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Mathematics concept
when it refers instead to a structure on vector spaces, it may be called a linear complex structure. A complex structure on a real vector space V {\displaystyle
Linear_complex_structure
Smooth manifold
mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an
Almost_complex_manifold
Topics referred to by the same term
A complex structure may refer to: Almost complex manifold Complex manifold Linear complex structure Generalized complex structure Complex structure deformation
Complex_structure
Mathematics concept
vector addition and real scalar multiplication) with the conjugate linear complex structure J {\displaystyle J} (different multiplication by i {\displaystyle
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
Group of 𝑛 × 𝑛 invertible matrices
More generally, the general linear group of degree n {\displaystyle n} over any field F {\displaystyle F} (such as the complex numbers), or a ring R {\displaystyle
General_linear_group
Data organization and storage formats
alternatively, user-defined) rule for comparing elements. A data structure is said to be linear if its elements form a sequence. Array Associative array Bit
List_of_data_structures
Study of abstract algebraic structures
of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra
Algebra_representation
Manifold
differential geometry and complex geometry, a complex manifold or a complex analytic manifold is a manifold with a complex structure, that is an atlas of charts
Complex_manifold
Number with a real and an imaginary part
ℂ, and gives an alternative complex structure on ℝ2. This is generalized by the notion of a linear complex structure. Hypercomplex numbers also generalize
Complex_number
Literary element
endings) through their actions. The overwhelmingly most typical story structure is a linear one, in which events proceed chronologically; they are presented
Story_structure
Structure group sub-bundle on a tangent frame bundle
real space of a complex vector space: it admits a linear complex structure. A real vector bundle admits an almost complex structure if and only if it
G-structure_on_a_manifold
Several equations of degree 1 to be solved simultaneously
approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.
System_of_linear_equations
Topological manifold with a piecewise linear structure on it
a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined
Piecewise_linear_manifold
Undeciphered writing system of ancient Crete
contains Linear A Unicode characters. Without proper rendering support, you may see question marks, boxes, or other symbols instead of Linear A. Linear A is
Linear_A
Topic in mathematics
the identical space – a real vector space with linear complex structure is identical data to a complex vector space – though it constructs the space differently
Complexification
Algebraic structure in linear algebra
In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled")
Vector_space
Branch of mathematics
application of linear algebra to function spaces. Linear algebra is also used in most sciences and fields of engineering, because linear structures are natural
Linear_algebra
Mathematical structure in differential geometry
instance, what are now called linear Poisson structures (i.e. Poisson brackets on a vector space which send linear functions to linear functions) correspond precisely
Poisson_manifold
Chemical compound of a transition metal and nitric oxide
kinds of nitrosyl complexes are known, which vary both in structure and coligand. Most complexes containing the NO ligand can be viewed as derivatives of
Metal_nitrosyl_complex
Speech coding algorithm
Code-excited linear prediction (CELP) is a linear predictive speech coding algorithm originally proposed by Manfred R. Schroeder and Bishnu S. Atal in
Code-excited linear prediction
Code-excited_linear_prediction
Construction in group theory
Gal(K / k) corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure. One may also define collineation
Projective_linear_group
System composed of many interacting components
any system. For a system to be complex most definitions require that the result of interactions are likely to be non-linear or asymmetrical---where small
Complex_system
Property of a differential manifold that includes complex structures
generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized
Generalized_complex_structure
Set of vectors used to define coordinates
structures and frames of reference. A basis B of a vector space V over a field F (such as the real numbers ℝ or the complex numbers ℂ) is a linearly independent
Basis_(linear_algebra)
Integral transform and linear operator
{\displaystyle L^{p}(\mathbb {R} )} , the Hilbert transform defines a linear complex structure on this Banach space. In particular, when p = 2, the Hilbert transform
Hilbert_transform
1966 mathematics textbook by Serge Lang
Linear Algebra is a 1966 mathematics textbook by Serge Lang. The third edition of 1987 covers fundamental concepts of vector spaces, matrices, linear
Linear_Algebra_(book)
the product produced. This deviation from linearity is described as the non-linear effect, NLE. The linearity can be expressed mathematically, as shown
Non-linear effects in enantioselective synthesis
Non-linear_effects_in_enantioselective_synthesis
algebraic structure of linear algebra Field – algebraic structure with addition, multiplication and division Groups – algebraic structure with a single
Outline_of_algebra
Mathematical set with some added structure
retains the same mathematical structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces
Space_(mathematics)
Mathematical concept
representation Orthogonal matrix Unitary matrix Hamiltonian mechanics Linear complex structure Williamson theorem Hamiltonian matrix Folland, G. B. (1989). Harmonic
Symplectic_matrix
Operation in algebra
can be interpreted either as a complex vector space (S-module) or as a real vector space with a linear complex structure (algebra representation of S as
Change_of_rings
Linear map from a vector space to its field of scalars
scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a
Linear_form
Algebraic variety with a group structure
are linear groups or abelian varieties; for instance, some group schemes occurring naturally in arithmetic geometry are neither. Chevalley's structure theorem
Algebraic_group
Representation of mathematical space
triangulable space. Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various applications
Triangulation_(topology)
Algebraic structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge
Hodge_structure
In projective geometry, a bijection between projective spaces that preserves collinearity
projective semi-linear structure". Correspondingly, the quotient group PΓL / PGL ≅ Gal(K/k) corresponds to "choices of linear structure", with the identity
Collineation
the map on fiber-level, then Jx2=−1 as a linear map. If E is a complex vector bundle, then the complex structure J can be defined by setting Jx to be the
Complex_vector_bundle
orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
Branch of geometry
symplectic space. This rotation is simply multiply-by-i of the standard linear complex structure on the symplectic space. In the plane, it exchanges a curve and
Contact_geometry
Affine space over the complex numbers
example is the Argand plane of complex numbers C {\displaystyle \mathbb {C} } itself. This has a canonical linear structure, and so "forgetting" the origin
Complex_affine_space
Geometric structure on a smooth manifold
codimension-one linear subspace Q p {\displaystyle Q_{p}} of the tangent space T p M {\displaystyle T_{p}M} ), a linear complex structure on it (that is, a linear function
Almost_contact_manifold
Ring that is also a vector space or a module
the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the
Associative_algebra
Number, approximately 3.14
the unique (positive) normalizing factor such that H defines a linear complex structure on the Hilbert space of square-integrable real-valued functions
Pi
Method to solve optimization problems
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical
Linear_programming
Mathematics concept
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces
Real_structure
Concept in differential geometry
Riemannian metric that preserves a complex structure. A complex structure is essentially an almost complex structure with an integrability condition, and
Hermitian_manifold
Mathematical function, in linear algebra
In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector
Linear_map
Mathematical group
mathematics, the symplectic group is the group of linear transformations that preserve the geometric structure of phase space, the space of position and momentum
Symplectic_group
Mathematical structure in abstract algebra
conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert
*-algebra
Vector space equipped with a bilinear product
set in the complex plane. These are also commutative. Incidence algebras are built on certain partially ordered sets. algebras of linear operators, for
Algebra_over_a_field
n} matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group. From the Jordan
Nilpotent_orbit
Statistical method
maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the observable
Partial least squares regression
Partial_least_squares_regression
Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
Mathematical concept
imaginary part of the standard complex (Hermitian) inner product on Cn (with the convention of the first argument being anti-linear). Let ω be an alternating
Symplectic_vector_space
Fundamental principle of physics
superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the
Superposition_principle
Differentiable manifold
differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an
CR_manifold
Flaw in mathematical modelling
Replacing this simple function with a new, more complex quadratic function, or with a new, more complex linear function on more than two independent variables
Overfitting
vector spaces over the normed fields of real or complex numbers in functional analysis." The term "linear topology" goes back to Lefschetz's work. For each
Linear_topology
Type of vector space in math
x,x\rangle } is a real number. The inner product is linear in its first argument. For all complex numbers a {\displaystyle a} and b , {\displaystyle b
Hilbert_space
Collection of mathematical theories
the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations
Spectral_theory
Notation for quantum states
notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces both in
Bra–ket_notation
Space of complex matrices with positive definite imaginary part
{\displaystyle \omega } . A compatible complex structure on V {\displaystyle V} is a linear complex structure on V {\displaystyle V} , such that ω ( J
Siegel_upper_half-space
System where changes of output are not proportional to changes of input
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change
Nonlinear_system
Concept in geometry
to the situation for complex manifolds, which always have a globally defined almost complex structure. A quaternionic structure on a smooth manifold M
Quaternionic_manifold
Archaeological site and Minoan palace complex in Heraklion, Crete
Ancient Greek: Κνωσσός, romanized: Knōssós; Greek: Κνωσός, romanized: Knōsós; Linear B: 𐀒𐀜𐀰 Ko-no-so) is an archaeological site and ancient urban centre in
Knossos
Array of numbers
widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces
Matrix_(mathematics)
Fundamental operation on complex numbers
} is a R {\textstyle \mathbb {R} } -linear transformation of V , {\textstyle V,} if one notes that every complex space V {\displaystyle V} has a real
Complex_conjugate
Set of lines described by homogeneous polynomial equations
line complex is a set of lines that can be specified by a list of homogeneous polynomial equations. That is, a projective variety of lines. A linear line
Line_complex
Statistical modeling method
independent variable) related via a linear combination. A linear model with exactly one explanatory variable is a simple linear regression; a model with two
Linear_regression
Anions composed of many iodine atoms
diverse structures. Most can be considered as associations of I2, I−, and I−3 units. Discrete polyiodides are usually linear. The more complex two- or
Polyiodide
Function acting on function spaces
most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators
Operator_(mathematics)
Branch of mathematics
The study of vector spaces and linear maps form a large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition
Algebra
vectors, matrices, or elements of algebraic structures. Algebraic analysis motivated by systems of linear partial differential equations, it is a branch
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Group of unitary matrices
of matrix multiplication. The unitary group is a subgroup of the general linear group GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )}
Unitary_group
Unsolved problem in geometry
description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before
Hodge_conjecture
Narrative technique
utilizes a non-linear structure, focusing on events throughout the life of the titular character rather than describing them in a linear narrative. From
Nonlinear_narrative
Topics referred to by the same term
algebra A over the real or complex numbers which at the same time is also a Banach space Operator algebra, continuous linear operators on a topological
Algebra_(disambiguation)
Subgroup of the group of invertible n×n matrices
viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic
Linear_algebraic_group
Archaeological horizon of Neolithic Europe
two large rondel complexes were discovered east of the Vistula near Toruń in Poland. A number of cultures ultimately replaced the Linear Pottery culture
Linear_Pottery_culture
Lie group whose manifold is complex and whose group operation is holomorphic
the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie
Complex_Lie_group
Universal construction of a complex Lie group from a real Lie group
group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition g = u •
Complexification_(Lie_group)
Algebraic curve in mathematics
structure of the torus. However, all real polynomials factorize completely into linear factors over the complex numbers, since the field of complex numbers
Elliptic_curve
Most widely known generalized inverse of a matrix
In mathematics, and in particular linear algebra, the Moore–Penrose inverse A + {\displaystyle A^{+}} of a matrix A {\displaystyle A} , often called
Moore–Penrose_inverse
Anion with formula OCN and charge –1
the silver cyanato complex, [Ag(NCO)2]−, has a linear structure as shown by X-ray crystallography. However, the crystal structure of silver cyanate shows
Cyanate
Mathematical model of the time dependence of a point in space
simple trigonometric functions (i.e. complex exponentials), and the behavior of all orbits can be classified. In a linear system the phase space is the N-dimensional
Dynamical_system
Intermolecular attraction between a hydrogen donor-and-acceptor pair
bonds also affect the aramid fibre, where hydrogen bonds stabilize the linear chains laterally. The chain axes are aligned along the fibre axis, making
Hydrogen_bond
Method for estimating new data outside known data points
periodic, etc. Linear extrapolation means creating a tangent line at the end of the known data and extending it beyond that limit. Linear extrapolation
Extrapolation
Line of thunderstorms along or ahead of a cold front
(which often are accompanied by abrupt and gusty wind shifts). Linear thunderstorm structures often contain heavy precipitation, hail, frequent lightning
Squall_line
Euclidean space without distance and angles
same linear combination, despite using different origins. While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i
Affine_space
Study of the response of buildings and structures to earthquakes
spectrum approach is no longer appropriate, and more complex analysis is often required, such as non-linear static analysis or dynamic analysis. Static procedures
Seismic_analysis
Branch of differential geometry and differential topology
on the structure of a symplectic manifold. The term "symplectic", as adopted into mathematics by Hermann Weyl, is a neo-Greek calque of "complex". Previously
Symplectic_geometry
Mathematical form
composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to
Product_(mathematics)
Mathematical concept
fractional linear transformations). Thus CPn carries the structure of a complex manifold of complex dimension n, and a fortiori the structure of a real
Complex_projective_space
Concepts from linear algebra
negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation
Eigenvalues_and_eigenvectors
System of resource-aware logic
Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities
Linear_logic
Differential form on a manifold which is permitted to have complex coefficients
play a role in the study of almost complex structures, the theory of spinors, and CR structures. Typically, complex forms are considered because of some
Complex_differential_form
Projective line over the real numbers
projective transformations, homographies, or linear fractional transformations. They form the projective linear group PGL(2, R). Each element of PGL(2, R)
Real_projective_line
Banach space structure of the space C(X) of continuous real- or complex-valued functions on X. If one is allowed to invoke the algebra structure of C(X) then
Banach–Stone_theorem
Group of unitary complex matrices with determinant of 1
\operatorname {M} (2,\mathbb {C} )} denotes the set of 2-by-2 complex matrices, is an injective real linear map (by considering C 2 {\displaystyle \mathbb {C} ^{2}}
Special_unitary_group
Family of pseudorandom number generators
(松本 眞). It is a form of linear-feedback shift register optimized for software implementation on a 32-bit machine. The structure is similar to the Mersenne
Well equidistributed long-period linear
Well_equidistributed_long-period_linear
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LINEAR COMPLEX-STRUCTURE
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