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Eduard Study (March 23, 1862 – January 6, 1930) was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. Year 1862 was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common year starting on Monday Events 1066 - Harold Godwinson is crowned King of England. 1205 - Philip of Swabia becomes King Year 1930 ( MCMXXX) was a Common year starting on Wednesday (link will display 1930 calendar of the Gregorian calendar. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. A mathematician is a person whose primary area of study and research is the field of Mathematics. Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect Ternary form is a structuring mechanism of a piece of music Along with several other Musical forms ternary form can also be applied to dance choreography Year 1889 ( MDCCCLXXXIX) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations He is also know for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.

Study was born in Coburg in the Duchy of Saxe-Coburg-Gotha. Coburg is a town located on the Itz River in Bavaria, Germany Saxe-Coburg and Gotha or Saxe-Coburg-Gotha (Sachsen-Coburg und Gotha served as the name of the two German duchies of Saxe-Coburg and Saxe-Gotha in

Contents

Career

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. The Ludwig Maximilian University of Munich (Ludwig-Maximilians-Universität München also known as LMU, is a University in Munich and with more Paul Albert Gordan, an expert in invariant theory was at Leipzig, and Study returned there as Privatdozent. Paul Albert Gordan ( 27 April 1837 &ndash 21 December 1912) was a German Mathematician, a student of Carl Jacobi Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect 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 Exhibition (see external link below), and took part in mathematics at Johns Hopkins University. The World's Columbian Exposition (also called The Chicago World's Fair) a World's Fair, was held in Chicago in 1893 to celebrate the 400th anniversary Back in Germany, in 1894, he was appointed extraordinary professor at Gottingen. 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. The University of Bonn ( German: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, Germany Rudolf Otto Sigismund Lipschitz ( May 14, 1832 &ndash October 7, 1903) was a German Mathematician and professor at the There he settled until retirement in 1927.

Euclidean space group and biquaternions

In 1891 Eduard Study published “Of Motions and Translations, in two parts”. It treats Euclidean space through the space group. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure The second part of his article constructs a seven-dimensional space out of “dual biquaternions”, that is numbers

q = a + bi + cj + dk \!

where a,b,c, and d are dual numbers and {1, i, j, k} multiply as in the quaternion group. A variety of dualities in mathematics are listed at Duality (mathematics. In Group theory, the quaternion group is a non-abelian group of order 8 He uses these conventions:

e_0 = 1,\ e_1 = i,\ e_2 = j,\ e_3 = k \!,
\epsilon _0 = \epsilon ,\ \epsilon _1 = \epsilon i,\ \epsilon _2 = \epsilon j,\ \epsilon _3 = \epsilon k \!.

The multiplication table is found on page 520 of volume 39 (1891) in Mathematische Annalen under the title “Von Bewegungen und Umlegungen, I. The Mathematische Annalen (abbreviated as Math Ann or Math Annal und II. Abhandlungen”. Eduard Study cites William Kingdon Clifford as an earlier source on these biquaternions. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and In 1901 Study published Geometrie der Dynamen to highlight the applications of this algebra. Due to Eduard Study’s profound and early exploitation of this eight-dimensional associative algebra, it is frequently referred to as Study Biquaternions. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Study’s achievement is celebrated, for example, in A History of Algebra (1985) by B. L. van der Waerden, who also cites Clifford’s earlier note. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich,

Since the space group is important in robotics, the Study biquaternions are a technical tool, now sometimes referred to as dual quaternions. See also Robot Robotics is the science and technology of Robots and their design manufacture and application In Ring theory, dual quaternions are a non-commutative ring constructed in the same way as the quaternions, except using Dual numbers For example, Joe Rooney has profiled the use of this algebra by several modelers of mechanics (see external link).

Hypercomplex numbers

In 1898 Eduard Study was the author of an article on hypercomplex numbers in the German Encyclopedia of Mathematics. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic This 34 page effort was expanded to 138 pages in 1908 by Elie Cartan, who succeeded in classifying Lie algebras. Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Cartan acknowledged Eduard Study’s essay in his title with the words “after Eduard Study”. In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads:

“[Study] defined the algebra °H of ‘semiquaternions’ with the units 1, i, ε, η having the properties i^2 = -1, \ \epsilon ^2 = 0, \ i \epsilon = - \epsilon i = \eta \! .
Semiquaternions are often called ‘Study’s quaternions’”.

Thus in the study of classical associative algebras over R there are two special ones: Study’s quaternions (4D) and Study’s biquaternions (8D). In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

Valence theory

Somewhat surprisingly Eduard Study is known by practitioners of quantum chemistry. Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in Like James Joseph Sylvester, Paul Gordan believed that invariant theory could contribute to the understanding of chemical valence. James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician Paul Albert Gordan ( 27 April 1837 &ndash 21 December 1912) was a German Mathematician, a student of Carl Jacobi In Chemistry, valence, also known as valency or valency number, is a measure of the number of Chemical bonds formed by the Atoms In 1900 Gordan and his student G. Alexejeff contributed an article on an analogy between the coupling problem for angular momenta and their work on invariant theory to the Zeitschrift für Physikalische Chemie (v. In Quantum mechanics, the procedure of constructing Eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling 35, p. 610). In 2006 Wormer and Paldus summarized Study's role as follows:

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 1890's. 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 Gordan and Clebsch. In Physics, the Clebsch-Gordan coefficients are sets of numbers that arise in Angular momentum coupling under the laws of Quantum mechanics.

Cited Publications

References

External references


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