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searching for Quaternion algebra 35 found (71 total)

Quaternionic vector space (349 words) [view diff] exact match in snippet view article find links to article

Since quaternion algebra is division ring, then module over quaternion algebra is called vector space. Because quaternion algebra is non-commutative,

Hyperbolic quaternion (2,107 words) [view diff] exact match in snippet view article find links to article

Furthermore, just as the quaternion algebra H can be viewed as a union of complex planes, so the hyperbolic quaternion algebra is a pencil of planes of

Klein quartic (3,263 words) [view diff] exact match in snippet view article find links to article

Γ(I) is a subgroup of the group of elements of unit norm in the quaternion algebra generated as an associative algebra by the generators i,j and relations

Rotor (mathematics) (839 words) [view diff] exact match in snippet view article

term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension"

Applications of dual quaternions to 2D geometry (1,473 words) [view diff] exact match in snippet view article find links to article

In this article, certain applications of the dual quaternion algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional

Irving Stringham (471 words) [view diff] exact match in snippet view article find links to article

Saxony, when he submitted an article on finite groups found in the quaternion algebra. Stringham began his professorship in mathematics at Berkeley in 1882

Versor (2,945 words) [view diff] exact match in snippet view article find links to article

multiplication, forms a group, and appears as a 3-sphere in the 4-dimensional quaternion algebra. Hamilton denoted the versor of a quaternion q by the symbol U q.

Alexander McAulay (599 words) [view diff] exact match in snippet view article find links to article

proceeded to Caius College, Cambridge, there taking up a study of the quaternion algebra. In 1883 he published an article "Some general theorems in quaternion

Opposite ring (3,532 words) [view diff] exact match in snippet view article find links to article

\end{aligned}}} which are not equal elements. The quaternion algebra H ( a , b ) {\displaystyle H(a,b)} over a field F {\displaystyle F}

Dual quaternion (4,786 words) [view diff] exact match in snippet view article find links to article

In 1898 Alexander McAulay used Ω with Ω2 = 0 to generate the dual quaternion algebra. However, his terminology of "octonions" did not stick as today's

Macbeath surface (648 words) [view diff] exact match in snippet view article find links to article

suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page

E. H. Moore (709 words) [view diff] exact match in snippet view article find links to article

over a finite field Moore determinant of a Hermitian matrix over a quaternion algebra "Obituary Record of Graduates of Yale University Deceased during the

Quaternionic manifold (1,493 words) [view diff] exact match in snippet view article find links to article

x {\displaystyle H_{x}} is isomorphic (as a real algebra) to the quaternion algebra H {\displaystyle \mathbb {H} } . The subbundle H {\displaystyle H}

Hasse invariant of an algebra (771 words) [view diff] exact match in snippet view article find links to article

are two Brauer classes, represented by the algebra R itself and the quaternion algebra H. It is convenient to assign invariant zero to the class of R and

Slerp (1,181 words) [view diff] exact match in snippet view article find links to article

ensuring that −90° ≤ Ω ≤ 90°. Slerp also has expressions in terms of quaternion algebra, all using exponentiation. Real powers of a quaternion are defined

Georg Scheffers (1,081 words) [view diff] exact match in snippet view article find links to article

e_{3}e_{1}-e_{1}e_{3}=2e_{2}.} In today's language, Scheffers' Qss has the quaternion algebra as a subalgebra. Scheffers anticipates the concepts of direct product

Lars Onsager (2,216 words) [view diff] exact match in snippet view article find links to article

(later called the Onsager algebra). The solution involved generalized quaternion algebra and the theory of elliptic functions, which he learned from A Course

Euler's rotation theorem (4,497 words) [view diff] exact match in snippet view article find links to article

Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the

Quaternion-Kähler symmetric space (264 words) [view diff] exact match in snippet view article find links to article

of the subalgebras of the octonion algebra O {\displaystyle \mathbb {O} } which are isomorphic to the quaternion algebra H {\displaystyle \mathbb {H} }

Alexander Macfarlane (1,418 words) [view diff] exact match in snippet view article find links to article

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

History of quaternions (2,294 words) [view diff] exact match in snippet view article find links to article

"biradials" to designate great circle arcs on the sphere. Then the quaternion algebra provided the foundation for spherical trigonometry introduced in chapter

Kempe's universality theorem (842 words) [view diff] exact match in snippet view article find links to article

algebraic curves. By focusing on parameterized algebraic curves, dual quaternion algebra can be used to factor the motion polynomial and obtain a drawing linkage

Kleinian group (2,344 words) [view diff] exact match in snippet view article find links to article

it is commensurable with the group norm 1 elements of an order of quaternion algebra A ramified at all real places over a number field k with exactly one

Quaternion group (3,716 words) [view diff] exact match in snippet view article find links to article

{C} )} . The quaternion group is a multiplicative subgroup of the quaternion algebra: H = R 1 + R i + R j + R k = C 1 + C j , {\displaystyle \mathbb {H}

Quaternionic analysis (2,820 words) [view diff] exact match in snippet view article find links to article

f(x)}{\operatorname {d} x}}:\mathbb {H} \rightarrow \mathbb {H} } is linear map of quaternion algebra H , {\displaystyle \ \mathbb {H} \ ,} and o : H → H {\displaystyle

Non-associative algebra (3,005 words) [view diff] exact match in snippet view article find links to article

than GF(2) (see previous section), and the sedenions. The hyperbolic quaternion algebra over R, which was an experimental algebra before the adoption of Minkowski

Split-complex number (4,144 words) [view diff] exact match in snippet view article find links to article

introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements "motors", a

William Rowan Hamilton (4,996 words) [view diff] exact match in snippet view article find links to article

dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along the

Symmetric space (4,599 words) [view diff] exact match in snippet view article find links to article

Space of subalgebras of the octonion algebra O {\displaystyle \mathbb {O} } which are isomorphic to the quaternion algebra H {\displaystyle \mathbb {H} }

Lagrange's identity (3,679 words) [view diff] exact match in snippet view article find links to article

x^{2}+\ y^{2}\ +\ z^{2}.} The multiplicativity of the norm in the quaternion algebra provides, for quaternions p and q: | p q | = | p | | q | . {\displaystyle

Classical Hamiltonian quaternions (5,252 words) [view diff] exact match in snippet view article find links to article

terminology. Hamilton's tensor is actually the absolute value on the quaternion algebra, which makes it a normed vector space. Hamilton defined tensor as

Spacetime algebra (7,332 words) [view diff] exact match in snippet view article find links to article

algebra to even subalgebra continues as algebra of physical space, quaternion algebra, complex numbers and real numbers. The even STA subalgebra Cl+(1,3)

Cross product (11,568 words) [view diff] exact match in snippet view article find links to article

special case of the multiplicativity |vw| = |v||w| of the norm in the quaternion algebra. It is a special case of another formula, also sometimes called Lagrange's

3D rotation group (11,446 words) [view diff] exact match in snippet view article find links to article

orthogonal matrices with determinant 1, by axis and rotation angle in quaternion algebra with versors and the map 3-sphere S3 → SO(3) (see quaternions and

Basil Hiley (11,814 words) [view diff] exact match in snippet view article find links to article

from each (2n-dimensional) phase space" and discussed relations of quaternion algebra, symplectic geometry and quantum mechanics. In 2011, de Gosson and