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Function theory with quaternion variable
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of
Quaternionic_analysis
Branch of mathematical analysis
quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis). A second instance involves functions of a motor variable
Hypercomplex_analysis
Four-dimensional number system
Springer-Verlag. ISBN 0-387-96980-2. Kravchenko, Vladislav (2003). Applied Quaternionic Analysis. Heldermann Verlag. ISBN 3-88538-228-8. Kuipers, Jack (2002). Quaternions
Quaternion
spin manifold. In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When n = 4, the Dirac operator is sometimes referred
Clifford_analysis
Mathematical operation
matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform
Cayley_transform
Algebraic structure designed for geometry
vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector
Geometric_algebra
geometry used to describe the physical phenomena of quantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must appear
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Fundamental construction of differential calculus
corresponds to the integral, whence the term differintegral. In quaternionic analysis, derivatives can be defined in a similar way to real and complex
Generalizations of the derivative
Generalizations_of_the_derivative
Generalization of the concept of directional derivative
Generalization of a derivative of a function between two Banach spaces Quaternionic analysis – Function theory with quaternion variable Semi-differentiability –
Gateaux_derivative
Swiss mathematician (1880–1950)
research on algebraic number theory and quaternion analysis proposing a definition of ‘regular’ for quaternionic functions similar to the definition of holomorphic
Rudolf_Fueter
Concept in linear algebra
A quaternionic matrix is a matrix whose elements are quaternions. The quaternions form a noncommutative ring, and therefore addition and multiplication
Quaternionic_matrix
Mathematical group
\operatorname {Sp} (n)} is given by the quaternionic skew-Hermitian matrices, the set of n × n {\displaystyle n\times n} quaternionic matrices that satisfy A + A
Symplectic_group
German mathematician
and quantum physics). In 1998 he was an Invited Speaker with talk Quaternionic analysis of Riemann surfaces and differential geometry at the International
Ulrich_Pinkall
Non-tensorial representation of the spin group
conditions. When S {\displaystyle S} is of quaternionic type, the representation carries an invariant quaternionic structure but no invariant real structure
Spinor
Quaternions with complex number coefficients
180, ISBN 978-0-521-37165-0 Kravchenko, Vladislav (2003), Applied Quaternionic Analysis, Heldermann Verlag, ISBN 3-88538-228-8 Lanczos, Cornelius (1949)
Biquaternion
German mathematician
specialising in differential geometry and known for her work on quaternionic analysis and Willmore surfaces. She works in England as a reader in mathematics
Katrin_Leschke
Italian mathematician
of Quaternionic Hyperfunctions, was supervised by Daniele C. Struppa. Sabadini is the author of multiple books in mathematics including: Analysis of Dirac
Irene_Sabadini
mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient
Hypertoric_variety
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
orthonormal k-frames in C n {\displaystyle \mathbb {C} ^{n}} and the quaternionic Stiefel manifold V k ( H n ) {\displaystyle V_{k}(\mathbb {H} ^{n})}
Stiefel_manifold
Hypercomplex number system
2025 – via GitHub. Imaeda, K.; Imaeda, M. (2000). "Sedenions: algebra and analysis". Applied Mathematics and Computation. 115 (2): 77–88. doi:10.1016/S0096-3003(99)00140-X
Sedenion
Manifold equipped with a quaternionic structure
structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. Every hyperkähler manifold is also hypercomplex
Hypercomplex_manifold
Every polynomial has a real or complex root
Eilenberg–Niven theorem, a generalization of the theorem to polynomials with quaternionic coefficients and variables Hilbert's Nullstellensatz, a generalization
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Finite simple group type not classified as Lie, cyclic or alternating
a type 2-3-3 triangle J2 is the group of automorphisms preserving a quaternionic structure (modulo its center). Consists of subgroups which are closely
Sporadic_group
Random matrix with gaussian entries
{\displaystyle M^{*}} is its transpose. If M {\displaystyle M} is complex or quaternionic, then M ∗ {\displaystyle M^{*}} is its conjugate transpose. λ 1 , …
Gaussian_ensemble
(pseudo-)Riemannian manifold whose geodesics are reversible
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
Symmetric_space
Matrix-valued random variable
{1}{Z_{{\text{GSE}}(n)}}}e^{-n\mathrm {tr} H^{2}}} on the space of n × n Hermitian quaternionic matrices, e.g. symmetric square matrices composed of quaternions, H =
Random_matrix
Topologist
Washington (September 1989). "Nonlinearly Equivalent Representations of Quaternionic 2-Groups" (PDF). Transactions of the American Mathematical Society. 315
Washington_Mio
Smooth manifold with an inner product on each tangent space
metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues of the real projective
Riemannian_manifold
Element of a unital algebra over the field of real numbers
{\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields a quaternionic matrix and its even subalgebra H ⊗ 2 ⊗ R C {\displaystyle \mathbb {H}
Hypercomplex_number
Study of complex manifolds and several complex variables
complex structures I , J , K {\displaystyle I,J,K} which satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − Id {\displaystyle
Complex_geometry
Geometric model of the physical space
5. ISBN 978-0-19-960139-4. Morais, João Pedro; et al. (2014). Real Quaternionic Calculus Handbook. Springer Science & Business Media. pp. 1–13. ISBN 978-3-0348-0622-0
Three-dimensional_space
Fiber bundle whose fibers are group torsors
S^{4n+3}} is a principal S p ( 1 ) {\displaystyle Sp(1)} -bundle over quaternionic projective space H P n {\displaystyle \mathbb {H} \mathbb {P} ^{n}}
Principal_bundle
Special mathematical functions defined on the surface of a sphere
certain spin representations of SO(3), with respect to the action by quaternionic multiplication. Spherical harmonics can be separated into two sets of
Spherical_harmonics
American scientist (1839–1903)
other physicists of the convenience of the vectorial approach over the quaternionic calculus of William Rowan Hamilton, which was then widely used by British
Josiah_Willard_Gibbs
Mnemonic for 3D vectors orientations and rotations
Following a substantial debate, the mainstream shifted from Hamilton's quaternionic system to Gibbs's three-vectors system. This transition led to the prevalent
Right-hand_rule
Menger sponge Newton fractal Nova fractal - derived from Newton fractal Quaternionic fractal - three dimensional complex quadratic map Sierpinski carpet Sierpinski
List_of_chaotic_maps
American mathematician
Zbl 0553.32008. Galicki, K.; Lawson, H. Blaine Jr. (1988). "Quaternionic reduction and quaternionic orbifolds". Mathematische Annalen. 282 (1): 1–21. doi:10
H._Blaine_Lawson
Hypercomplex number system
basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element): ( I , j , k ) , ( i
Octonion
Generalization of a polytope in real space
triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection. Quaternionic polytope Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation
Complex_polytope
Representation theory
the Weyl group of A. The group G = SL(2,C) acts transitively on the quaternionic upper half space H 3 = { x + y i + t j ∣ t > 0 } {\displaystyle {\mathfrak
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Russian-French mathematician
Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978, Gromov introduced the
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Correspondence between quaternions and 3D rotations
{\displaystyle {\vec {u}}} that specifies a rotation as to axial vectors. In quaternionic formalism the choice of an orientation of the space corresponds to order
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
operators on an infinite-dimensional real, complex or quaternionic Hilbert space. The quaternionic space is defined as all sequences x = (xi) with xi in
Jordan_operator_algebra
Equations describing classical electromagnetism
and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used. Maxwell's equations are partial differential equations
Maxwell's_equations
Four-dimensional associative algebra over the reals
2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature, k is replaced
Split-quaternion
Concept in differential geometry
Date incompatibility (help) Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc., 71, 3, 1 (3): 526–7, doi:10
Holonomy
representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation Schur's
List of representation theory topics
List_of_representation_theory_topics
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
sometimes denoted H(A,σ). 1. The set of self-adjoint real, complex, or quaternionic matrices with multiplication ( x y + y x ) / 2 {\displaystyle (xy+yx)/2}
Jordan_algebra
Matrix representing a Euclidean rotation
\mathrm {SO} (3).} For a detailed account of the SU(2)-covering and the quaternionic covering, see spin group SO(3). Many features of these cases are the
Rotation_matrix
Double cover Lie group of the special orthogonal group
simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact SO(p) × SO(q)
Spin_group
Smooth manifold
vanishing pure spinor then M is a generalized Calabi–Yau manifold. Almost quaternionic manifold – Concept in geometryPages displaying short descriptions of
Almost_complex_manifold
Particular projective representations of the orthogonal or special orthogonal groups
that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure. There is an invariant complex antilinear
Spin_representation
Mathematics term
≥ 2. For n ≥ 2, the noncompact Lie group Sp(n, 1) of isometries of a quaternionic hermitian form of signature (n,1) is a simple Lie group of real rank
Kazhdan's_property_(T)
researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial
List_of_women_in_mathematics
Quaternion of norm 1 (unit quaternion)
binary icosahedral group. A hyperbolic versor is a generalization of quaternionic versors to indefinite orthogonal groups, such as Lorentz group. It is
Versor
Mathematical concept
and Kottwitz (2005) Harry Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics, 1657, Springer, 1997
Shimura_variety
Mathematical concept
Sabadini; M Shapiro; F Sommen (eds.). Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed.). Birkhäuser. p. 168. ISBN 978-3-7643-9892-7
Seven-dimensional cross product
Seven-dimensional_cross_product
American mathematician (1936–2023)
Mathematica, v. 141 (2005), pp. 1504–1530. arXiv:math/0402283 Complex forms of quaternionic symmetric spaces, in Complex, contact and symmetric manifolds, Progress
Joseph_A._Wolf
Representations of finite groups, particularly on vector spaces
complex conjugate representations of G . {\displaystyle G.} Definition. A quaternionic representation is a (complex) representation V , {\displaystyle V,} which
Representation theory of finite groups
Representation_theory_of_finite_groups
Theorem in quantum mechanics
measurements are defined must be a real or complex Hilbert space, or a quaternionic module. (Gleason's argument is inapplicable if, for example, one tries
Gleason's_theorem
Form of differential geometry
the quaternionic projective plane is not its systolically optimal metric, in contrast with the 2-systole in the complex case. While the quaternionic projective
Systolic_geometry
U(N) to U(N – 1) states that Example. The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations
Restricted_representation
Type of Dirichlet series associated to number field extensions
algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root number is the subject of significant research
Artin_L-function
Mathematical result in differential geometry
that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so
Atiyah–Singer_index_theorem
Italian mathematician (1911–1999)
; Pontecorvo, M., eds. (1999), Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics. Dedicated to the Memory of André
Enzo_Martinelli
Structure group sub-bundle on a tangent frame bundle
(1997). "Canonical connections for almost-hypercomplex structures". Complex Analysis and Geometry. Pitman Research Notes in Mathematics Series. Vol. 366. Longman
G-structure_on_a_manifold
Low-rank isomorphisms in mathematics
)\times \mathrm {SL} (2,\mathbf {R} )\to \mathrm {SO} (2,2).} On the quaternionic real form one recovers the compact case S U ( 2 ) × S U ( 2 ) → S O (
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
Spin representations of the SO(3) group
constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that
Spinors_in_three_dimensions
S2CID 119288632. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8
Snub_24-cell
Riemannian manifold equipped with a differential p-form
Sci. Paris. 260: 5445–5448. Kraines, Vivian Yoh (1965). "Topology of quaternionic manifolds". Bull. Amer. Math. Soc. 71, 3, 1 (3): 526–527. doi:10
Calibrated_geometry
Canadian-American mathematician
1994 Thompson, Robert C. (1997). "The upper numerical range of a quaternionic matrix is not a complex numerical range". Linear Algebra and Its Applications
Robert_Charles_Thompson
are polynomials. Positive matrix A matrix with all positive entries. Quaternionic matrix A matrix whose entries are quaternions. Random matrix A matrix
List_of_named_matrices
Discrete subgroup in a locally compact topological group
1)} (groups of matrices with quaternion coefficients which preserve a "quaternionic quadratic form" of signature ( n , 1 ) {\displaystyle (n,1)} ) for n
Lattice_(discrete_subgroup)
Complex vector of electromagnetic fields
transition is made: With the advent of spinor calculus that superseded the quaternionic calculus, the transformation properties of the Riemann-Silberstein vector
Riemann–Silberstein_vector
QUATERNIONIC ANALYSIS
QUATERNIONIC ANALYSIS
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Close inspection, A review, Analysis
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Hindu
Analysis
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Hindu
Close inspection, A review, Analysis
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Samiksha | ஸமீகà¯à®·à®¾
Analysis
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Hindu
Analysis
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Indian
Analysis
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Muslim
Analysis
Biblical
a guard of four soldiers,...and delivered him to four quaternions of soldiers to guard him...
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Sameksha | ஸமேகà¯à®·à®¾
Analysis
Sameksha | ஸமேகà¯à®·à®¾
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Tamil
Sameeksha | ஸமீகà¯à®·à®¾Â
Analysis
Sameeksha | ஸமீகà¯à®·à®¾Â
Girl/Female
Indian, Telugu
Review; Analysis
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Hindu
Analysis
QUATERNIONIC ANALYSIS
QUATERNIONIC ANALYSIS
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Arabic, Muslim
Al-ameeh was a Great Worshipper who Worshipped Long in the Night Sometimes
Surname or Lastname
English
English : patronymic from a short form of the personal name Jeffrey.
Boy/Male
British, English, Finnish, French, German, Latin, Polish, Swedish
From Aurehanus which is Derived from the Latin Aurum; Fair; Golden Haired; Little Golden One
Girl/Female
Gujarati, Hindu, Indian, Kannada, Sanskrit, Tamil, Telugu
The Goddess of Wealth and Knowledge
Boy/Male
Latin
Curly-haired.
Boy/Male
Australian, Hebrew, Italian, Swiss
Told by God; God has Hearkened
Girl/Female
Hindu, Indian
Morning
Boy/Male
Hindu
Prince
Surname or Lastname
English
English : variant of Arundel.Perhaps an altered spelling of Swedish Arendall.
Boy/Male
Hindu, Indian, Marathi
A Cluster of Blossoms
QUATERNIONIC ANALYSIS
QUATERNIONIC ANALYSIS
QUATERNIONIC ANALYSIS
QUATERNIONIC ANALYSIS
QUATERNIONIC ANALYSIS
n.
The turning factor of a quaternion.
n.
The art or process of making a compound by putting the ingredients together, as contrasted with analysis; thus, water is made by synthesis from hydrogen and oxygen; hence, specifically, the building up of complex compounds by special reactions, whereby their component radicals are so grouped that the resulting substances are identical in every respect with the natural articles when such occur; thus, artificial alcohol, urea, indigo blue, alizarin, etc., are made by synthesis.
n.
The science of spectrum analysis in any or all of its relations and applications.
n.
A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.
n.
An apparatus for determining the amount of nitrogen or some of its compounds in any substance subjected to analysis; an azotometer.
n.
The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
n.
A word of four syllables; a quadrisyllable.
n.
The science of blowpipe analysis.
a.
Of or pertaining to the spectrum; made by the spectrum; as, spectral colors; spectral analysis.
n.
The combination of separate elements of thought into a whole, as of simple into complex conceptions, species into genera, individual propositions into systems; -- the opposite of analysis.
a.
Incapable of further analysis; incapable of further division or separation; constituent; elemental; as, an ultimate constituent of matter.
v. t.
A very small quantity of an element or compound in a given substance, especially when so small that the amount is not quantitatively determined in an analysis; -- hence, in stating an analysis, often contracted to tr.
n.
Chemical analysis.
n.
The number four.
n.
The separation of a compound substance, by chemical processes, into its constituents, with a view to ascertain either (a) what elements it contains, or (b) how much of each element is present. The former is called qualitative, and the latter quantitative analysis.
n.
The number four; a collection of four things; a quaternion.
v. t.
To reduce to a normal standard; to calculate or adjust the strength of, by means of, and for uses in, analysis.
n.
A rare metallic element of the boron group, whose existence was predicted under the provisional name ekaboron by means of the periodic law, and subsequently discovered by spectrum analysis in certain rare Scandinavian minerals (euxenite and gadolinite). It has not yet been isolated. Symbol Sc. Atomic weight 44.
n.
In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.
v. t.
To divide into quaternions, files, or companies.