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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_complex_structure
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
Smooth manifold
is closed under the Lie bracket. A generalized almost complex structure integrates to a generalized complex structure if the subspace is closed under the
Almost_complex_manifold
Concept in differential geometry
bracket. This bracket is used to describe the integrability of generalized complex structures. A more general example arises from a Lie algebroid A {\displaystyle
Courant_algebroid
It is a generalization of the Lie bracket from an operation on the tangent bundle
the Courant bracket is the integrability condition of a generalized almost complex structure. Let X {\displaystyle X} and Y {\displaystyle Y} be vector
Courant_bracket
British professor of theoretical physics (born 1957)
Hull is known for his work on string theory, M-theory, and generalized complex structures. Edward Witten drew partially from Hull's work for his development
Chris_Hull
Type of topological space
different dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology
CW_complex
British mathematician
his article on generalized Calabi–Yau manifolds, he introduced the notion of generalized complex manifolds, providing a single structure that incorporates
Nigel_Hitchin
Class of spinors constructed using Clifford algebras
in 10D, superstrings, generalized complex structures and parametrizing solutions of integrable hierarchies. Consider a complex vector space V {\displaystyle
Pure_spinor
Algebraic structure
Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined
Hodge_structure
In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of
Generalized_Jacobian
Swedish-American sociologist (1937–2025)
generalized game theory is especially open to the use of concepts such as incomplete information and bounded rationality. Proponents of generalized game
Tom_R._Burns
Type of mathematical space
When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties
Generalized_flag_variety
Objects extending the notion of functions
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory
Generalized_function
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
Theory in theoretical physics
Topological string is closely related to Hitchin's ideas on generalized complex structure, Hitchin system, and ADHM construction etc.. The 2-dimensional
Topological_string_theory
Nonspecific long-lasting anxiety
Generalized anxiety disorder (GAD) is an anxiety disorder characterized by excessive, uncontrollable, and often irrational worry about events or activities
Generalized_anxiety_disorder
Academic concept
complexities that artists draw from. The term structure of feeling seeks to describe a complex experience, since structures of feeling capture something formative
Structure_of_feeling
In image analysis, the generalized structure tensor (GST) is an extension of the Cartesian structure tensor to curvilinear coordinates. It is mainly used
Generalized_structure_tensor
Mathematical structure in differential geometry
ISSN 1945-001X. Bailey, Michael (2013-08-01). "Local classification of generalize complex structures". Journal of Differential Geometry. 95 (1). arXiv:1201.4887
Poisson_manifold
In statistics, hierarchical generalized linear models extend generalized linear models by relaxing the assumption that error components are independent
Hierarchical generalized linear model
Hierarchical_generalized_linear_model
Generalization of a polytope in real space
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real
Complex_polytope
Classifications of epileptic seizures
focal seizures, generalized-onset seizures became generalized seizures, and unknown-onset seizures became unknown whether focal or generalized. Other changes
Seizure_types
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
Mathematical structure in abstract algebra
generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers
*-algebra
Algebraic structure used in topology
generalized homology theory or a generalized cohomology theory, defined below. There are generalized cohomology theories such as K-theory or complex cobordism
Cohomology
Study of complex manifolds and several complex variables
complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry
Complex_geometry
Concept in mathematical physics
articles of the Hitchin functional. As with Hitchin's introduction of generalized complex manifolds, this is an example of a mathematical tool found useful
Hitchin_functional
Vector satisfying some of the criteria of an eigenvector
eigenvectors to have complex values. The set spanned by all generalized eigenvectors for a given λ {\displaystyle \lambda } forms the generalized eigenspace for
Generalized_eigenvector
Method of detecting shapes within images
was invented by Richard Duda and Peter Hart in 1972, who called it a "generalized Hough transform" after the related 1962 patent of Paul Hough. The transform
Hough_transform
Study of triangles in other spaces than the Euclidean plane
orthoschemes - right simplexes (right triangles generalized to n dimensions) - studied by Schoute who called the generalized trigonometry of n Euclidean dimensions
Generalized_trigonometry
Mechanical and acoustical property
more complex systems, including multi-degree-of-freedom systems and elastic continua such as beams and plates, the concept has been generalized to describe
Transmissibility_(vibration)
Data structure
In computing, GiST or Generalized Search Tree, is a data structure and API that can be used to build a variety of disk-based search trees. GiST is a generalization
GiST
Analytical method for social structure
regular or generalized. Using blockmodeling, a network can be analyzed using newly created blockmodels, which transforms large and complex network into
Blockmodeling
Field of mathematics which studies incidence structures
ordinary n-gon that contains them both. Generalized 3-gons are projective planes. Generalized 4-gons are called generalized quadrangles. By the Feit-Higman theorem
Incidence_geometry
Most widely known generalized inverse of a matrix
= A + {\textstyle A^{+}AA^{+}=A^{+}} , it is called a generalized reflexive inverse. Generalized inverses always exist but are not in general unique. Uniqueness
Moore–Penrose_inverse
ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra
List_of_cohomology_theories
Number with a real and an imaginary part
alternative complex structure on R 2 . {\displaystyle \mathbb {R} ^{2}.} This is generalized by the notion of a linear complex structure. Hypercomplex
Complex_number
Research field in deep learning
is a research field that extends deep learning to handle complex, non-Euclidean data structures. Traditional deep learning models, such as convolutional
Topological_deep_learning
Branch of mathematics studying (smooth) functions of manifolds
admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture;
Geometric_topology
Genetic condition causing non-cancerous tumours
Tuberous sclerosis complex (TSC) is a rare, multi-system genetic disorder that causes the growth of benign tumors in various organs, including the brain
Tuberous_sclerosis
Expression denoting a set of sets in formal semantics
we can define the model of a generalized quantifier as follows. Given a first-order language augmented with generalized quantifiers Q 1 , Q 2 , … {\displaystyle
Generalized_quantifier
Interrelated entities that form a whole
work by pushing on it). In 1850, the German physicist Rudolf Clausius generalized this picture to include the concept of the surroundings and began to
System
Monster and modular connection
Ryba's conjecture should generalize to Tate cohomology of composite order elements, and the nature of any connections to generalized moonshine and other moonshine
Monstrous_moonshine
structures defined on them. It differs from other branches of topology as the topological spaces do not have to be similar to manifolds. Generalized trigonometry
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Protein complex
The membrane attack complex (MAC) or terminal complement complex (TCC) is a complex of proteins typically formed on the surface of pathogen cell membranes
Complement membrane attack complex
Complement_membrane_attack_complex
Mathematical structure
after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes
Building_(mathematics)
Tensor related to gradients
detection Edge detection Lucas-Kanade method Affine shape adaptation Generalized structure tensor J. Bigun and G. Granlund (1986), Optimal Orientation Detection
Structure_tensor
Dissociative disorder
DSM-IV codes), which has several other subtypes: selective amnesia, generalized amnesia, continuous amnesia, and systematized amnesia, in addition to
Dissociative_fugue
Topological manifold with a piecewise linear structure on it
manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension
Piecewise_linear_manifold
Practice of publicly funded construction projects as propaganda
generalized outside of the context of Imelda and Ferdinand Marcos and the Philippines. The term is a play-on-words alluding to the "Oedipus complex"
Edifice_complex
Basic unit of life forms
September 2025. "Prokaryote structure". khanacademy. Retrieved 2025-08-16. Knoll, Andrew H. (2011). "The Multiple Origins of Complex Multicellularity". Annual
Cell_(biology)
over X {\displaystyle X} with a complex linear structure on the stable normal bundle. Complex bordism is a generalized homology theory, corresponding to
Complex_cobordism
Construction for adding objects to a Hilbert space
space theory. This was supplied by the apparatus of distributions, and a generalized eigenfunction theory was developed in the years after 1950. A rigged
Rigged_Hilbert_space
Matrix factorisation in mathematics
T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition. The generalized eigenvalues λ {\displaystyle
Schur_decomposition
Computational quantum mechanical modelling method to investigate electronic structure
chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular
Density_functional_theory
Positive integer of the form (2^(2^n))+1
primes today are generalized Fermat primes. Generalized Fermat numbers can be prime only for even a, because if a is odd then every generalized Fermat number
Fermat_number
Generalization of the notion of convergence that is found in general topology
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain
Convergence_space
Mathematics award
original on 31 August 2006. Retrieved 24 August 2006. UNESCO (2015). A Complex Formula: Girls and Women in Science, Technology, Engineering and Mathematics
Fields_Medal
Formation of a 3-center 2-electron bond between a transition metal and C–H bond
La Placa, Sam J.; Ibers, James A. (1965). "A Five-Coordinated d6 Complex: Structure of Dichlorotris(triphenylphosphine)ruthenium(II)". Inorg. Chem. 4
Agostic_interaction
Type of mathematical functions
a Stein manifold, and more generalized Stein space. C n {\displaystyle \mathbb {C} ^{n}} is also considered to be a complex projective variety, a Kähler
Function of several complex variables
Function_of_several_complex_variables
Class of algorithms
following algorithms are described for a binary tree, but they may be generalized to other trees as well. 0 Traversal method: 1 Previous node Restart Start
Tree_traversal
Numerical method for solving boundary value problems
The proper generalized decomposition (PGD) is an iterative numerical method for solving boundary value problems (BVPs), that is, partial differential
Proper generalized decomposition
Proper_generalized_decomposition
Chaos 2D Strange Attractor Tutorial A new chaotic system and beyond: The generalized Lorenz-like system Gingerbreadman map Mira Fractals Half-inverted tearing
List_of_chaotic_maps
A generalized probabilistic theory (GPT) is a general framework to describe the operational features of arbitrary physical theories. A GPT must specify
Generalized probabilistic theory
Generalized_probabilistic_theory
Italian British network scientist and mathematical physicist
Multilayer Networks: Structure and Function (Oxford University Press, 2018) and Higher-Order Networks: An introduction to Simplicial Complexes (Cambridge University
Ginestra_Bianconi
Type of social exchange
exchange resources with each other, generalized exchange naturally involves more than two parties. Examples of generalized exchange include; matrilateral cross-cousin
Generalized_exchange
Type of Riemannian manifold
, g ) {\displaystyle (M,g)} endowed with three integrable almost complex structures I , J , K {\displaystyle I,J,K} that are Kähler with respect to the
Hyperkähler_manifold
Rock formation in Minnesota
the complex abuts and intrudes into the Rove Formation, an older structure of sedimentary rocks. Gabbro and diabase structures of the Duluth Complex trend
Duluth_Complex
System responsible for combining morphemes into complex structures
paradigm are: Arc pair grammar Generalized phrase structure grammar (GPSG) Generative semantics Head-driven phrase structure grammar (HPSG) Lexical functional
Syntax
Structure-preserving map between two algebraic structures of the same type
algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set
Homomorphism
Fractal named after mathematician Benoit Mandelbrot
is well-known, even outside mathematics, for how it exhibits complex fractal structures when visualized and magnified, despite having a relatively simple
Mandelbrot_set
Image edge detection algorithm
ISSN 1687-9309. Scharr, Hanno (2007). "Optimal Filters for Extended Optical Flow". Complex Motion. Lecture Notes in Computer Science. Vol. 3417. Berlin, Heidelberg:
Sobel_operator
complex structure Hermitian manifold Newlander–Nirenberg theorem Generalized complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex
List of differential geometry topics
List_of_differential_geometry_topics
Branch of geometry that studies combinatorial properties and constructive methods
also generalize the higher-dimensional analogs and the finite structures are sometimes called finite geometries. Formally, an incidence structure is a
Discrete_geometry
Feature detection algorithm in computer vision
the Hessian, or more generally considering a more general family of generalized scale-space interest points. Recently, a slight variation of the descriptor
Scale-invariant feature transform
Scale-invariant_feature_transform
Type of vector space in math
square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition
Hilbert_space
Mental illness with multiple personality states
to severe symptoms is a result of different etiologies and biological structures. Other terms used in the literature, including personality, personality
Dissociative identity disorder
Dissociative_identity_disorder
Two-dimensional Turing machine with emergent behavior
runs on a square lattice of black and white cells. The idea has been generalized in several different ways, such as turmites which add more colors and
Langton's_ant
Manifold with Riemannian, complex and symplectic structure
manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by
Kähler_manifold
types of domains, for instance lineally convex domains which can be generalized using convex analysis. A great deal is already known about these domains
Complex_convexity
Image processing method
horizontal strokes (being the one-dimensional cells of an abstract cell complex corresponding to the image) mostly compose a connected sequence representing
Edge_detection
Array of numbers
bilinear forms, generalized the theorem to all dimensions (1898). Also at the end of the 19th century, the Gauss–Jordan elimination (generalizing a special
Matrix_(mathematics)
Electroencephalogram wave pattern during seizures
Bursts of generalized spike-and-wave discharges lasting two seconds or longer is considered an absence seizure. Absence seizures are generalized epileptic
Spike-and-wave
Term in mathematics
are the Jacobians of curves. The Picard variety, the Albanese variety, generalized Jacobian, and intermediate Jacobians are generalizations of the Jacobian
Jacobian_variety
Phylum of invertebrate animals
systems are mainly open. The "generalized" mollusc's feeding system consists of a rasping "tongue", the radula, and a complex digestive system in which exuded
Mollusca
Matrix decomposition
{\displaystyle k} . That is, it is the space of generalized eigenvectors (in the first sense), where a generalized eigenvector is any vector which eventually
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Recognition. 2016. Oswald, Martin Ralf, Jan Stühmer, and Daniel Cremers. "Generalized connectivity constraints for spatio-temporal 3d reconstruction." European
4D_reconstruction
Belgian mathematician
Hodge structures, and many geometric and topological tools were critical to its investigations. His work in complex singularity theory generalized Milnor
Pierre_Deligne
Fundamental operation on complex numbers
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but with opposite
Complex_conjugate
Riemannian manifold with SU(n) holonomy
the special unitary group, in a manner that is compatible with the complex structure. M {\displaystyle M} has a Kähler metric with global holonomy contained
Calabi–Yau_manifold
Area of mathematics using condensed sets
whereas condensed mathematics can be developed strictly within ZFC. Generalized space Pyknotic set "Math 205 – Topics in Number Theory – Lecture 1 video"
Condensed_mathematics
Theoretical attempts to unify the forces of nature
differential geometric structure for space-time, Erwin Schrödinger from 1940 to 1951 thoroughly investigated pure-affine formulations of generalized gravitational
Classical unified field theories
Classical_unified_field_theories
Overview of mechanics based on the least action principle
in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta
Analytical_mechanics
Statistical distribution of complex random variables
distribution (a complex normal distribution is a bivariate normal distribution) Generalized chi-squared distribution Wishart distribution Complex random variable
Complex_normal_distribution
Theoretical electronic band structure model in which the potential is periodic and weak
the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures. The periodic potential
Empty_lattice_approximation
Abstract mathematical system of two types of objects and a relation between them
geometric objects. Some examples of incidence structures 1. Fano plane 2. Non-uniform structure 3. Generalized quadrangle 4. Möbius–Kantor configuration 5
Incidence_structure
Image edge detection algorithm
threshold values to filter out the false edges. However, as the image gets complex, different local areas will need very different threshold values to accurately
Canny_edge_detector
Mix of funds used to start and sustain a business
includes common stock and retained earnings. In practice, the capital structure may be complex and include other sources of capital. Financial analysts use some
Capital_structure
Fields of natural science related to Earth
science can be generalized into planetary science, a more recent branch of science. Geology is broadly the study of Earth's structure, substance, and
Earth_science
travel, tourism, insurance
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
GENERALIZED COMPLEX-STRUCTURE
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