Whereas 2 × 3 and 3 × 2 both equal 6, order matters for quaternion multiplication.
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To see quaternion multiplication in action, watch the newly released video below by the popular math animator 3Blue19203Brown.
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On the other hand, if you find yourself wondering, "WTF is a quaternion?!" you'll probably still love this course.
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You need two full rotations of the phone or arrow to bring the associated quaternion back to its initial state.
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While a phone or arrow turns all the way around in 360 degrees, the quaternion describing this 360-degree rotation only turns 180 degrees up in four-dimensional space.
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Upside-down arrows produce spurious negative signs that can wreak havoc in physics, so nearly 40 years after Hamilton's bridge vandalism, physicists went to war with one another to keep the quaternion system from becoming standard.
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Deciding the fourth dimension was entirely too much trouble, Gibbs decapitated Hamilton's creation by lopping off the a term altogether: Gibbs' quaternion-spinoff kept the i, j, k notation, but split the unwieldy rule for multiplying quaternions into separate operations for multiplying vectors that every math and physics undergraduate learns today: the dot product and the cross product.
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Vectors and scalars can be added. When a vector is added to a scalar, a completely different entity, a quaternion is created. A vector plus a scalar is always a quaternion even if the scalar is zero. If the scalar added to the vector is zero then the new quaternion produced is called a right quaternion.
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The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces.. The Hurwitz quaternion order was studied in 1967 by Goro Shimura,. but first explicitly described by Noam Elkies in 1998.. For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature).
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Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇. In 1878, Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product.
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The vector part of a unit quaternion represents the radius of the 2-sphere corresponding to the axis of rotation, and its magnitude is the sine of half the angle of rotation. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Also, the space of unit quaternions is "flat" in any infinitesimal neighborhood of a given unit quaternion.
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Shading by Quaternion Interpolation. WSCG'05. pp. 53-56. 2005. which uses Quaternion interpolation of the normals with the advantage that the normal will always have unit length and the computationally heavy normalization is avoided.
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For example, matrix multiplication and quaternion multiplication are both non-commutative.
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There is a catenoid of idempotents in the split-quaternion ring.
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There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
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Quaternion is a poetry style in which the theme is divided into four parts.
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As confirmed by experimental results, simultaneous solutions have less error than separable quaternion solutions.
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He developed the idea of a characteristic root of a quaternion matrix (an eigenvalue) and shows that they must exist. He also shows that a quaternion matrix is unitarily-equivalent to a triangular matrix. In 1956 he became a Senior Mathematician at Stanford Research Institute.
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The dihedral groups are both very similar to and very dissimilar from the quaternion groups and the semidihedral groups. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of maximal class, that is those groups of order 2n+1 and nilpotency class n.
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Initially called The Quaternion Walk, now called The Hamilton Walk, takes place in October each year.
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The generalized quaternion groups have the property that every abelian subgroup is cyclic., p. 101, exercise 1 It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above., Theorem 11.6, p. 262 Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group.
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Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that generate the elements of order two in the Brauer group of F. For some fields, including algebraic number fields, every element of order 2 in its Brauer group is represented by a quaternion algebra. A theorem of Alexander Merkurjev implies that each element of order 2 in the Brauer group of any field is represented by a tensor product of quaternion algebras.Lam (2005) p.
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A quaternion algebra over a field F is a four-dimensional central simple F-algebra. A quaternion algebra has a basis 1, i, j, ij where i^2, j^2 \in F^\times and ij = -ji. A quaternion algebra is said to be split over F if it is isomorphic as an F-algebra to the algebra of matrices M_2(F); a quaternion algebra over an algebraically closed field is always split. If \sigma is an embedding of F into a field E we shall denote by A \otimes_\sigma E the algebra obtained by extending scalars from F to E where we view F as a subfield of E via \sigma.
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Michael J. Crowe devotes chapter six of his book A History of Vector Analysis to the various published views, and notes the hyperbolic quaternion: :Macfarlane constructed a new system of vector analysis more in harmony with Gibbs–Heaviside system than with the quaternion system. ...he...defined a full product of two vectors which was comparable to the full quaternion product except that the scalar part was positive, not negative as in the older system. In 1899 Charles Jasper Joly noted the hyperbolic quaternion and the non-associativity property while ascribing its origin to Oliver Heaviside. The hyperbolic quaternions, as the Algebra of Physics, undercut the claim that ordinary quaternions made on physics.
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Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space; or, more generally, as the quotient of two vectors. A quaternion can be represented as the sum of a scalar and a vector. It can also be represented as the product of its tensor and its versor.
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To every quaternion algebra A, one can associate a quadratic form N (called the norm form) on A such that :N(xy) = N(x)N(y) for all x and y in A. It turns out that the possible norm forms for quaternion F-algebras are exactly the Pfister 2-forms.
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It is a theorem of Frobenius that there are only two real quaternion algebras: 2×2 matrices over the reals and Hamilton's real quaternions. In a similar way, over any local field F there are exactly two quaternion algebras: the 2×2 matrices over F and a division algebra. But the quaternion division algebra over a local field is usually not Hamilton's quaternions over the field. For example, over the p-adic numbers Hamilton's quaternions are a division algebra only when p is 2.
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Her thesis Certain quaternary quadratic forms and diophantine equations by generalized quaternion algebras earned her a doctorate degree in 1927.
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When Slerp is applied to unit quaternions, the quaternion path maps to a path through 3D rotations in a standard way. The effect is a rotation with uniform angular velocity around a fixed rotation axis. When the initial end point is the identity quaternion, Slerp gives a segment of a one-parameter subgroup of both the Lie group of 3D rotations, SO(3), and its universal covering group of unit quaternions, S3. Slerp gives a straightest and shortest path between its quaternion end points, and maps to a rotation through an angle of 2Ω.
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The icosians lie in the golden field, (a + b) + (c + d)i + (e + f)j + (g + h)k, where the eight variables are rational numbers. This quaternion is only an icosian if the vector (a, b, c, d, e, f, g, h) is a point on a lattice L, which is isomorphic to an E8 lattice. More precisely, the quaternion norm of the above element is (a + b)2 + (c + d)2 + (e + f)2 + (g + h)2. Its Euclidean norm is defined as u + v if the quaternion norm is u + v.
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Twenty Years of the Hamilton Walk by Fiacre Ó Cairbre, Department of Mathematics, National University of Ireland, Maynooth (2005), Irish Math. Soc. Bulletin 65 (2010) The quaternion involved abandoning commutativity, a radical step for the time. Not only this, but Hamilton also invented the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the 'scalar' part, and the remaining three as the 'vector' part.
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In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.
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A quaternion algebra over a field F is a four-dimensional central simple F-algebra. A quaternion algebra has a basis 1, i, j, ij where i^2, j^2 \in F^\times and ij = -ji. A quaternion algebra is said to be split over F if it is isomorphic as an F-algebra to the algebra of matrices M_2(F). If \sigma is an embedding of F into a field E we shall denote by A \otimes_\sigma E the algebra obtained by extending scalars from F to E where we view F as a subfield of E via \sigma.
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Such a set of four numbers is called a quaternion. While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers. Rotation calculation via quaternions has come to replace the use of direction cosines in aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.
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Understanding the meaning of the subtraction symbol is critical in quaternion theory, because it leads to an understanding of the concept of a vector.
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In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra that is the intersection of two maximal orders.
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139 In particular, over p-adic fields the construction of quaternion algebras can be viewed as the quadratic Hilbert symbol of local class field theory.
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In mathematics, hypercomplex analysis is the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. The first instance is functions of a quaternion variable, where the argument is a quaternion. A second instance involves functions of a motor variable where arguments are split-complex numbers. In mathematical physics, there are hypercomplex systems called Clifford algebras.
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The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3 dimensions. This Julia set is cross sectioned at the plane. Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable.
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The Quaternions can be generalized into further algebras called quaternion algebras. Take to be any field with characteristic different from 2, and and to be elements of ; a four- dimensional unitary associative algebra can be defined over with basis and , where , and (so ). Quaternion algebras are isomorphic to the algebra of 2×2 matrices over or form division algebras over , depending on the choice of and .
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His research interests are Yang-Mills gauge theories, supersymmetry, supergravity, quaternion and octonion algebras, spin structures, generalised theories of gravity, cosmological solutions, integrable systems and phase space quantisation.
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Let F be a field of characteristic not equal to 2. A biquaternion algebra over F is a tensor product of two quaternion algebras.Lam (2005) p.60Szymiczek (1997) p.
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The exchanges of e1 and e4 alternate signs an even number of times, and show the dual unit ε commutes with the quaternion basis elements i, j, and k.
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The hyperbolic orbifold of minimal volume can be obtained as the surface associated to a particular order, the Hurwitz quaternion order, and it is compact of volume \pi/21.
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Corrado Segre (1912) continued the development with that ring. Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.Arthur Conway (1911) "On the application of quaternions to some recent developments of electrical theory", Proceedings of the Royal Irish Academy 29:1–9, particularly page 9 In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.
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Classical quaternion notation had only one concept of multiplication. Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation. Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two quaternions.
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P.R. Girard's 1984 essay The quaternion group and modern physics discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely , the Lorentz group, the general theory of relativity group, the Clifford algebra and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics of rigid body motion.
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In the complex numbers, , there are just two numbers, i and −i, whose square is −1 . In there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in . To see this, let be a quaternion, and assume that its square is −1. In terms of , and , this means :a^2 - b^2 - c^2 - d^2 = -1, :2ab = 0, :2ac = 0, :2ad = 0.
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To satisfy the last three equations, either or , and are all 0. The latter is impossible because a is a real number and the first equation would imply that Therefore, and In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere. Only negative real quaternions have infinitely many square roots.
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In mathematics, Brandt matrices are matrices, introduced by , that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra. calculated the traces of the Brandt matrices. Let O be an order in a quaternion algebra with class number H, and Ii,...,IH invertible left O-ideals representing the classes. Fix an integer m.
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November 2011 Yefremov is known for his work in theoretical physics, among others, for his fundamental work on the role of quaternion geometry space in quantum mechanics and field theory.
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The corresponding quaternion formula , where , or, in expanded form: :x'i + y'j + z'k = (a + bi + cj + dk)(xi + yj + zk)(a - bi - cj - dk) is known as the Hamilton–Cayley formula.
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The Quaternion Eagle, hand-coloured woodcut (c. 1510) by Hans Burgkmair. One rendition of the coat of the empire was the "Quaternion Eagle" (so named after the imperial quaternions) printed by David de Negker of Augsburg, after a 1510 woodcut by Hans Burgkmair. It showed a selection of 56 shields of various Imperial States in groups of four on the feathers of a double-headed eagle supporting, in place of a shield, Christ on the Cross.
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The integral curves or fibers respectively are certain pairwise linked great circles, the orbits in the space of unit norm quaternions under left multiplication by a given unit quaternion of unit norm.
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They all have unit magnitude and therefore lie in the unit quaternion group Sp(1). The 120 elements in 4-dimensional space match the 120 vertices the 600-cell, a regular 4-polytope.
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On page 173 Macfarlane expands on his greater theory of quaternion variables. By way of contrast he notes that Felix Klein appears not to look beyond the theory of Quaternions and spatial rotation.
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The surface's Fuchsian group can be constructed as the principal congruence subgroup of the (2,3,7) triangle group in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing the ideal \langle 2 \rangle in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its systole is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations.
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Furthermore, the fraternity's chapter at Furman carries a unique flag which bears a red rose in the upper right-hand corner. On campus today the only known active secret society is The Quaternion Club, although many are rumored to exist. Quaternion, which dates back to 1903, taps four juniors and four seniors each year in the late winter or early spring. The selection process is guarded but is thought to be controlled by current Quaternions currently in residence at the school.
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Page numbers are carried from previous publications, and the reader is presumed familiar with quaternions. The first paper is "Principles of the Algebra of Physics" where he first proposes the hyperbolic quaternion algebra, since "a student of physics finds a difficulty in principle of quaternions which makes the square of a vector negative." The second paper is "The Imaginary of the Algebra". Similar to Homersham Cox (1882/83), Macfarlane uses the hyperbolic versor as the hyperbolic quaternion corresponding to the versor of Hamilton.
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For odd prime p, the p-adic Hamilton quaternions are isomorphic to the 2×2 matrices over the p-adics. To see the p-adic Hamilton quaternions are not a division algebra for odd prime p, observe that the congruence x2 \+ y2 = −1 mod p is solvable and therefore by Hensel's lemma — here is where p being odd is needed — the equation :x2 \+ y2 = −1 is solvable in the p-adic numbers. Therefore the quaternion :xi + yj + k has norm 0 and hence doesn't have a multiplicative inverse. One way to classify the F-algebra isomorphism classes of all quaternion algebras for a given field, F is to use the one-to-one correspondence between isomorphism classes of quaternion algebras over F and isomorphism classes of their norm forms.
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A Riemannian symmetric space which is additionally equipped with a parallel subbundle of End(TM) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called quaternion-Kähler symmetric space. An irreducible symmetric space G/K is quaternion-Kähler if and only if isotropy representation of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non- compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with p = 2 or q = 2 (these are isomorphic), BDI with p = 4 or q = 4, CII with p = 1 or q = 1, EII, EVI, EIX, FI and G.
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This quaternion solution and the calculation of the optimal isometry in the d-dimensional case were both extended to infinite sets and to the continuous case in the appendix A of another paper of Petitjean.
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In classical quaternion literature the equation : q^2=-1 was thought to have infinitely many solutions that were called geometrically real. These solutions are the unit vectors that form the surface of a unit sphere. A geometrically real quaternion is one that can be written as a linear combination of i, j and k, such that the squares of the coefficients add up to one. Hamilton demonstrated that there had to be additional roots of this equation in addition to the geometrically real roots.
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"Memorial for Captain Ferber"Flight 15 July 1911 Gaston Combebiac wrote that Feber ought to be considered a member of the Quaternion Society when he contributed a biographical note to the Society's Bulletin:Gaston Combebiac (1912) Ferdinand Ferber, Bulletin of the Quaternion Society, link from HathiTrust :After all, since his intuition, at once mathematical and realistic, having not failed to recognize the advantages presented by the use of vector calculus for certain physical applications of mathematics, we must rank him among the members of our Association.
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In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense.
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The system is therefore nonholonomic. The anholonomy may be represented by the doubly unique quaternion (q and −q) which, when applied to the points that represent the sphere, carries points B and R to their new positions.
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Subplanes of projective spaces of geometrical dimension at least 3 are necessarily Desarguesian, see §1 or §16 or. Therefore, all compact connected projective spaces can be coordinatized by the real or complex numbers or the quaternion field.
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Another way of looking at this group is with quaternion multiplication. Every rotation in four dimensions can be achieved by multiplying by a pair of unit quaternions, one before and one after the vector. These quaternion are unique, up to a change in sign for both of them, and generate all rotations when used this way, so the product of their groups, S3 × S3, is a double cover of SO(4), which must have six dimensions. Although the space we live in is considered three-dimensional, there are practical applications for four-dimensional space.
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A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion. In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The rotation matrices have therefore 6 out of 16 independent components. Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.
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The "latitude" on the hypersphere will be half of the corresponding angle of rotation, and the neighborhood of any point will become "flatter" (i.e. be represented by a 3D Euclidean space of points) as the neighborhood shrinks. This behavior is matched by the set of unit quaternions: A general quaternion represents a point in a four-dimensional space, but constraining it to have unit magnitude yields a three-dimensional space equivalent to the surface of a hypersphere. The magnitude of the unit quaternion will be unity, corresponding to a hypersphere of unit radius.
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A Hurwitz integer is called irreducible if it is not 0 or a unit and is not a product of non-units. A Hurwitz integer is irreducible if and only if its norm is a prime number. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not primes in the usual sense of commutative algebra: it is possible for an irreducible quaternion to divide a product ab without dividing either a or b. Every Hurwitz quaternion can be factored as a product of irreducible quaternions.
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A fully robust approach will use a different algorithm when , the trace of the matrix , is negative, as with quaternion extraction. When is zero because the angle is zero, an axis must be provided from some source other than the matrix.
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If the Einstein constant of a simply connected manifold with holonomy in Sp(n) Sp(1) is zero, where n\geq 2, then the holonomy is actually contained in Sp(n), and the manifold is hyperkähler. We will exclude this case from the definition by declaring quaternion-Kähler to mean not only that the holonomy group is contained in Sp(n) Sp(1), but also that the manifold has non-zero (constant) scalar curvature. With this convention, quaternion-Kähler manifolds can thus be naturally divided into those for which the Ricci curvature is positive, and those for which it is instead negative.
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This approach utilizes the multiplicative formulation of the error quaternion, which allows for the unity constraint on the quaternion to be better handled. It is also common to use a technique known as dynamic model replacement, where the angular rate is not estimated directly, but rather the measured angular rate from the gyro is used directly to propagate the rotational dynamics forward in time. This is valid for most applications as gyros are typically far more precise than one's knowledge of disturbance torques acting on the system (which is required for precise estimation of the angular rate).
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Through the Imperial Eagles the ideal of the durable unity of the Holy Roman Empire took decorative shape and demonstrates the emotional relationship of a broad public to the Empire. The imperial eagle was mostly pictured in the form of a quaternion eagle which related the theory of quaternions to one of the most important symbols of the Empire. As the structure of the Empire was in need of explanation, even for contemporaries, the quaternion model was meant to depict the fabric of the Empire. It was developed in the 14th century and remained popular until the end of the Empire.
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Alexander Macfarlane called the structure of split-quaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900.Alexander Macfarlane (1900) Application of space analysis to curvilinear coordinates , Proceedings of the International Congress of Mathematicians, Paris, page 306, from International Mathematical Union The unit sphere was considered in 1910 by Hans Beck.Hans Beck (1910) Ein Seitenstück zur Mobius'schen Geometrie der Kreisverwandschaften, Transactions of the American Mathematical Society 11 For example, the dihedral group appears on page 419. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.
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Since the direction r in space is arbitrary, this hyperbolic quaternion multiplication can express any Lorentz boost using the parameter a called rapidity. However, the hyperbolic quaternion algebra is deficient for representing the full Lorentz group (see biquaternion instead). Writing in 1967 about the dialogue on vector methods in the 1890s, a historian commented :The introduction of another system of vector analysis, even a sort of compromise system such as Macfarlane's, could scarcely be well received by the advocates of the already existing systems and moreover probably acted to broaden the question beyond the comprehension of the as-yet uninitiated reader.
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The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic . The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals.
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Similarly, one can regard S4n+3 as lying in Hn+1 (quaternionic n-space) and factor out by unit quaternion (= S3) multiplication to get the quaternionic projective space HPn. In particular, since S4 = HP1, there is a bundle S7 → S4 with fiber S3.
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The pose can be described by means of a rotation and translation transformation which brings the object from a reference pose to the observed pose. This rotation transformation can be represented in different ways, e.g., as a rotation matrix or a quaternion.
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There has been a great deal of research in applying the principles of computer-aided geometric design (CAGD) to the problem of computer-aided motion design. In recent years, it has been well established that rational Bézier and rational B-spline based curve representation schemes can be combined with dual quaternion representation of spatial displacements to obtain rational Bézier and B-spline motions. Ge and Ravani, developed a new framework for geometric constructions of spatial motions by combining the concepts from kinematics and CAGD. Their work was built upon the seminal paper of Shoemake, in which he used the concept of a quaternion for rotation interpolation.
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In 1883 he published an article "Some general theorems in quaternion integration".A. McAulay (1883) Messenger of Mathematics 13:26 to 37 McAulay took his degree in 1886, and began to reflect on the instruction of students in quaternion theory. In an article "Establishment of the fundamental properties of quaternions"McAulay (1888) Messenger of Mathematics 18:131 to 136 he suggested improvements to the texts then in use. He also wrote a technical articleA. McAulay (1888) "The transformation of multiple surface integrals into multiple line integrals", Messenger of Mathematics 18:139 to 45 on integration. Departing for Australia, he lectured at Ormond College, University of Melbourne from 1893 to 1895.
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In the Middle Ages, a quire (also called a "gathering") was most often formed of 4 folded sheets of vellum or parchment, i.e. 8 leaves, 16 sides. The term "quaternion" (or sometimes quaternum) designates such a quire. A quire made of a single folded sheet (i.e.
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The homography group of the ring of integers Z is modular group . Ring homographies have been used in quaternion analysis, and with dual quaternions to facilitate screw theory. The conformal group of spacetime can be represented with homographies where A is the composition algebra of biquaternions.
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This Euclidean norm defines a quadratic form on L, under which the lattice is isomorphic to the E8 lattice. This construction shows that the Coxeter group H_4 embeds as a subgroup of E_8. Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.
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A4 symmetry, or [3,3,3] is order 120, with Conway quaternion notation +1/60[I×].21. Its abstract structure is the symmetric group S5. Three forms with symmetric Coxeter diagrams have extended symmetry, 3,3,3 of order 240, and Conway notation ±1/60[I×].2, and abstract structure S5×C2.
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The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ... In this passage is the operator giving the scalar part of a quaternion, and is the "tensor", now called norm, of a quaternion. A modern view of the unification of the sphere and hyperboloid uses the idea of a conic section as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in four-dimensional space with points determined by quadratic forms. First consider the conical hypersurface :P = \lbrace p \ : \ w^2 = x^2 + y^2 + z^2 \rbrace and :H_r = \lbrace p \ :\ w = r \rbrace , which is a hyperplane.
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When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2IL of quaternion left-multiplications and as the subgroup 2IR of quaternion right- multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion −Id.
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With Montserrat Alsina, Bayer is the author of the book Quaternion Orders, Quadratic Forms, and Shimura Curves (American Mathematical Society, 2004). As well as quaternion algebras, Eichler orders, quadratic forms, and Shimura curves (the subject of the book), other topics in her research include automorphic forms, diophantine equations, elliptic curves, modular curves, and zeta functions. Beyond number theory, with Jordi Guàrdia and Artur Travesa she is the author of Arrels germàniques de la matemàtica contemporània: amb una antologia de textos matemàtics de 1850 a 1950 (Institut d'Estudis Catalans, 2012), on the history of mathematics in Germany from the mid-19th century to the mid-20th century. In total she is an author or editor of 19 books.
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The book by Hervé Jacquet and Langlands on presented a theory of automorphic forms for the general linear group , establishing among other things the Jacquet–Langlands correspondence showing that functoriality was capable of explaining very precisely how automorphic forms for related to those for quaternion algebras. This book applied the adelic trace formula for and quaternion algebras to do this. Subsequently, James Arthur, a student of Langlands while he was at Yale, successfully developed the trace formula for groups of higher rank. This has become a major tool in attacking functoriality in general, and in particular has been applied to demonstrating that the Hasse–Weil zeta functions of certain Shimura varieties are among the -functions arising from automorphic forms.
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In computer graphics, Slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed motion along a unit-radius great circle arc, given the ends and an interpolation parameter between 0 and 1\.
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The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2. The Schur multipliers of the finite simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of considerable recent interest.
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Up until Genesis 3 the Genesis figures had been using TriAx Weight Maps, where many other industry platforms were using Dual Quaternion. This changed in Genesis 3, released in 2015, to allow Daz 3D figures to be more compatible with other 3D software platforms as well as Game Development platforms.
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Over a field k, Azumaya algebras are completely classified by the Artin-Wedderburn theorem since they are the same as central simple algebras. These are algebras isomorphic to the matrix ring M_n(D) for some division algebra D over k. For example, quaternion algebras provide examples of central simple algebras.
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Given any 2 × 2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring M(2,C) is isomorphicLeonard Dickson (1914) Linear Algebras, §13 "Equivalence of the complex quaternion and matric algebras", page 13, via HathiTrust to the biquaternion ring.
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The split-biquaternions form an associative ring as is clear from considering multiplications in its basis {1, ω, i, j, k, ωi, ωj, ωk}. When ω is adjoined to the quaternion group one obtains a 16 element group :( {1, i, j, k, −1, −i, −j, −k, ω, ωi, ωj, ωk, −ω, −ωi, −ωj, −ωk}, × ).
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To apply angular changes, the orientation is modified by a delta angle/axis rotation. The resulting orientation must be re-normalized to prevent floating-point error from successive transformations from accumulating. For matrices, re- normalizing the result requires converting the matrix into its nearest orthonormal representation. For quaternions, re-normalization requires performing quaternion normalization.
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In group theory, a Dedekind group is a group G such that every subgroup of G is normal. All abelian groups are Dedekind groups. A non-abelian Dedekind group is called a Hamiltonian group. The most familiar (and smallest) example of a Hamiltonian group is the quaternion group of order 8, denoted by Q8.
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Given a transformation matrix T_l , the joint position at the T-pose can be transferred to its corresponding position in the world coordination. In many works, the 3D joint rotation is expressed as a normalized quaternion [x,y,z,w] due to its continuity that can facilitate gradient-based optimization in the parameter estimation.
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On page 665 of Elements of Quaternions Hamilton defines a biquaternion to be a quaternion with complex number coefficients. The scalar part of a biquaternion is then a complex number called a biscalar. The vector part of a biquaternion is a bivector consisting of three complex components. The biquaternions are then the complexification of the original (real) quaternions.
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For , if Q has diagonalization with non-zero a and b (which always exists if Q is non- degenerate), then is isomorphic to a K-algebra generated by elements x and y satisfying , and . Thus is isomorphic to the (generalized) quaternion algebra . We retrieve Hamilton's quaternions when , since . As a special case, if some x in V satisfies , then .
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In mathematics, the Shimura subgroup Σ(N) is a subgroup of the Jacobian of the modular curve X0(N) of level N, given by the kernel of the natural map to the Jacobian of X1(N). It is named after Goro Shimura. There is a similar subgroup Σ(N,D) associated to Shimura curves of quaternion algebras.
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The cross product can also be described in terms of quaternions. In general, if a vector is represented as the quaternion , 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.
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It has the property that the absolute value of a quaternion is equal to the square root of the determinant of the matrix image of . The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group . Thus, as a Lie group is isomorphic to .
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The simplest example is m = 1, n = 2, when π1(M) is the quaternion group of order 8. Prism manifolds are uniquely determined by their fundamental groups: if a closed 3-manifold has the same fundamental group as a prism manifold M, it is homeomorphic to M. Prism manifolds can be represented as Seifert fiber spaces in two ways.
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The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elements z3 and x2 = y2, and the quotient by the center is the tetrahedral group, equivalently, the alternating group A4. When m = 1 this group is the binary tetrahedral group.
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The former method represents the classical stochastic approach while the latter implements a linear fractal model. Using recursion allowed programmers to create complex images through simple direction. Three dimensional fractals are generated in a variety of ways including by using quaternion algebra. Fractals emerge from fluid dynamics modelling simulations as turbulence when contour advection is used to study chaotic mixing.
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This provides the correspondence of Cl(R) with dual quaternion algebra. To see this, compute : \varepsilon ^2 = (e_1 e_2 e_3 e_4)^2 = e_1 e_2 e_3 e_4 e_1 e_2 e_3 e_4 = -e_1 e_2 e_3 (e_4 e_4 ) e_1 e_2 e_3 = 0 , and : \varepsilon i = (e_1 e_2 e_3 e_4) e_2 e_3 = e_1 e_2 e_3 e_4 e_2 e_3 = e_2 e_3 (e_1 e_2 e_3 e_4) = i\varepsilon.
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Knott continued his work as a mathematician, including quaternion methods of his professor and mentor Peter Guthrie Tait. When the tight constraints of a single linear algebra began to be felt in the 1890s, and revisionists began publishing, Knott contributed the pivotal article "Recent Innovations in Vector Theory". As M.J. Crowe describes,M.J. Crowe (1967) A History of Vector Analysis, esp. pp.
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The initial tangent vector is parallel transported to each tangent along the curve; thus the curve is, indeed, a geodesic. In the tangent space at any point on a quaternion Slerp curve, the inverse of the exponential map transforms the curve into a line segment. Slerp curves not extending through a point fail to transform into lines in that point's tangent space.
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Cartan famously asked the following question: given a division ring D and a proper sub- division-ring S that is not contained in the center, does each inner automorphism of D restrict to an automorphism of S? The answer is negative: this is the Cartan–Brauer–Hua theorem. A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.
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He was called to positions as professor ordinarius in 1908 at Basel, in 1913 at the Technische Hochschule Karlsruhe, and in 1916 at the University of Zurich. From 1920 to 1922 he was the rector of the University of Zurich. Fueter did research on algebraic number theory and quaternion analysis. He also published a proof of the Fueter–Pólya theorem with George Pólya.
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Quaternions are a concise method of representing the automorphisms of three- and four-dimensional spaces. They have the technical advantage that unit quaternions form the simply connected cover of the space of three-dimensional rotations. For this reason, quaternions are used in computer graphics,Ken Shoemake (1985), Animating Rotation with Quaternion Curves, Computer Graphics, 19(3), 245–254. Presented at SIGGRAPH '85.
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For instance, he shows "a Hamilton group of order 2a has quaternion groups as subgroups". In 2005 Horvat et al used this structure to count the number of Hamiltonian groups of any order where o is an odd integer. When then there are no Hamiltonian groups of order n, otherwise there are the same number as there are Abelian groups of order o.
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The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818). In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four- dimensional space of quaternion imaginaries, in which three of the dimensions are analogous to the imaginary numbers in the complex field.
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The gatherings are sewn together at the spine, done in such a way that two or more stretches of thread are visible along each gathering's innermost fold. In medieval manuscripts, a gathering, or quire, was most often formed of 4 folded sheets of vellum or parchment, i.e. 8 leaves, 16 sides. The term quaternion (or sometimes quaternum) designates such a unit.
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Hamilton needed a way to distinguish between two different types of double quaternions, the associative biquaternions and the octaves. He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion. Hamilton 1853 pg 740See a hard copy of Lectures on quaternions, appendix B, half of the hyphenated word double quaternion has been cut off in the online Edition See Hamilton's talk to the Royal Irish Academy on the subject observed in reply that they were not associative, which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it; Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845 – as an appendix to a paper on a different subject.
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Dynamics are usually continuous, as in the case of Continuous-Time CNN (CT-CNN) processors, but can be discrete, as in the case of Discrete-Time CNN (DT-CNN) processors. Each cell has one output, by which it communicates its state with both other cells and external devices. Output is typically real-valued, but can be complex or even quaternion, i.e. a Multi- Valued CNN (MV-CNN).
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Symplectic gauge groups could also be considered. For example, (which is called in the article symplectic group) has a representation in terms of quaternion unitary matrices which has a dimensional real representation and so might be considered as a candidate for a gauge group. has 32 charged bosons and 4 neutral bosons. Its subgroups include so can at least contain the gluons and photon of .
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Alexander McAulay (9 December 1863 – 6 July 1931) was the first professor of mathematics and physics at the University of Tasmania, Hobart, Tasmania. He was also a proponent of dual quaternions, which he termed "octonions" or "Clifford biquaternions". McAulay was born on 9 December 1863 and attended Kingswood School in Bath. He proceeded to Caius College, Cambridge, there taking up a study of the quaternion algebra.
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Next he used complex quaternions (biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions (split- biquaternions) as an instance of Clifford algebra.
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Guttormsen is a graduate of the jazz program at Trondheim Musikkonservatorium (1990). He was first recognized as bassist in the Farmers Market, where he has contributed since the start in 1991. He has also been a member of bands like The Source (1995–2005) and Quaternion, and has played within Hemisfair. In recent years he has also contributed within Håkon Storm-Mathisen Quartet and Silje Nergaard Band.
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It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates.
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He taught himself the theory of space groups, including the quaternion method, which became the mathematical basis of a lengthy paper on crystal structure for which he won a joint prize with Ronald G.W. Norrish in his third year. At Cambridge, he also became known as "Sage", a nickname given to him about 1920 by a young woman working in Charles Kay Ogden's Bookshop at the corner of Bridge Street.
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Four operations are of fundamental importance in quaternion notation. : + − ÷ × In particular it is important to understand that there is a single operation of multiplication, a single operation of division, and a single operations of addition and subtraction. This single multiplication operator can operate on any of the types of mathematical entities. Likewise every kind of entity can be divided, added or subtracted from any other type of entity.
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Then the scalars of that vector space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part.
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The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines.James Cockle (1849) On a New Imaginary in Algebra 34:37–47, London-Edinburgh-Dublin Philosophical Magazine (3) 33:435–9, link from Biodiversity Heritage Library. William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions.
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The Green line initially was provisioned with a works depot just past the Sandyford terminus and opposite the old Stillorgan Railway station building. The depot could stable 32 trams. When the Green line was extended to Broombridge a further depot was constructed over part of the old Liffey Junction site. It was named Hamilton Depot in honour of William Rowan Hamilton who developed the quaternion mathematical number system.
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As for mathematics, the hyperbolic quaternion is another hypercomplex number, as such structures were called at the time. By the 1890s Richard Dedekind had introduced the ring concept into commutative algebra, and the vector space concept was being abstracted by Giuseppe Peano. In 1899 Alfred North Whitehead promoted Universal algebra, advocating for inclusivity. The concepts of quasigroup and algebra over a field are examples of mathematical structures describing hyperbolic quaternions.
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The only green shown in the arms of the states of the Holy Roman Empire in the Quaternion Eagle by Hans Burgkmair (c. 1510) are the crancelin of Saxony and the Zirbelnuss of Augsburg. The three lions rampant, verts of the Marquessate of Franchimont are attested in the 16th century. Siebmachers Wappenbuch of 1605 shows a number of green heraldic devices in the coat of arms of cities.
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Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. takes advantage of the odd dimension to change a Householder reflection to a rotation by negation, and uses that to aim the axis of a uniform planar rotation. Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere.
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As complex numbers use an imaginary unit to complement the real line, Hamilton considered the vector to be the imaginary part of a quaternion: :The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.W. R. Hamilton (1846) London, Edinburgh & Dublin Philosophical Magazine 3rd series 29 27 Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including Augustin Cauchy, Hermann Grassmann, August Möbius, Comte de Saint-Venant, and Matthew O'Brien. Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s.
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Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix. There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors.
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The cause of gimbal lock is representing an orientation as three axial rotations with Euler angles. A potential solution therefore is to represent the orientation in some other way. This could be as a rotation matrix, a quaternion (see quaternions and spatial rotation), or a similar orientation representation that treats the orientation as a value rather than three separate and related values. Given such a representation, the user stores the orientation as a value.
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In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form Q may be taken as a diagonal form :Σ aixi2. Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras :(ai, aj) for i < j.
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In mathematical finite group theory, the classical involution theorem of classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic. extended the classical involution theorem to groups of finite Morley rank. A classical involution t of a finite group G is an involution whose centralizer has a subnormal subgroup containing t with quaternion Sylow 2-subgroups.
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Since the homomorphism is a local isometry, we immediately conclude that to produce a uniform distribution on SO(3) we may use a uniform distribution on . In practice: create a four-element vector where each element is a sampling of a normal distribution. Normalize its length and you have a uniformly sampled random unit quaternion which represents a uniformly sampled random rotation. Note that the aforementioned only applies to rotations in dimension 3.
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The dual tree hypercomplex wavelet transform (HWT) developed in consists of a standard DWT tensor and wavelets obtained from combining the 1-D Hilbert transform of these wavelets along the n-coordinates. In particular a 2-D HWT consists of the standard 2-D separable DWT tensor and three additional components: For the 2-D case, this is named dual tree quaternion wavelet transform (QWT). The total redundancy in M-D is tight frame.
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The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Matematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy. In 1897, Hensel introduced p-adic numbers.
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A scientific society, the Quaternion Society was an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the academic world that were experimenting with quaternions and other hypercomplex number systems. The guiding light was Alexander Macfarlane who served as its Secretary initially, and became President in 1909. The Association published a Bibliography in 1904 and a Bulletin (annual report) from 1900 to 1913.
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The "Quaternion eagle", representing the estates of the Holy Roman Empire (1510). Arms of Charles V, Holy Roman Emperor in the choir of the Cathedral of Cordoba, 16th century. Royal coats of arms in Siebmachers Wappenbuch (1605) In 1484, during the reign of Richard III, the various heralds employed by the crown were incorporated into the College of Arms, through which all new grants of arms would eventually be issued.Fox- Davies, A Complete Guide to Heraldry, p. 38.
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The so-called classical groups generalise the examples 1 and 2 above. They arise as linear algebraic groups, that is, as subgroups of GLn defined by a finite number of equations. Basic examples are orthogonal, unitary and symplectic groups but it is possible to construct more using division algebras (for example the unit group of a quaternion algebra is a classical group). Note that the projective groups associated to these groups are also linear, though less obviously.
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Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation,Johnson, 2015 orbifold notation,Conway, 2008 and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts.
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Dickstein wrote many mathematical books and founded the journal Wiadomości Mathematyczne (Mathematical News), now published by the Polish Mathematical Society. He was a bridge between the times of Cauchy and Poincaré and those of the Lwów School of Mathematics. He was also thanked by Alexander Macfarlane for contributing to the Bibliography of Quaternions (1904) published by the Quaternion Society. He was also one of the personalities, who contributed to the foundation of the Warsaw Public Library in 1907.
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On page 95 Clifford deconstructed the quaternion product of William Rowan Hamilton into two separate "products" of two vectors: vector product and scalar product, anticipating the complete severance seen in Vector Analysis (1901). Elements of Dynamic was the debut of the term cross-ratio for a four-argument function frequently used in geometry. Clifford uses the term twist to discuss (pages 126 to 131) the screw theory that had recently been introduced by Robert Stawell Ball.
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In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.
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When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the Elements of Quaternions, a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition was prepared by Charles Jasper Joly, when the book was split into two volumes, the first appearing 1899 and the second in 1901. The subject index and footnotes in this second edition improved the Elements accessibility.
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Therefore, if momentum desaturations can be reduced or eliminated, a larger fraction of propellant can be used to maintain the spacecraft in its desired orbit, and it will have a longer operational lifetime. Typically spacecraft rotations are performed as quaternion rotations or about a fixed axis (Euler's rotation theorem) usually referred to as an eigenaxis. Rotations about an eigenaxis result in the smallest angle between two orientations. Moreover, eigenaxis rotations are performed with a fixed rotation rate or maneuver rate.
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A versor is a quaternion with a tensor of 1. Alternatively, a versor can be defined as the quotient of two equal-length vectors. In general a versor defines all of the following: a directional axis; the plane normal to that axis; and an angle of rotation. When a versor and a vector which lies in the plane of the versor are multiplied, the result is a new vector of the same length but turned by the angle of the versor.
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Rezero uses a full 3D model that includes yaw motion too. Kugle uses a fully coupled quaternion- based 3D model which couples the motion of all axes. The ballbots (CMU Ballbot, BallIP, NXT Ballbot, Adelaide Ballbot, Rezero) use linear feedback control approaches to maintain balance and achieve motion. The CMU Ballbot uses an inner balancing control loop that maintains the body at desired body angles and an outer loop controller that achieves desired ball motions by commanding body angles to the balancing controller.
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The Kugle robot is tested with both linear feedback controllers (LQR) and non-linear sliding mode controllers to show the benefit of its coupled dynamic quaternion model. A ballbot is a shape-accelerated underactuated system. The inclination angles of a ballbot is thus dynamically linked to the resulting accelerations of the ball and robot leading to an underactuated system. The CMU Ballbot plans motions in the space of body lean angles in order to achieve fast, dynamic and graceful ball motions.
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For a textbook on quaternions, lecturers and students relied on Tait and Kelland's Introduction to Quaternions which had editions in 1873 and 1882. It fell to Knott to prepare a third edition in 1904. By then the Universal Algebra of Alfred North Whitehead (1898) presumed some grounding in quaternions as students encountered matrix algebra. In Knott's introduction to his textbook edition he says "Analytically the quaternion is now known to take its place in the general theory of complex numbers and continuous groups,...".
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The coquaternions were initially introduced (under that name)James Cockle (1849), On Systems of Algebra involving more than one Imaginary, Philosophical Magazine (series 3) 35: 434,5, link from Biodiversity Heritage Library in 1849 by James Cockle in the London-Edinburgh- Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 BibliographyA. Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, from Cornell University Historical Math Monographs, entries for James Cockle, pp. 17–18 of the Quaternion Society.
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A unit vector in ℝ3 was called a right versor by W. R. Hamilton, as he developed his quaternions ℍ ⊂ ℝ4. In fact, he was the originator of the term vector, as every quaternion q = s + v has a scalar part s and a vector part v. If v is a unit vector in ℝ3, then the square of v in quaternions is –1. Thus by Euler's formula, \exp (\theta v) = \cos \theta + v \sin \theta is a versor in the 3-sphere.
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Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for nonlinear σ-models with supersymmetry. Compact Einstein manifolds have been much studied in differential geometry, and many examples are known, although constructing them is often challenging. Compact Ricci-flat manifolds are particularly difficult to find: in the monograph on the subject by the pseudonymous author Arthur Besse, readers are offered a meal in a starred restaurant in exchange for a new example.
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Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley. The Latin term for turn is versor, which is a quaternion that can be visualized as an arc of a great circle. The product of two versors can be compared to a spherical triangle where two sides add to the third. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation.
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Latimer earned his PhD in 1924 from the University of Chicago under Leonard Dickson with thesis Arithmetic of Generalized Quaternion Algebras. He was an assistant professor at Tulane University for 2 years, before becoming a mathematics professor at the University of Kentucky in 1927. After 20 years at the University of Kentucky, he resigned in 1947 and became a professor at Emory University. Latimer was an amateur photographer; some of his photographs are preserved in the archives of the University of Kentucky and Emory University.
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In this case, each axis cuts the unit circle in the plane (which is the one-dimensional sphere) at two points that are each other's antipodes. For N = 4, the Bingham distribution is a distribution over the space of unit quaternions. Since a unit quaternion corresponds to a rotation matrix, the Bingham distribution for N = 4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von Mises–Fisher distribution. These distributions are for example used in geology, crystallography and bioinformatics.
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In 1930 Brenner earned a B.A. degree with major in chemistry from Harvard University. In graduate study there he was influenced by Hans Brinkmann, Garrett Birkhoff, and Marshall Stone. He was granted the Ph.D. in February 1936. Brenner later described some of his reminiscences of his student days at Harvard and of the state of American mathematics in the 1930s in an article for American Mathematical Monthly.Brenner (1979) "Student Days", American Mathematical Monthly 86: 359–6 In 1951 Brenner published his findings about matrices with quaternion entries.
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What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by Heaviside, FitzGerald, Lodge and Hertz. The original equations used Hamilton's more expressive quaternion notation, a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today. In the late 1880s there was a debate over the relative merits of vector analysis and quaternions. According to Heaviside the electromagnetic potential field was purely metaphysical, an arbitrary mathematical fiction, that needed to be "murdered".
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Because these lines are longer, irregular, and frequently enjambed ("as the / dead fountains"), it is quite clear that the symmetry of syllables is not meant to be audible. Moore's use of end-rhyme is telling. Only 2 lines in each stanza are rhymed: these are emphasized for the reader by indentation, but hidden from the listener by radical enjambment ("fawn- / brown" and "coxcomb- / tinted"). Elizabeth Daryush, known for her use of syllabic verse, used the quaternion form for her celebrated syllabic verse poem 'Accentedal'.
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All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and G is isoclinic with G×A if and only if A is abelian. The dihedral, quasidihedral, and quaternion groups of order 2n are isoclinic for n≥3, in more detail. Isoclinism divides p-groups into families, and the smallest members of each family are called stem groups.
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Quaternions are also used in one of the proofs of Lagrange's four- square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.
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The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reportedThomas Kirkman (1848) "On Pluquaternions and Homoid Products of n Squares", London and Edinburgh Philosophical Magazine 1848, p 447 Google books link on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.
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As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field, over Q, of the polynomial :x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36. The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.
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In 1935, George Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép product of the regular subgroup and a point stabilizer. He gives PSL(2,11) and the alternating group of degree 5 as examples, and of course every alternating group of prime degree is an example. This same paper gives a number of examples of groups which cannot be realized as Zappa–Szép products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
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This formulation uses the negative sign so the correspondence with quaternions is easily shown. Denote a set of orthogonal unit vectors of R3 as e1, e2, and e3, then the Clifford product yields the relations : e_2 e_3 = -e_3 e_2, \,\,\, e_3 e_1 = -e_1 e_3,\,\,\, e_1 e_2 = -e_2 e_1, and : e_1 ^2 = e_2^2 = e_3^2 = -1. The general element of the Clifford algebra Cl(R) is given by : A = a_0 + a_1 e_1 + a_2 e_2 + a_3 e_3 + a_4 e_2 e_3 + a_5 e_3 e_1 + a_6 e_1 e_2 + a_7 e_1 e_2 e_3. The linear combination of the even degree elements of Cl(R) defines the even subalgebra Cl(R) with the general element : q = q_0 + q_1 e_2 e_3 + q_2 e_3 e_1 + q_3 e_1 e_2. The basis elements can be identified with the quaternion basis elements i, j, k as : i= e_2 e_3, j = e_3 e_1, k = e_1 e_2, which shows that the even subalgebra Cl(R) is Hamilton's real quaternion algebra. To see this, compute : i^2 = (e_2 e_3)^2 = e_2 e_3 e_2 e_3 = - e_2 e_2 e_3 e_3 = -1, and : ij = e_2 e_3 e_3 e_1 = -e_2 e_1 = e_1 e_2 = k.
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Anton III Wierix 1606). The ten quaternions are shown underneath the emperor flanked by the prince-electors (Archbishop of Trier, Archbishop of Cologne, Archbishop of Mainz; King of Bohemia, Count Palatine, Duke of Saxony, Margrave of Brandenburg). A " Quaternion Eagle" (each quaternion being represented by four coats of arms on the imperial eagle's remiges) Hans Burgkmair, c. 1510. Twelve quaternions are shown, as follows (eight dukes being divided into two quaternions called "pillars" and "vicars", respectively c.f. Christian Knorr von Rosenroth, Anführung zur Teutschen Staats-Kunst (1672), p. 669.): Seill ("pillars"), Vicari ("vicars"), Marggrauen (margraves), Lantgrauen (landgraves), Burggrauen (burggraves), Grauen (counts), Semper freie (nobles), Ritter (knights), Stett (cities), Dörfer (villages), Bauern (peasants), Birg (castles). The so- called imperial quaternions (German: Quaternionen der Reichsverfassung "quaternions of the imperial constitution"; from Latin quaterniō "group of four soldiers") were a conventional representation of the Imperial States of the Holy Roman Empire which first became current in the 15th century and was extremely popular during the 16th century.Hans Legband, "Zu den Quaternionen der Reichsverfassung", Archiv für Kulturgeschichte 3 (1905), 495–498. Ernst Schubert, "Die Quaternionen", Zeitschrift für historische Forschung 20 (1993), 1–63.
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Another important interpretation of the Brauer group of a field K is that it classifies the projective varieties over K that become isomorphic to projective space over an algebraic closure of K. Such a variety is called a Severi–Brauer variety, and there is a one-to-one correspondence between the isomorphism classes of Severi–Brauer varieties of dimension n−1 over K and the central simple algebras of degree n over K.Gille & Szamuely (2006), section 5.2. For example, the Severi–Brauer varieties of dimension 1 are exactly the smooth conics in the projective plane over K. For a field K of characteristic not 2, every conic over K is isomorphic to one of the form ax2 \+ by2 = z2 for some nonzero elements a and b of K. The corresponding central simple algebra is the quaternion algebraGille & Szamuely (2006), Theorem 1.4.2. :(a,b) = K\langle i,j\rangle/(i^2=a, j^2=b, ij=-ji). The conic is isomorphic to the projective line P1 over K if and only if the corresponding quaternion algebra is isomorphic to the matrix algebra M(2, K).
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If the order of q and β were to be reversed the result would not in general be α. The quaternion q can be thought of as an operator that changes β into α, by first rotating it, formerly an act of version and then changing the length of it, formerly call an act of tension. Also by definition the quotient of two vectors is equal to the numerator times the reciprocal of the denominator. Since multiplication of vectors is not commutative, the order cannot be changed in the following expression.
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In mathematics, a split-biquaternion is a hypercomplex number of the form :q = w + xi + yj + zk \\! where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight- dimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the split-complex numbers form the ring.
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In 1927, after earning her doctorate, she was engaged as an instructor of mathematics at Northwestern University in Evanston, Illinois, where she spent the remainder of her career. In 1930, she was promoted to assistant professor of mathematics, and in 1938 she was named associate professor. She retired from Northwestern University in 1964 and was named professor emeritus. During her professional career she published many mathematics papers such as "Generalized Quaternion Algebras and the Theory of Numbers" and "Representation of Integers in the Form x2 \+ 2y2 \+ 3z2 \+ 6w2", both in the American Journal of Mathematics.
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In the 19th century, the notion of a single "brown people" was sometimes superseded by multiple "brown peoples." Cust mentions Grammar in 1852 denying that there was one single "brown race", but in fact, several races speaking distinct languages. The 1858 Cyclopaedia of India and of eastern and southern Asia notes that Keane was dividing the "brown people" into quaternion: a western branch that he termed the Malay, a north-western group that he termed the Micronesian, and the peoples of the eastern archipelagos that he termed the Maori and the Polynesian.
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Originally, the eagle was represented with the holy cross or a picture of the crucified Jesus on its chest. The cross symbolized the Christian foundation of the Empire with the Imperial eagle protecting the church. Since the beginning of the 17th century, the crucified Jesus has generally been replaced with a representation of the Empire's orb. A total of 56 coats of arms of 'electors', as well as of estates of the empire and of imperial cities are depicted on the wings in quaternion formation as a symbol of the Imperial Constitution.
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The model divided the classes of the Empire into fictitious groups of four, the quaternions, whose members shared one common feature: hence, the group of worldly electors, the margraves, et cetera. This, however, often caused misleading and inappropriate constellations due to the endeavour to come up to the quaternion. Nevertheless, the success of the model was not affected. Early modern drinking culture, in which toasting was a very important custom, resulted in the beaker being connected to the "Imperial Eagle" as an expression of solidarity between the owner and the Empire.
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Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere , and SU(2) is the universal cover of SO(3).
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To do this, Sachs found, requires (2) generalizing Einstein's Mach principle, positing that all manifestations of matter, not only inertial mass, derive from the interaction of matter. From this, (3) quantum mechanics can be seen to emerge via the correspondence principle, as a nonrelativistic approximation for a theory of inertia in relativity. The result is a continuous quaternion-based formalism modeling all manifestations of matter. Sachs called the transformation symmetry group that Einstein sought in completing general covariance 'the Einstein group', which approaches the Poincaré group towards the flat spacetime of special relativity.
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"Physics at the British Association" Nature 56:461,2 (# 1454) A system of national secretaries was announced in the AMS Bulletin in 1899: Alexander McAulay for Australasia, Victor Schlegel for Germany, Joly for Great Britain and Ireland, Giuseppe Peano for Italy, Kimura for Japan, Aleksandr Kotelnikov for Russia, F. Kraft for Switzerland, and Arthur Stafford Hathaway for the USA. For France the national secretary was Paul Genty, an engineer with the division of Ponts et Chaussees, and a quaternion collaborator with Charles-Ange Laisant, author of Methode des Quaterniones (1881).
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Quaternion Slerps are commonly used to construct smooth animation curves by mimicking affine constructions like the de Casteljau algorithm for Bézier curves. Since the sphere is not an affine space, familiar properties of affine constructions may fail, though the constructed curves may otherwise be entirely satisfactory. For example, the de Casteljau algorithm may be used to split a curve in affine space; this does not work on a sphere. The two-valued Slerp can be extended to interpolate among many unit quaternions, but the extension loses the fixed execution-time of the Slerp algorithm.
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400px A representation of the UV mapping of a cube. The flattened cube net may then be textured to texture the cube. UV mapping is the 3D modeling process of projecting a 2D image to a 3D model's surface for texture mapping. The letters "U" and "V" denote the axes of the 2D texture because "X", "Y", and "Z" are already used to denote the axes of the 3D object in model space, while "W" (in addition to XYZ) is used in calculating quaternion rotations, a common operation in computer graphics.
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For p ≠ 2, one is a semi-direct product of Cp×Cp with Cp, and the other is a semi- direct product of Cp2 with Cp. The first one can be described in other terms as group UT(3,p) of unitriangular matrices over finite field with p elements, also called the Heisenberg group mod p. For p = 2, both the semi-direct products mentioned above are isomorphic to the dihedral group Dih4 of order 8. The other non-abelian group of order 8 is the quaternion group Q8.
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The relationship of quaternions to each other within the complex subplanes of can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions and commute (i.e., ) only if they lie in the same complex subplane of , the profile of as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. This method of commutative subrings is also used to profile the split-quaternions, which as an algebra over the reals are isomorphic to 2 × 2 real matrices.
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For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as a 16‑tuple. Although these uses treat ‑uple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from medieval Latin plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".OED, s.v.
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Working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially, he realized an equivalence relation on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane. The term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum of a Real number (also called scalar) and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments.
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The same argument eliminates the possibility of the coefficient field being the reals or the -adic numbers, because the quaternion algebra is still a division algebra over these fields. However it does not eliminate the possibility that the coefficient field is the field of -adic numbers for some prime , because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space. Grothendieck and Michael Artin managed to construct suitable cohomology theories over the field of -adic numbers for each prime , called -adic cohomology.
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Another source of complexity is the generality of Kempe's application to all algebraic curves. By focusing on parameterized algebraic curves, dual quaternion algebra can be used to factor the motion polynomial and obtain a drawing linkage.G.Hegedus, Z. Li, J. Schicho, H. P. Schrocker (2015), From the Fundamental Theorem of Algebra to Kempe’s Universality Theorem This has been extended to provide movement of the end-effector, but again for parameterized curves. M. Gallet, C. Koutschan, Z. Li, G. Regensburger, J. Schicho, and N. Villamiza (2017), Planar Linkages Following a Prescribed Motion, Mathematics of Computation, 86(303), pages 473-506.
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The dihedral group of order 8, G, satisfies: ℧(G) = Z(G) = [ G, G ] = Φ(G) = Soc(G) is the unique normal subgroup of order 2, typically realized as the subgroup containing the identity and a 180° rotation. However Ω(G) = G is the entire group, since G is generated by reflections. This shows that Ω(G) need not be the set of elements of order p. The quaternion group of order 8, H, satisfies Ω(H) = ℧(H) = Z(H) = [ H, H ] = Φ(H) = Soc(H) is the unique subgroup of order 2, normally realized as the subgroup containing only 1 and −1.
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Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions. The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the “Plate trick”).
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Examples include matrix algebras and quaternion algebras. A quasigroup is a structure in which division is always possible, even without an identity element and hence inverses. In an integral domain, where not every element need have an inverse, division by a cancellative element a can still be performed on elements of the form ab or ca by left or right cancellation, respectively. If a ring is finite and every nonzero element is cancellative, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, and division by any nonzero element is possible.
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Then there were two semesters spent in the Queen's University Belfast, where he lectured on mathematical mechanics. During this time, he wrote a long article giving the quaternion equations for the special relativity and for the motion of a charged particle emitting electromagnetic radiation. After the Second World War broke out, on 5 September 1939, Weiss expressed his desire to work for national defense. However, at that time, he did not have British citizenship, so on 12 May 1940, during a visit to Cambridge he was interned and in July sent to a special camp in Quebec.
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Quaternion Plaque on Broom Bridge The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843. However, in 1840, Benjamin Olinde Rodrigues had already reached a result that amounted to their discovery in all but name. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He failed to find a useful 3-dimensional system (in modern terminology, he failed to find a real, three-dimensional skew-field), but in working with four dimensions he created quaternions.
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Today, the quaternions are used in computer graphics, control theory, signal processing, and orbital mechanics, mainly for representing rotations/orientations. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining quaternion transformations is more numerically stable than combining many matrix transformations. In control and modelling applications, quaternions do not have a computational singularity (undefined division by zero) that can occur for quarter-turn rotations (90 degrees) that are achievable by many Air, Sea and Space vehicles.
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In the Euclidean plane, the real projective plane, higher-dimensional Euclidean spaces or real projective spaces, or spaces with coordinates in an ordered field, the Sylvester–Gallai theorem shows that the only possible Sylvester–Gallai configurations are one-dimensional: they consist of three or more collinear points. was inspired by this fact and by the example of the Hesse configuration to ask whether, in spaces with complex-number coordinates, every Sylvester–Gallai configuration is at most two-dimensional. repeated the question. answered Serre's question affirmatively; simplified Kelly's proof, and proved analogously that in spaces with quaternion coordinates, all Sylvester–Gallai configurations must lie within a three-dimensional subspace.
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The doubling method has come to be called the Cayley–Dickson construction. In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras). In 1931 Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate split-octonions.Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402 Adrian Albert also used the gamma in 1942 when he showed that Dickson doubling could be applied to any field with the squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms.
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Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, E.C. Dade, J.A. Green and J.G. Thompson, among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block. Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a dihedral group, semidihedral group or (generalized) quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann.
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This problem may be overcome by use of a fourth gimbal, actively driven by a motor so as to maintain a large angle between roll and yaw gimbal axes. Another solution is to rotate one or more of the gimbals to an arbitrary position when gimbal lock is detected and thus reset the device. Modern practice is to avoid the use of gimbals entirely. In the context of inertial navigation systems, that can be done by mounting the inertial sensors directly to the body of the vehicle (this is called a strapdown system) and integrating sensed rotation and acceleration digitally using quaternion methods to derive vehicle orientation and velocity.
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Founded in 1985 by poet and fiction writer John F. Deane, it is now run by poet and editor Pat Boran and manager Raffaela Tranchino. At present the press publishes approximately 8 new book-length publications each year, concentrating on contemporary poetry from Ireland but also regularly issuing anthologies and individual volumes by European writers in translation (often in bilingual formats). Dedalus also represents Thomas Kinsella's Peppercanister series of pamphlets, Iggy McGovern's occasional Quaternion Press and Pat Boran's own imprint, Orange Crate Books. According to MEAS report providing statistics for Irish poetry publications, Dedalus Press in 2018 was the joint-fourth most prolific poetry press on the Island of Ireland.
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The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors. Just the previous year, in Ireland, William Rowan Hamilton had discovered quaternions. It was not until English mathematician William Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector as it is known today was fully understood. Around this time Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication.
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The possibility of gravitational waves was also discussed by Heaviside using the analogy between the inverse-square law in gravitation and electricity.A gravitational and electromagnetic analogy,Electromagnetic Theory, 1893, 455-466 Appendix B. This was 25 years before Einstein's paper on this subject With quaternion multiplication, the square of a vector is a negative quantity, much to Heaviside’s displeasure. As he advocated abolishing this negativity, he has been credited by C. J. Joly with developing hyperbolic quaternions, though in fact that mathematical structure was largely the work of Alexander Macfarlane. He invented the Heaviside step function, using it to calculate the current when an electric circuit is switched on.
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It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular. The primes p such that every supersingular elliptic curve in characteristic p can be defined over the prime subfield F_p rather than F_{p^m} are called supersingular primes.
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Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as gravitational instantons in quantum theories of gravity. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose Weyl tensor is self-dual, and it is usually assumed that the metric is asymptotic to the standard metric of Euclidean 4-space (and are therefore complete but non-compact). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) hyperkähler manifolds in the Ricci-flat case, and quaternion Kähler manifolds otherwise. Higher-dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as string theory, M-theory and supergravity.
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Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow -subgroup.
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In mathematical finite group theory, a p-group of symplectic type is a p-group such that all characteristic abelian subgroups are cyclic. According to , the p-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an extraspecial group with a group that is cyclic, dihedral, quasidihedral, or quaternion. gives a proof of this result. The width n of a group G of symplectic type is the largest integer n such that the group contains an extraspecial subgroup H of order p1+2n such that G = H.CG(H), or 0 if G contains no such subgroup.
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Hart's career as a mathematics popularizer began in 2010 with a video series about "doodling in math class". After these recreational mathematics videos — which introduced topics like fractal dimension — grew popular, they were featured in The New York Times and on National Public Radio, eventually gaining the support of the Khan Academy and making videos for the educational site as their "Resident Mathemusician". Many of Hart's videos combined mathematics and music, such as "Twelve tones", which was called "deliriously and delightfully profound" by Salon. Together with Henry Segerman, Hart wrote "The Quaternion Group as a Symmetry Group", which was included in the anthology The Best Writing on Mathematics 2015.
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William Thomson, Lord Kelvin (1904) Molecular dynamics and the wave theory of light, twenty lectures transcribed by A.S. Hathaway, Cambridge University Press, link from Internet Archive In 1987 Hathaway's original transcription from 1884 was published when Johns Hopkins Center for the History and Philosophy of Science decided to commemorate the centennial of Kelvin's lectures.Robert Kargon and Peter Achinstein (1987) Kelvin’s Baltimore Lectures and Modern Theoretical Physics: historical and philosophical perspectives, MIT Press In Terre Haute, Indiana Hathaway taught at Rose Polytechnic Institute until 1920 and published A Primer on Quaternions in 1896. He became the U.S. national secretary for the international Quaternion Society in 1899.
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One of the features of Hamilton's quaternion system was the differential operator del which could be used to express the gradient of a vector field or to express the curl. These operations were applied by Clerk Maxwell to the electrical and magnetic studies of Michael Faraday in Maxwell's Treatise on Electricity and Magnetism (1873). Though the del operator continues to be used, the real quaternions fall short as a representation of spacetime. On the other hand, the biquaternion algebra, in the hands of Arthur W. Conway and Ludwik Silberstein, provided representational tools for Minkowski space and the Lorentz group early in the twentieth century.
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The non-triviality of the (additional) conjugacy conclusion can be illustrated with the Klein four-group V as the non-example. Any of the three proper subgroups of V (all of which have order 2) is normal in V; fixing one of these subgroups, any of the other two remaining (proper) subgroups complements it in V, but none of these three subgroups of V is a conjugate of any other one, because V is Abelian. The quaternion group has normal subgroups of order 4 and 2 but is not a [semi]direct product. Schur's papers at the beginning of the 20th century introduced the notion of central extension to address examples such as C_4 and the quaternions.
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The ring of endomorphisms of an elliptic curve can be of one of three forms:the integers Z; an order in an imaginary quadratic number field; or an order in a definite quaternion algebra over Q.Silverman (1989) p. 102 When the field of definition is a finite field, there are always non-trivial endomorphisms of an elliptic curve, coming from the Frobenius map, so the complex multiplication case is in a sense typical (and the terminology is not often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the Hodge conjecture.
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Brougham (Broom) Bridge, Dublin, which says: > Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton > in a flash of genius discovered the fundamental formula for quaternion > multiplication > > & cut it on a stone of this bridge. In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.
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By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge: : Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. From 1844 to 1850 Philosophical Magazine communicated Hamilton's exposition of quaternions.W.R. Hamilton(1844 to 1850) On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, link to David R. Wilkins collection at Trinity College, Dublin In 1853 he issued Lectures on Quaternions, a comprehensive treatise that also described biquaternions.
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Octonions were developed independently by Arthur Cayley in 1845 Penrose 2004 pg 202 and John T. Graves, a friend of Hamilton's. Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold [the three imaginary units], why should you stop there?" Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843 presenting a kind of double quaternionSee Penrose Road to Reality pg. 202 'Graves discovered that there exists a kind of double quaternion...' that is called an octonion, and showed that they were what we now call a normed division algebra; Graves called them octaves.
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Visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can be visualized by the associated tilings, as depicted at right: the (2,3,7) tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane.
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Malcolm D. Shuster (31 July 1943 – 23 February 2012) was an American physicist and aerospace engineer, whose work contributed significantly to spacecraft attitude determination. In 1977 he joined the Attitude Systems Operation of the Computer Sciences Corporation in Silver Spring, Maryland, during which time he developed the QUaternion ESTimator (QUEST) algorithm for static attitude determination. He later, with F. Landis Markley, helped to develop the standard implementation of the Kalman filter used in spacecraft attitude estimation. During his career, he authored roughly fifty technical papers on subjects in physics and spacecraft engineering, many of which have become seminal within the field of attitude estimation, and held teaching assignments at Johns Hopkins University, Howard University, Carnegie-Mellon University and Tel-Aviv University.
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Much of the structure of a finite group is carried in the structure of its so-called local subgroups, the normalizers of non-identity p-subgroups. The large elementary abelian subgroups of a finite group exert control over the group that was used in the proof of the Feit–Thompson theorem. Certain central extensions of elementary abelian groups called extraspecial groups help describe the structure of groups as acting on symplectic vector spaces. Richard Brauer classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and John Walter, Daniel Gorenstein, Helmut Bender, Michio Suzuki, George Glauberman, and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.
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For X0(N) and X1(N), the level structure is, respectively, a cyclic subgroup of order N and a point of order N. These curves have been studied in great detail, and in particular, it is known that X0(N) can be defined over Q. The equations defining modular curves are the best-known examples of modular equations. The "best models" can be very different from those taken directly from elliptic function theory. Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves. Remark: quotients of H that are compact do occur for Fuchsian groups Γ other than subgroups of the modular group; a class of them constructed from quaternion algebras is also of interest in number theory.
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The stabilizer of 3 points is the projective special unitary group PSU(3,22), which is solvable. The stabilizer of 4 points is the quaternion group. Likewise, M24 has a maximal simple subgroup of order 6072 isomorphic to PSL2(F23). One generator adds 1 to each element of the field (leaving the point N at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(N), and the other is the order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M24 sends an element x of F23 to 4x4 − 3x15 (which sends perfect squares via x^4 and non-perfect squares via 7 x^4); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).
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The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group: this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus showed that it has a normal subgroup of index 1 or 2 that is the product of SL(2,5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5).
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The Genesis platform has gone through several versions since the launch in 2011: Genesis 2: One of the shortcomings of the Genesis platform was that although it allowed extremely flexibility in the shape of characters and clothing, it also toned down some of the elements of what made a male or female figure unique. Genesis 2 changed this by splitting the Genesis base figure into two separate base figures: Genesis 2 Male and Genesis 2 Female. Genesis 3: Up until Genesis 3, the Genesis figures had been using TriAx Weight Maps, where many other industry platforms were using Dual Quaternion. This changed in Genesis 3 to allow Daz 3D figures to be more compatible with other 3D software platforms as well as Game Development platforms.
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R.S. Ball (1883) The Boundaries of Astronomy Part I and Part II Ball expounded the tides in Time and Tide: a Romance of the MoonSee Project Gutenberg In 1892 he was appointed Lowndean Professor of Astronomy and Geometry at Cambridge University at the same time becoming director of the Cambridge Observatory. He was a fellow of King's College, Cambridge. In 1900 Cambridge University Press published A Treatise on the Theory of ScrewsR.S. Ball (1900) A Treatise on the Theory of Screws, weblink from Cornell University Historical Math Monographs That year he also published The Story of the HeavensThe Story of the Heavens is available from Project Gutenberg (external link) Much in the limelight, he stood as President of the Quaternion Society.
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In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L. E. J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics. The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Edinburgh Mathematical Society in 1883, the Circolo Matematico di Palermo in 1884, and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy.
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After being spoiled by the action of vandals and some visitors, the plaque was moved to a different place, higher, under the railing of the bridge. The text on the plaque reads: > Here as he walked by > on the 16th of October 1843 > Sir William Rowan Hamilton > in a flash of genius discovered > the fundamental formula for > quaternion multiplication > i² = j² = k² = ijk = −1 > & cut it on a stone of this bridge. > Given the historical importance of the bridge with respect to mathematics, mathematicians from all over the world have been known to take part in the annual commemorative walk from Dunsink Observatory to the site. Attendees have included Nobel Prize winners Murray Gell-Mann, Steven Weinberg and Frank Wilczek, and mathematicians Sir Andrew Wiles, Sir Roger Penrose and Ingrid Daubechies.
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The computation of the homology groups of the product of two topological spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined. The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to de Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product. The resulting rather severe write-up of the topic (by Bourbaki) entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups).
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Daryush took her father's experiment in syllabic verse a step farther by making it less experimental; whereas Bridges' syllable count excluded elidable syllables, producing some variation in the total number of pronounced syllables per line, Daryush's was strictly aural, counting all syllables actually sounded when the poem was read aloud. It is for her successful experiments with syllabic meter that Daryush is best known to contemporary readers, as exampled in her poem Accentedal in the quaternion form.'Biography of Elizabeth Daryush' MyPoeticSide.com Yvor Winters, the poet and critic, considered Daryush more successful in writing syllabics than was her father, noting that her poem Still-Life was her finest syllabic experiment,Winters, Yvor, Primitivism and Decadence: A Study of American Experimental Poetry Arrow Editions , New York , 1937 and also a companion-piece to Children of Wealth.
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Sophus Lie was less than a year old when Hamilton first described quaternions, but Lie's name has become associated with all groups generated by exponentiation. The set of versors with their multiplication has been denoted Sl(1,q) by Robert Gilmore in his text on Lie theory.Robert Gilmore (1974) Lie Groups, Lie Algebras and some of their Applications, chapter 5: Some simple examples, pages 120–35, Wiley Gilmore denotes the real, complex, and quaternion division algebras by r, c, and q, rather than the more common R, C, and H. Sl(1,q) is the special linear group of one dimension over quaternions, the "special" indicating that all elements are of norm one. The group is isomorphic to SU(2,c), a special unitary group, a frequently used designation since quaternions and versors are sometimes considered anachronistic for group theory.
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In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H. The Bender–Suzuki theorem, proved by extending work of , classifies the groups G with a strongly embedded subgroup H. It states that either # G has cyclic or generalized quaternion Sylow 2-subgroups and H contains the centralizer of an involution # or G/O(G) has a normal subgroup of odd index isomorphic to one of the simple groups PSL2(q), Sz(q) or PSU3(q) where q≥4 is a power of 2 and H is O(G)NG(S) for some Sylow 2-subgroup S. revised Suzuki's part of the proof. extended Bender's classification to groups with a proper 2-generated core.
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Note the negative sign is introduced to simplify the correspondence with quaternions. Denote a set of mutually orthogonal unit vectors of R4 as e1, e2, e3 and e4, then the Clifford product yields the relations :e_m e_n = -e_n e_m, \,\,\, m e n, and :e_1 ^2 = e_2^2 =e_3^2 = -1, \,\, e_4^2 = 0. The general element of the Clifford algebra has 16 components. The linear combination of the even degree elements defines the even subalgebra with the general element : H = h_0 + h_1 e_2 e_3 + h_2 e_3 e_1 + h_3 e_1 e_2 + h_4 e_4 e_1 + h_5 e_4 e_2 + h_6 e_4 e_3 + h_7 e_1 e_2 e_3 e_4. The basis elements can be identified with the quaternion basis elements i, j, k and the dual unit ε as : i = e_2 e_3, j = e_3 e_1, k = e_1 e_2, \,\, \varepsilon = e_1 e_2 e_3 e_4.
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He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by Ernst Witt. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular , based on the Selberg trace formula for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory. In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the length spectrum, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.
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Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions and the sedenions; they are also useful tool in number theory, particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group. The study of integral quaternions began with Rudolf Lipschitz in 1886, whose system was later simplified by Leonard Eugene Dickson; but the modern system was published by Adolf Hurwitz in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions all four of whose coordinates are half-integers.
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According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation : suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). This event marks the discovery of the quaternion group. A plaque under the bridge was unveiled by the Taoiseach Éamon de Valera, himself a mathematician and student of quaternions,De Valera School of Mathematics and Statistics University of St Andrews, Scotland on 13 November 1958. Since 1989, the National University of Ireland, Maynooth has organised a pilgrimage called the Hamilton Walk, in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery.
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About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra: :Historically, the first non- associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras... In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit , and for quaternions and writes a Cayley number . Denoting the quaternion conjugate by , the product of two Cayley numbers is :(q + Qe)(r + Re) = (qr - R'Q) + (Rq + Q r')e . The conjugate of a Cayley number is , and the quadratic form is , obtained by multiplying the number by its conjugate.
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The book has eight chapters: the first on the origins of vector analysis including Ancient Greek and 16th and 17th century influences; the second on the 19th century William Rowan Hamilton and quaternions; the third on other 19th and 18th century vectorial systems including equipollence of Giusto Bellavitis and the exterior algebra of Hermann Grassmann. Chapter four is on the general interest in the 19th century on vectorial systems including analysis of journal publications as well as sections on major figures and their views (e.g., Peter Guthrie Tait as an advocate of Quaternions and James Clerk Maxwell as a critic of Quaternions); the fifth chapter describes the development of the modern system of vector analysis by Josiah Willard Gibbs and Oliver Heaviside. In chapter six, "Struggle for existence", Michael J. Crowe delves into the zeitgeist that pruned down quaternion theory into vector analysis on three-dimensional space.
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There are various connections between the lower central series (LCS) and upper central series (UCS) , particularly for nilpotent groups. Most simply, a group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is trivial) if and only if the UCS stabilizes at the first step (the center is the entire group). More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree (and is called the nilpotency class of the group). However, the LCS and UCS of a nilpotent group may not necessarily have the same terms. For example, while the UCS and LCS agree for the cyclic group C2 and quaternion group Q8 (which are C2 ⊵ {e} and Q8 ⊵ {1, -1} ⊵ {1} respectively), the UCS and LCS of their direct product C2 × Q8 do not: its lower central series is C2 × Q8 ⊵ {e} × {-1, 1} ⊵ {e} × {1}, while the upper central series is C2 × Q8 ⊵ C2 × {-1, 1} ⊵ {e} × {1}.
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She was one of a handful of American women poets who produced imaginative verse in the two centuries that mark the beginning of an American poetic literary tradition. Previous colonial American women poets, Anne Bradstreet and Jane Colman Turell, focused primarily on religion and family life. Brewster's 21 poems vary widely in theme and form: the more than 1100 lines include letters, farewells to friends who are moving, epithalamiums, eulogies, scriptural paraphrases, a love poem, a quaternion, a dream (in prose), and meditations. While she does write about more conventional religious and family themes, her work is also the first to tackle radical subject matterBert, 71 for a woman of the eighteenth-century and reflects a shift from those themes to focus on the evils of war, military invasion and conquest and its cumulative effect on a nation and its citizens; and locates a woman's voice alongside those of the male founders of the country.
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Below the groups explained above are arranged by abstract group type. The smallest abstract groups that are not any symmetry group in 3D, are the quaternion group (of order 8), Z3 × Z3 (of order 9), the dicyclic group Dic3 (of order 12), and 10 of the 14 groups of order 16. The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C2, Ci, Cs. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group. Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 2n + 1 elements of order 2, and there are three with 2n + 3 elements of order 2 (for each n ≥ 2 ).
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This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley–Hamilton theorem). Hamilton also invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.
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A finite- dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion. The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F. One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).
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The textbook presentations in , , , , all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of splitting called p-nilpotence. During the course of the Alperin–Brauer–Gorenstein theorem classifying finite simple groups with quasi- dihedral Sylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups: the 2-nilpotent groups, the Q-type groups whose focal subgroup is a generalized quaternion group of index 2, the D-type groups whose focal subgroup a dihedral group of index 2, and the QD-type groups whose focal subgroup is the entire quasi-dihedral group. In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the Q-type have 2 classes of involutions and one class of cyclic subgroup of order 4; the QD-type have one class each of involutions and cyclic subgroups of order 4. In other words, finite groups with quasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns.
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