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In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local
Complex_dimension
Property of a mathematical space
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify
Dimension
Analysis of the dimensions of different physical quantities
engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical
Dimensional_analysis
Branch of mathematics
contained in U.) Complex dynamics has been effectively developed in any dimension. This section focuses on the mappings from complex projective space
Complex_dynamics
Branch of mathematics studying functions of a complex variable
The theory of several complex variables generalizes one-variable complex function theory to more than one complex dimension. While many of the techniques
Complex_analysis
Mathematics concept
has complex dimension n {\displaystyle n} , then V {\displaystyle V} must have real dimension 2 n {\displaystyle 2n} . That is, a finite-dimensional space
Linear_complex_structure
Study of complex manifolds and several complex variables
\mathbb {R} ^{2n}} , every complex manifold of dimension n {\displaystyle n} is in particular a smooth manifold of dimension 2 n {\displaystyle 2n} , which
Complex_geometry
Concept in algebraic geometry
since a complex curve has real dimension 2): Kodaira dimension − ∞ {\displaystyle -\infty } corresponds to positive curvature, Kodaira dimension 0 corresponds
Kodaira_dimension
a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line
Complex_line
Manifold
us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R2n, whereas it is "rare" for a complex manifold to have a holomorphic
Complex_manifold
Type of mathematical functions
the n dimensional Cauchy–Riemann equations. For one complex variable, every domain is the domain of holomorphy of some function. For several complex variables
Function of several complex variables
Function_of_several_complex_variables
Mathematical manifold theory
then their wedge product is necessarily zero because C has only one complex dimension; consequently, the cup product of their cohomology classes is zero
Hodge_theory
Type of vector space in mathematics
A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n
Lagrangian_Grassmannian
Model of the extended complex plane plus a point at infinity
itself. As a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane C {\displaystyle
Riemann_sphere
Dianalytic manifold of complex dimension 1
In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein
Klein_surface
Geometric space with four dimensions
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible
Four-dimensional_space
2-dimensional complex projective space
^{2},} is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates ( Z 1
Complex_projective_plane
2D surface which extends indefinitely
adding more structure, one may view the plane as a 1-dimensional complex manifold, called the complex line. Many fundamental tasks in mathematics, geometry
Plane_(mathematics)
Number of vectors in any basis of the vector space
is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there
Dimension_(vector_space)
Type of topological space
k} -dimensional complex. The topology of the CW complex is the quotient topology defined by these gluing maps. An infinite-dimensional CW complex can
CW_complex
Mathematical concept
account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space
Complex_projective_space
Manifold with Riemannian, complex and symplectic structure
a Kähler manifold X {\displaystyle X} is a Hermitian manifold of complex dimension n {\displaystyle n} such that for every point p {\displaystyle p}
Kähler_manifold
Real-valued number of spatial dimensions
space-filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of approximately 1
Fractal_dimension
Mathematical space
a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast
4-manifold
Type of smooth complex surface of kodaira dimension 0
name "K3 surface" In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and
K3_surface
Mathematical space with two coordinates
or finite. Some two-dimensional mathematical spaces are not used to represent physical positions, like an affine plane or complex plane. The most basic
Two-dimensional_space
Unsolved problem in geometry
conjecture. Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2 n {\displaystyle 2n} , so
Hodge_conjecture
Algebraic structure in linear algebra
vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal
Vector_space
Theory in algebraic geometry
groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension n has real dimension 2n, so these higher cohomology
Weil_cohomology_theory
Space with one dimension
^{1}(K),} is a one-dimensional space. In particular, if the field is the complex numbers C , {\displaystyle \mathbb {C} ,} then the complex projective line
One-dimensional_space
{2z}{w+i}}\right).} In the projective model, the complex hyperbolic space identifies with the complex unit ball of dimension n {\displaystyle n} , and its boundary
Complex_hyperbolic_space
Riemannian manifold with SU(n) holonomy
of a complex torus of complex dimension 2, which have vanishing first integral Chern class but non-trivial canonical bundle. For a compact complex n {\displaystyle
Calabi–Yau_manifold
Type of mathematical set
mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts) such
Simplicial_complex
Topics referred to by the same term
topological spaces: Complex dimension Hausdorff dimension Inductive dimension Lebesgue covering dimension Packing dimension Isoperimetric dimension Measurements
Dimension_(disambiguation)
Structure in data warehousing
In data management and data warehousing, a slowly changing dimension (SCD) is a dimension that stores data which, while generally stable, may change over
Slowly_changing_dimension
Term in mathematics
in algebraic geometry. Suppose X {\displaystyle X} is a complex manifold of complex dimension n {\displaystyle n} and let O ( X ) {\displaystyle {\mathcal
Stein_manifold
Geometric space with five dimensions
A five-dimensional (5D) space is a mathematical or physical space that has five independent dimensions. In physics and geometry, such a space extends
Five-dimensional_space
Topological space that locally resembles Euclidean space
complex geometry. A one-complex-dimensional manifold is called a Riemann surface. An n {\displaystyle n} -dimensional complex manifold has dimension 2
Manifold
to as Hurwitz curves, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2). The Fuchsian group of a Hurwitz surface
Hurwitz_surface
Tabletop role-playing web series
Dimension 20 is an actual play show produced by and broadcast on Dropout, and created and generally hosted by Brennan Lee Mulligan as the show's regular
Dimension_20
Infinitely detailed mathematical structure
arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales
Fractal
a complex projective space has cup-length equal to its complex dimension. In what follows, vertical bars around an element denote its dimension in the
Cohomology_ring
Theory in theoretical physics
are not spheres vanish unless the complex dimension of the spacetime is three, and so spacetimes with complex dimension three are the most interesting.
Topological_string_theory
248-dimensional exceptional simple Lie group
group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This
E8_(mathematics)
Property of a differential manifold that includes complex structures
generalized complex structures also play a leading role in physical string theory, as supersymmetric flux compactifications, which relate 10-dimensional physics
Generalized_complex_structure
Parametrizes complex structures on a surface
differentials on X {\displaystyle X} . The space of those is a complex space of complex dimension 3 g − 3 {\displaystyle 3g-3} , and the image of Teichmüller
Teichmüller_space
Number with a real and an imaginary part
upon convention and style considerations. The complex numbers also form a real vector space of dimension two, with { 1 , i } {\displaystyle \{1,i\}} as
Complex_number
Mathematical theorem of complex manifolds
smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} is any Stein manifold of dimension n {\displaystyle
Andreotti–Frankel_theorem
Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element X ∈ g {\displaystyle
Regular element of a Lie algebra
Regular_element_of_a_Lie_algebra
cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their higher-dimensional counterparts
Cubical_complex
One-dimensional complex manifold
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied
Riemann_surface
Polygon in complex space, or which self-intersects
ib} are called imaginary numbers. A complex number lies in a complex plane having one real and one imaginary dimension, which may be represented as an Argand
Complex_polygon
Group of 𝑛 × 𝑛 invertible matrices
field of complex numbers, GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} , is a complex Lie group of complex dimension n 2 {\displaystyle
General_linear_group
Geometric model of the physical space
rarely, tri-dimensional space. Most commonly, it means the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which
Three-dimensional_space
Data modeling concept
descriptive (dimension) tables Developers often don't normalize dimensions due to several reasons: Normalization makes the data structure more complex Performance
Dimensional_modeling
Property of algebraic varieties and complex manifolds
variety V of complex dimension n it is the number of linearly independent holomorphic n-forms to be found on V. This definition, as the dimension of H0(V,Ωn)
Geometric_genus
Kind of complex manifold
circles). Here N must be the even number 2n, where n is the complex dimension of M. All such complex structures can be obtained as follows: take a lattice Λ
Complex_torus
Chinese-American mathematician (born 1949)
Calabi−Yau manifold with complex dimension three should be foliated by special Lagrangian tori, which are certain types of three-dimensional minimal submanifolds
Shing-Tung_Yau
Differentiable manifold
is one-dimensional) and L ∩ L ¯ = { 0 } {\displaystyle L\cap {\bar {L}}=\{0\}} since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates
CR_manifold
Geometric space with seven dimensions
generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions
Seven-dimensional_space
On the preimage of points in a manifold under the action of a smooth map
g − 1 ( y ) {\displaystyle g^{-1}(y)} is a complex submanifold of X {\displaystyle X} of complex dimension n − m . {\displaystyle n-m.} Fiber (mathematics) –
Preimage_theorem
Finding linear approximation of function at given point
equation (L-function) Quasilinearization The linearization problem in complex dimension one dynamical systems at Scholarpedia Linearization. The Johns Hopkins
Linearization
Generalization of a manifold
is usually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces. Conifolds
Conifold
78-dimensional exceptional simple Lie group
unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78
E6_(mathematics)
Algebra of eight complex dimensions
the exceptional symmetric domain of dimension 27. The second exceptional symmetric domain (of complex dimension 16) lives in the space M 2 , 1 ( O C
Bioctonion
Curve defined as zeros of polynomials
A complex projective algebraic curve resides in n-dimensional complex projective space CPn. This has complex dimension n, but topological dimension, as
Algebraic_curve
Manifold with inversion symmetry
{\displaystyle ({\mathfrak {m}},J)} is a real vector space with a complex structure J, whose complex dimension is given in the table. Correspondingly, there is a graded
Hermitian_symmetric_space
Method for producing composition algebras
twice the dimension. Hurwitz's theorem states that the reals, complex numbers, quaternions, and octonions are the only finite-dimensional normed division
Cayley–Dickson_construction
Topological space in mathematics
Rosenlicht gave an example of a non-paracompact complex manifold of complex dimension 2. Lexicographic order topology on the unit square List of topologies
Long_line_(topology)
133-dimensional exceptional simple Lie group
complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This
E7_(mathematics)
Mathematical group
\operatorname {Sp} (2n,\mathbb {F} )} is a real or complex Lie group of real or complex dimension n ( 2 n + 1 ) {\displaystyle n(2n+1)} , respectively
Symplectic_group
Differential form on a manifold which is permitted to have complex coefficients
applies. Suppose that M is a complex manifold of complex dimension n. Then there is a local coordinate system consisting of n complex-valued functions z1, .
Complex_differential_form
Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Theorem stating that smooth algebraic curve has minimum genus its homology class
result because algebraic curves (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic
Thom_conjecture
Gives general conditions under which sheaf cohomology groups with indices > 0 are zero
Kunihiko Kodaira's result is that if M is a compact Kähler manifold of complex dimension n, L any holomorphic line bundle on M that is positive, and KM is
Kodaira_vanishing_theorem
Study of the topology of a complex manifold
holomorphic map from an ( k + 1 ) {\displaystyle (k+1)} -dimensional projective complex manifold to the projective line P1. Also suppose that all critical
Picard–Lefschetz_theory
Four-dimensional number system
{\displaystyle \mathbb {R,C} } (complex numbers) and H {\displaystyle \mathbb {H} } (quaternions) which have dimension 1, 2, and 4 respectively.[citation
Quaternion
Describes the objects of a given type, up to some equivalence
Mathematical classification of surfaces of algebraic surfaces (complex dimension two, real dimension four) Nielsen–Thurston classification – Characterizes homeomorphisms
Classification_theorem
Mathematical object
description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets
Abstract_simplicial_complex
Topological space of dimension zero
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several
Zero-dimensional_space
Method in evaluating divergent integrals
meromorphic functions of a complex parameter d, the analytic continuation of the number of spacetime dimensions. Dimensional regularization writes a Feynman
Dimensional_regularization
Theorem in algebraic geometry
decomposition theorem. Let X {\displaystyle X} be an n {\displaystyle n} -dimensional complex projective algebraic variety in C P N {\displaystyle \mathbb {C}
Lefschetz_hyperplane_theorem
Universal construction of a complex Lie group from a real Lie group
as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations
Complexification_(Lie_group)
Representation learning technique
a representation learning technique that maps complex, high-dimensional data into a lower-dimensional vector space of numerical vectors. It also denotes
Embedding_(machine_learning)
Compact complex manifold in algebraic geometry
compact complex manifold such that the field of meromorphic functions on each component M has transcendence degree equal to the complex dimension of the
Moishezon_manifold
Topologically invariant definition of the dimension of a space
Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
Lebesgue_covering_dimension
Relation between genus, degree, and dimension of function spaces over surfaces
theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions
Riemann–Roch_theorem
Spacetime with complexified coordinates
gravitation and electromagnetism within a complex 4-dimensional Riemannian geometry. The line element ds2 is complex-valued, so that the real part corresponds
Complex_spacetime
Mathematical classification of surfaces
irregularity is defined as the dimension of the Picard variety and the Albanese variety and denoted by q. For complex surfaces (but not always for surfaces
Enriques–Kodaira classification
Enriques–Kodaira_classification
Element of a unital algebra over the field of real numbers
Hurwitz's theorem says finite-dimensional real composition algebras are the reals R {\displaystyle \mathbb {R} } , the complexes C {\displaystyle \mathbb
Hypercomplex_number
Geometric space with eight dimensions
field, such as an eight-dimensional complex vector space, which has 16 real dimensions. It may also refer to an eight-dimensional manifold such as an 8-sphere
Eight-dimensional_space
Dimensionality measure in chaos theory
measuring dimension (e.g. the Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage
Correlation_dimension
Mathematical result in differential geometry
operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined
Atiyah–Singer_index_theorem
Non-associative algebras with positive-definite quadratic form
have dimension 2(N − 2)/2. The space on which the Vi's act can be complexified. It will have complex dimension N. It breaks up into some of complex irreducible
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Differentiable manifold
right. Complex nilmanifolds are usually not homogeneous, as complex varieties. In complex dimension 2, the only complex nilmanifolds are a complex torus
Nilmanifold
In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective
Iitaka_dimension
Mathematical space
{\displaystyle \mathbf {P} (V)} of one dimension lower than V {\displaystyle V} . When V {\displaystyle V} is a real or complex vector space, Grassmannians are
Grassmannian
Vector bundles theorem
concrete in the early 1980s. A direct correspondence when the dimension of the base complex manifold is one was explained in the work of Atiyah and Bott
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Mathematical concept
done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic
Infinity
Topic in mathematics
the tensor product of V {\displaystyle V} with the complex numbers (thought of as a 2-dimensional vector space over the reals): V C = V ⊗ R C . {\displaystyle
Complexification
COMPLEX DIMENSION
COMPLEX DIMENSION
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Girl/Female
Tamil
Complete
Boy/Male
Indian
Complete
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Girl/Female
Tamil
Complete
Boy/Male
Tamil
Complete
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Boy/Male
Tamil
Complete
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Boy/Male
Tamil
Complete
Girl/Female
Muslim
Complex, Zigzag, Curling
Girl/Female
Tamil
Complete
Girl/Female
Bengali, Indian
Good Complex
Girl/Female
Tamil
Complete
Boy/Male
Indian
Complete
Girl/Female
Tamil
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Complete
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Girl/Female
Hindu, Indian
Complex
Girl/Female
Tamil
Shesha Harani | ஷேஷ ஹரணீÂ
Complete
COMPLEX DIMENSION
COMPLEX DIMENSION
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Wise; Intelligent
Girl/Female
Muslim/Islamic
Old Ancient
Girl/Female
Indian
Beautiful, Splendor, Derived from Indira - Goddess laxmis name
Girl/Female
French
An Old French surname meaning pilferer or ferret.
Boy/Male
Arabic, Muslim
Brave; Lion; Terrible; Held in Awe; Another Name for God
Girl/Female
Australian, Danish, Finnish, Greek, Russian, Ukrainian
Defender of Man
Girl/Female
Muslim
Leader
Girl/Female
Hindu, Indian
Treasure
Boy/Male
Indian, Telugu
Wishes
Boy/Male
Hindu, Indian, Tamil
Happy; Joyful
COMPLEX DIMENSION
COMPLEX DIMENSION
COMPLEX DIMENSION
COMPLEX DIMENSION
COMPLEX DIMENSION
imp. & p. p.
of Compile
v. t.
To bring to a state in which there is no deficiency; to perfect; to consummate; to accomplish; to fulfill; to finish; as, to complete a task, or a poem; to complete a course of education.
a.
See Couple-close.
pl.
of Couple-close
a.
Finished; ended; concluded; completed; as, the edifice is complete.
a.
Not complex; uncompounded; simple.
a.
That which joins or links two things together; a bond or tie; a coupler.
n.
One who couples; that which couples, as a link, ring, or shackle, to connect cars.
n.
One who compiles; esp., one who makes books by compilation.
a.
One of the pairs of plates of two metals which compose a voltaic battery; -- called a voltaic couple or galvanic couple.
n.
One who complies, yields, or obeys; one of an easy, yielding temper.
imp. & p. p.
of Comply
a.
Repeatedly compound; made up of complex constituents.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
n.
Two taken together; a pair or couple; especially two lines of verse that rhyme with each other.
a.
Intricate; entangled; complicated; complex.
n.
A complex; an aggregate of parts; a complication.
a.
Complex, complicated.
imp. & p. p.
of Couple
adv.
In a complex manner; not simply.