surface]]. The lines '''A'''(''u'') are the ''generators'' or ''rulings'' of the surface." Guggenheimer also shows the representation of the Euclidean motions in '''R'''<sup>3</sup> by orthogonal dual matrices.
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| ==Hermitian form metric==
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| In 1905 Study wrote "Kürzeste Weg im complexen Gebiet" (Shortest path in complex domains) for [[Mathematische Annalen]] (60:321–378). Some of its contents were anticipated by [[Guido Fubini]] a year before. The distance Study refers to is a [[Sesquilinear form#Hermitian form|Hermitian form]] on [[complex projective space]]. Since then this [[Metric (mathematics)|metric]] has been called the [[Fubini–Study metric]]. Study was careful in 1905 to distinguish the hyperbolic and elliptic cases in Hermitian geometry.
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| ==Valence theory==
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| Somewhat surprisingly Eduard Study is known by practitioners of [[quantum chemistry]]. Like [[James Joseph Sylvester]], [[Paul Gordan]] believed that invariant theory could contribute to the understanding of [[valence (chemistry)|chemical valence]]. In 1900 Gordan and his student G. Alexejeff contributed an article on an analogy between the [[angular momentum coupling|coupling problem for angular momenta]] and their work on invariant theory to the ''Zeitschrift für Physikalische Chemie'' (v. 35, p. 610). In 2006 Wormer and Paldus summarized Study's role as follows:<ref>Paul E.S. Wormer and [[Josef Paldus]] (2006) [http://www.theochem.ru.nl/files/dbase/aqc-51-59-2006.pdf Angular Momentum Diagrams] Advances in Quantum Chemistry, v. 51, pp. 51–124</ref>
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| : The analogy, lacking a physical basis at the time, was criticised heavily by the '''mathematician E. Study''' and ignored completely by the chemistry community of the 1890s. After the advent of quantum mechanics it became clear, however, that chemical valences arise from electron-spin couplings ... and that electron spin functions are, in fact, binary forms of the type studied by [[Clebsch-Gordan coefficients|Gordan and Clebsch]].
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| ==Cited publications==
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| * ''Sphärische Trigonometrie, orthogonale Substitutionen, und elliptische Functionen: Eine Analytisch-Geometrische Untersuchung.'' Leipzig, Germany: Teubner, 1893.
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| * ''Aeltere und neuere Untersuchungen uber Systeme complexer Zahlen'', Mathematical Papers Chicago Congress.
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| * ''Einleitung in die Theorie der Invarianten'' (1923).
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| * [http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN360504671&DMDID=dmdlog116 Theorie der allgemeinen und höheren komplexen Grossen] in ''Encyklopädie der mathematischen Wissenschaften'', weblink to [[University of Göttingen]].
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| ==References==
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| {{Reflist}}
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| * Werner Burau (1970) "Eduard Study" in [[Dictionary of Scientific Biography]].
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| * E.A. Weiss (1930) "E. Study", ''Sitzungsberichte der Berliner mathematischen Gesellschaft'' 10:52–77.
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| ==External links==
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| * {{MacTutor Biography|id=Study}}
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| * [http://genealogy.math.ndsu.nodak.edu/id.php?id=19466 Eduard Study] at [[Mathematics Genealogy Project]]
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| *[http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248563 Photo of Study]
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| *[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study_-_appendix.pdf Appendix to Geometrie der Dynamen on the foundations of kinematics] (English translation)
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| *[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study-analytical_kinematics.pdf "Foundations and goals of analytical kinematics"] (English translation)
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| *[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study_-_a_new_branch_of_geometry.pdf "A New Branch of Geometry"](English translation)
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| *[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study_-_non-euclidian_and_line_geometry.pdf "On non-Euclidian and line geometry"] (English translation)
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| {{Authority control|VIAF=19752182}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Study, Eduard
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = German mathematician
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| | DATE OF BIRTH = March 23, 1862
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| | PLACE OF BIRTH =
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| | DATE OF DEATH = January 6, 1930
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Study, Eduard}}
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| [[Category:1862 births]]
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| [[Category:1930 deaths]]
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| [[Category:19th-century German mathematicians]]
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| [[Category:20th-century mathematicians]]
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| [[Category:German mathematicians]]
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| [[Category:People from Coburg]]
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| [[Category:University of Bonn faculty]]
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