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Branch of mathematical analysis
In mathematics, hypercomplex analysis is the extension of complex analysis to the hypercomplex numbers. The first instance is functions of a quaternion
Hypercomplex_analysis
Element of a unital algebra over the field of real numbers
In mathematics, the hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The
Hypercomplex_number
Branch of mathematics
Mathematics portal Arithmetization of analysis Constructive analysis History of calculus Hypercomplex analysis Multiple rule-based problems Multivariable
Mathematical_analysis
Branch of mathematics studying functions of a complex variable
quantum mechanics as wave functions. Complex geometry Hypercomplex analysis List of complex analysis topics Monodromy theorem Riemann–Roch theorem Runge's
Complex_analysis
Hypercomplex number system
octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter
Octonion
Four-dimensional number system
"Some new aspects in hypercomplex analysis". In Breaz, Daniel; Rassias, Michael Th. (eds.). Advancements in Complex Analysis: From Theory to Practice
Quaternion
Branch of number theory
p-adic Teichmüller theory Hypercomplex analysis p-adic quantum mechanics Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions. Graduate
P-adic_analysis
Academic subfield of computer science
Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory
Theory_of_computation
Branch of mathematics
Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many
Abstract_algebra
Form of entertainment in mathematics
Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory
Recreational_mathematics
Topics referred to by the same term
Hypercomplex may refer to: Hypercomplex cell Hypercomplex analysis Hypercomplex manifold Hypercomplex number This disambiguation page lists articles associated
Hypercomplex
Branch of mathematics
techniques of real analysis and discrete mathematics. It has close connections to convex analysis, optimization and functional analysis and important applications
Geometry
trigonometry. Hypercomplex analysis the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Mathematics independent of applications
professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent. After Weierstrass, by the end of
Pure_mathematics
Branch of algebraic geometry
numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. In 1996, the proof of
Arithmetic_geometry
operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis. Clifford analysis has analogues of Cauchy transforms, Bergman
Clifford_analysis
Area of mathematics
problems of statistical physics. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces
Dynamical_systems_theory
Italian mathematician
is an Italian mathematician specializing in complex analysis, hypercomplex analysis and the analysis of superoscillations. She is a professor of mathematics
Irene_Sabadini
Study of Lie groups, Lie algebras and differential equations
bracket in this algebra is twice the cross product of ordinary vector analysis. Another elementary 3-parameter example is given by the Heisenberg group
Lie_theory
Manifold equipped with a quaternionic structure
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a
Hypercomplex_manifold
Branch of elementary mathematics
it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and the examination of
Arithmetic
Branch of mathematics
simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found. This section introduces
Order_theory
Branch of mathematics
has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. As a study of systems of polynomial equations
Algebraic_geometry
Type of functional equation (mathematics)
Porter, Ronald I. (1978). "XIX Differential Equations". Further elementary analysis (4th ed.). London: Bell & Hyman. ISBN 978-0-7135-1594-7. Teschl, Gerald
Differential_equation
Mathematics of varieties with integer coordinates
Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory
Diophantine_geometry
Mathematical concept
of a hypercomplex variable". In Irene Sabadini; M Shapiro; F Sommen (eds.). Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed
Seven-dimensional cross product
Seven-dimensional_cross_product
Algebraic variety that is a moduli space for principally polarized abelian varieties
(PDF). In Arthur, James; Ellwood, David; Kottwitz, Robert (eds.). Harmonic Analysis, the Trace Formula, and Shimura Varieties. Clay Mathematics Proceedings
Siegel_modular_variety
Mathematical functions of split-complex numbers
of conventional complex analysis have an interpretation given with motor variables, and more generally in hypercomplex analysis. Let D = { z = x + j y
Motor_variable
Geometric system with a finite number of points
Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory
Finite_geometry
Area of geometry, about angles and lengths
Complex Analysis. Springer. p. 63. ISBN 978-3-642-59273-7. Silvia Maria Alessio (9 December 2015). Digital Signal Processing and Spectral Analysis for Scientists:
Trigonometry
Used to count, measure, and label
are explicitly referred to as numbers (such as the p-adic numbers and hypercomplex numbers) while others are not, but this is more a matter of convention
Number
Basic concepts of algebra
algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined
Elementary_algebra
Museum in Manhattan, New York
Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory
National Museum of Mathematics
National_Museum_of_Mathematics
Quaternions with complex number coefficients
biquaternions with non-zero square modulus. Biquaternion algebra Hypercomplex number Hypercomplex analysis Joachim Lambek MacFarlane's use Quotient ring Quaternion
Biquaternion
particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations
Numerical_algebraic_geometry
Italian mathematician (1924–2018)
fields of complex analysis of several variables and in differential geometry: he is known for his contribution to hypercomplex analysis, notably for extending
Giovanni_Battista_Rizza
Hyeokho Choi; Baraniuk, R.G. (2004). "Directional hypercomplex wavelets for multidimensional signal analysis and processing". 2004 IEEE International Conference
Wavelet for multidimensional signals analysis
Wavelet_for_multidimensional_signals_analysis
Day of the year
Sabadini, Irene; Shapiro, Michael; Sommen, Franciscus (2009-04-21). Hypercomplex Analysis. Springer Science & Business Media. ISBN 978-3-7643-9893-4. "World's
December_26
Functions of complex quaternions
Biquaternion Quaternion Biquaternion algebra Quaternion algebra Hypercomplex number Hypercomplex analysis Stillwell, John (2010). Mathematics and Its History Third
Biquaternion_functions
Book on the history of mathematics by Michael J. Crowe
hypercomplex numbers" twenty-five years after his book was first published. The book has eight chapters: the first on the origins of vector analysis including
A_History_of_Vector_Analysis
imaginary numbers, and sums and differences of real and imaginary numbers. Hypercomplex numbers include various number-system extensions: quaternions ( H {\displaystyle
List_of_types_of_numbers
Algebraic object
Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory
Ring_of_modular_forms
Concept in linear algebra
1007/s00006-018-0881-8. Sprössig, W. (2020). "Some new aspects in hypercomplex analysis". Advancements in Complex Analysis: From Theory to Practice. Springer. pp. 497–518
Quaternionic_matrix
Branch of mathematics
quaternion difference p – q also produces a segment equipollent to pq. Other hypercomplex number systems also used the idea of a linear space with a basis. Arthur
Linear_algebra
Method for producing composition algebras
(2015). "An unified approach for developing rationalized algorithms for hypercomplex number multiplication". Przegląd Elektrotechniczny. 1 (2). Wydawnictwo
Cayley–Dickson_construction
Fractal named after mathematician Benoit Mandelbrot
been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power α {\displaystyle \alpha } of the iterated
Mandelbrot_set
Real-valued number of spatial dimensions
ecology, electrochemical processes, image analysis, biology and medicine, neuroscience, network analysis, physiology, physics, and Riemann zeta zeros
Fractal_dimension
Commutative, associative algebra of two complex dimensions
hypercomplex numbers. In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine. A tessarine is a hypercomplex
Bicomplex_number
Number with a real and an imaginary part
^{2}.} This is generalized by the notion of a linear complex structure. Hypercomplex numbers also generalize R , {\displaystyle \mathbb {R} ,} C , {\displaystyle
Complex_number
Mathematical encyclopedia begun by Felix Klein
useful in analytic geometry, and the del operator in analysis. Explorative articles on hypercomplex numbers, mentioned by Bottazzini and Gray, written by
Klein's Encyclopedia of Mathematical Sciences
Klein's_Encyclopedia_of_Mathematical_Sciences
Real numbers adjoined with a nil-squaring element
Application of Dual Algebra to Kinematic Analysis", Computational Methods in Mechanical Systems: Mechanism Analysis, Synthesis, and Optimization, NATO ASI
Dual_number
Algebra based on a vector space with a quadratic form
generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected
Clifford_algebra
Involutive change of basis in linear algebra
symmetric, involutive, linear operation on 2m real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real).
Hadamard_transform
Array of numbers
linear algebra, partially due to their use in the classification of the hypercomplex number systems of the previous century. The inception of matrix mechanics
Matrix_(mathematics)
Hungarian and American mathematician and physicist (1903–1957)
"the cold, wet, rain-wet streets of Göttingen" after class discussing hypercomplex number systems and their representations. Von Neumann's habilitation
John_von_Neumann
Algebra of eight complex dimensions
Algebra ? J. D. Edmonds (1978) Nine-vectors, complex octonion/quaternion hypercomplex numbers, Lie groups and the ‘real’ world, Foundations of Physics 8(3-4):
Bioctonion
Number of vectors in any basis of the vector space
theorem for vector spaces Itzkov, Mikhail (2009). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer. p. 4
Dimension_(vector_space)
Reals with an extra square root of +1 adjoined
page on the topic of: Split binarions Minkowski space Split-quaternion Hypercomplex number Vladimir V. Kisil (2012) Geometry of Mobius Transformations: Elliptic
Split-complex_number
they also began the use of hypercomplex numbers. Karl Weierstrass and others carried out the arithmetization of analysis for functions of real and complex
19th_century_in_science
Special interest group of mathematicians (1899 to 1913)
the academic world that were experimenting with quaternions and other hypercomplex number systems. The group's guiding light was Alexander Macfarlane who
Quaternion_Association
Non-associative algebras with positive-definite quadratic form
A.S. (1989), "Normed algebras with an identity. Hurwitz's theorem.", Hypercomplex numbers. An elementary introduction to algebras, Trans. A. Shenitzer
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Delimited medium where some stimuli can evoke neuronal responses
of cells in the visual cortex into simple cells, complex cells, and hypercomplex cells. Simple cell receptive fields are elongated, for example with an
Receptive_field
Arithmetical operation
commutative for matrices and quaternions. Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions
Multiplication
surpassed in the 19th century through considerations of parameter space and hypercomplex numbers. Abel and Galois's investigations into the solutions of various
History_of_mathematics
Geometric model of the physical space
came with William Rowan Hamilton's development of the quaternions, a hypercomplex number system. For this purpose, Hamilton coined the terms scalar and
Three-dimensional_space
Branch of algebra
theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative
Ring_theory
since some novelty in the subject lingered there. Research turned to hypercomplex numbers more generally. For instance, Thomas Kirkman and Arthur Cayley
History_of_quaternions
cryptographer, mathematician, and professor of acoustics Irene Sabadini, Italian hypercomplex analyst Flora Sadler (1912–2000), Scottish mathematician and astronomer
List_of_women_in_mathematics
Subspace of n-space whose dimension is (n-1)
Arrangement of hyperplanes Supporting hyperplane theorem "Excerpt from Convex Analysis, by R.T. Rockafellar" (PDF). u.arizona.edu. Beutelspacher, Albrecht; Rosenbaum
Hyperplane
Mathematical model combining space and time
appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily
Spacetime
Correspondence between quaternions and 3D rotations
Patrick J. Ryan, Cambridge University Press, Cambridge, 1987. I.L. Kantor. Hypercomplex numbers, Springer-Verlag, New York, 1989. Andrew J. Hanson. Visualizing
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Greek mathematician and university professor (1857–1917)
Karl Weierstrass's hypercomplex numbers theorem. In 1883, Stefanos proved that the theorem fails when three-dimensional hypercomplex numbers are applied
Cyparissos_Stephanos
Mutation of quaternions where unit vectors square to +1
on physics. As for mathematics, the hyperbolic quaternion is another hypercomplex number, as such structures were called at the time. By the 1890s Richard
Hyperbolic_quaternion
Transformation of a geometric space preserving structure
transformations of spacetime by use of biquaternions. Early in the 20th century, hypercomplex number systems were examined. Later their automorphism groups led to
Motion_(geometry)
Australian mathematician (1863 to 1931)
quaternions to dual quaternions, McAulay made a special study of this hypercomplex number system. In 1898 McAulay published, through Cambridge University
Alexander_McAulay
Property of a space in which the local dimensionality is the same everywhere
variable. Equidimensional equations play an important rule in dimensional analysis. Wirthmüller, Klaus. A Topology Primer: Lecture Notes 2001/2002 (PDF).
Equidimensionality
American historian of mathematics (born 1938)
History of Exact Sciences, 7: 142–170 ISSN 0003-9519 JSTOR 41133320 1972: "Hypercomplex numbers, Lie groups and the creation of group representation theory"
Thomas_W._Hawkins_Jr.
Generalization of a rectangle for higher dimensions
Fundamentals of Real Analysis. CRC Press. ISBN 9780824784539. Retrieved 23 May 2014. Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill.
Hyperrectangle
Geometric space with six dimensions
(2001), pp. 86–89 Josiah Willard Gibbs, Edwin Bidwell Wilson (1901). Vector analysis: a text-book for the use of students of mathematics and physics. Yale University
Six-dimensional_space
Geometric space with four dimensions
source of the science of vector analysis in three dimensions as recounted by Michael J. Crowe in A History of Vector Analysis. The study of Minkowski space
Four-dimensional_space
Möbius transformation generalized to rings other than the complex numbers
Springer-Verlag ISBN 0-387-90872-2. Geoffry Fox (1949) Elementary Theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane
Linear fractional transformation
Linear_fractional_transformation
View of mathematicians to consolidate two or more theories into a more generalized one
then studying their consequences. Thus, for example, the studies of "hypercomplex numbers", such as considered by the Quaternion Association, were put
Unifying theories in mathematics
Unifying_theories_in_mathematics
complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex manifold Quaternion-Kähler manifold Symplectic topology Symplectic space
List of differential geometry topics
List_of_differential_geometry_topics
German mathematician (1868–1942)
Hausdorff wrote other works on optics, on non-Euclidean geometry, and on hypercomplex number systems, as well as two papers on probability theory. However
Felix_Hausdorff
Method of determining fractal dimension
FracLac: online user guide and software ImageJ and FracLac box counting plugin; free user-friendly open source software for digital image analysis in biology
Minkowski–Bouligand_dimension
Matrices important in quantum mechanics and the study of spin
the differential and integral calculus of vectors". Elements of Vector Analysis. New Haven, CT: Tuttle, Moorehouse & Taylor. p. 67. In fact, however, the
Pauli_matrices
Faster-than-light travel in science fiction
November 2021. Muir, John Kenneth (15 September 2015). A History and Critical Analysis of Blake's 7, the 1978-1981 British Television Space Adventure. McFarland
Hyperspace
ISBN 978-0-521-28274-1, p. 104 f. Igor V. Kanatchikov: De Donder–Weyl theory and a hypercomplex extension of quantum mechanics to field theory, arXiv:hep-th/9810165
De_Donder–Weyl_theory
Hypercomplex number system
) ( e 6 − e 15 ) {\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})} . All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction
Sedenion
split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples. In commutative algebra, the
Topological_ring
American-Israeli mathematician
relativity, representations of quantum theory on hypercomplex Hilbert modules, group theory and functional analysis and stochastic theories of irreversible quantum
Lawrence_Paul_Horwitz
One hundred years, from 1801 to 1900
they also began the use of hypercomplex numbers. Karl Weierstrass and others carried out the arithmetization of analysis for functions of real and complex
19th_century
German polymath, linguist and mathematician (1809–1877)
Roger (February 2005). "2. An Ancient Theorem and a Modern Question, 11. Hypercomplex numbers". The Road to Reality: A Complete Guide to the Laws of the Universe
Hermann_Grassmann
French mathematician (1869–1951)
modern terminology, they are: Lie theory Representations of Lie groups Hypercomplex numbers, division algebras Systems of PDEs, Cartan–Kähler theorem Theory
Élie_Cartan
Invariant measure of fractal dimension
exponent" of the Master theorem for solving recurrence relations in the analysis of algorithms. Space-filling curves like the Peano curve have the same
Hausdorff_dimension
Anticommutating number
definition of Grassmann numbers allows mathematical analysis to be performed, in analogy to analysis on complex numbers. That is, one may define superholomorphic
Grassmann_number
Fundamental space of geometry
Hilbert space, a generalization to infinite dimension, used in functional analysis Position space, an application in physics It may depend on the context
Euclidean_space
Property of a mathematical space
the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the real numbers, it is sometimes
Dimension
Geometric model of the planar projection of the physical universe
ISBN 978-0-07-154352-1. M.R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis (Schaum's Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7. Mathematical
Euclidean_plane
Russian mathematician (1861–1941)
interest concentrated on so-called higher complex numbers (nowadays called hypercomplex numbers). His studies resulted in his article "Über Systeme höherer komplexer
Theodor_Molien
HYPERCOMPLEX ANALYSIS
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n.
In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.
n.
Chemical analysis.
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.
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.
An apparatus for determining the amount of nitrogen or some of its compounds in any substance subjected to analysis; an azotometer.
a.
Of or pertaining to the spectrum; made by the spectrum; as, spectral colors; spectral analysis.
n.
The science of spectrum analysis in any or all of its relations and applications.
v. t.
To consider by a separate act of attention or analysis.
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.
That which indicates the condition of acidity, alkalinity, or the deficiency, excess, or sufficiency of a standard reagent, by causing an appearance, disappearance, or change of color, as in titration or volumetric analysis.
n.
A rare alkaline metal found in mineral water; -- so called from the two characteristic blue lines in its spectrum. It was the first element discovered by spectrum analysis, and is the most strongly basic and electro-positive substance known. Symbol Cs. Atomic weight 132.6.
n.
Any original inherent constituent which characterizes a substance, or gives it its essential properties, and which can usually be separated by analysis; -- applied especially to drugs, plant extracts, etc.
n.
An instrument for ascertaining the strength of an indigo solution, as in volumetric analysis.
v. t.
To reduce to a normal standard; to calculate or adjust the strength of, by means of, and for uses in, analysis.
n.
Analysis into primary or elemental parts.
n.
The science of blowpipe 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.
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.
n.
Anything which resounds; specifically, a vessel in the form of a cylinder open at one end, or a hollow ball of brass with two apertures, so contrived as to greatly intensify a musical tone by its resonance. It is used for the study and analysis of complex sounds.
a.
Incapable of further analysis; incapable of further division or separation; constituent; elemental; as, an ultimate constituent of matter.