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Difference between the dimensions of mathematical object and a sub-object
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of
Codimension
Generalizations of codimension-1 subvarieties of algebraic varieties
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common
Divisor_(algebraic_geometry)
Subspace of n-space whose dimension is (n-1)
between a subspace and its ambient space is known as its codimension. A hyperplane has codimension 1. In geometry, a hyperplane of an n-dimensional space
Hyperplane
Differentiable function whose derivative is everywhere injective
bundle, so it cannot immerse in codimension 0 (in R 2 {\displaystyle \mathbb {R} ^{2}} ), though it embeds in codimension 1 (in R 3 {\displaystyle \mathbb
Immersion_(mathematics)
Study of sudden qualitative behavior changes caused by small parameter changes
stable manifolds of the saddle. In three or more dimensions, higher codimension bifurcations can occur, producing complicated, possibly chaotic dynamics
Bifurcation_theory
In mathematics, a partition of a manifold into submanifolds
submanifolds are called the leaves of the foliation. The 3-sphere has a famous codimension-1 foliation called the Reeb foliation. The submanifolds are required
Foliation
Basic question in geometry and topology
the middle dimension has codimension more than 2: when the codimension is 2, one encounters knot theory, but when the codimension is more than 2, embedding
Classification_of_manifolds
Unsolved problem in geometry
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular
Hodge_conjecture
Branch of geometry
integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the
Contact_geometry
In mathematics, dimension of a ring
or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal. In a Noetherian ring, every prime
Krull_dimension
something happens, it happens in a particular codimension". For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds
Purity_(algebraic_geometry)
Number of vectors in any basis of the vector space
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Dimension_(vector_space)
American mathematician (born 1951)
studies and received a Ph.D. in 1973 for his doctoral dissertation titled Codimension-Two Surgery, written under the supervision of William Browder. After
Michael_Freedman
Branching out of a mathematical structure
something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example
Ramification_(mathematics)
Fundamental space of geometry
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Euclidean_space
Way of decomposing a topological space
boundary (top dimension and codimension 1 boundary) and manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners), real or complex
Thom–Mather_stratified_space
To find the minimal surface with a given boundary
perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed. The theory
Plateau's_problem
Quadratic form related to curvatures of surfaces
surface). The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values
Second_fundamental_form
is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one
Algebraic_cycle
Inequalities in number theory and matrix theory
B {\textstyle W_{B}} with codimension j − 1 {\textstyle j-1} . Now W A ∩ W B {\textstyle W_{A}\cap W_{B}} has codimension ≤ i + j − 2 {\textstyle \leq
Weyl's_inequality
Geometric space with five dimensions
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Five-dimensional_space
Y} ). A codimension-1 subvariety Z ⊂ X {\displaystyle Z\subset X} is said to be exceptional if f ( Z ) {\displaystyle f(Z)} has codimension at least
Exceptional_divisor
functorial, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form
Adequate_equivalence_relation
Difference between two dimensions
linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map V → Q {\displaystyle V\to Q}
Relative_dimension
Manifold or algebraic variety of dimension n in a space of dimension n+1
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Hypersurface
Generalized sphere of dimension n (mathematics)
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
N-sphere
Branch of mathematics studying (smooth) functions of manifolds
embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension
Geometric_topology
orthogonal foliations of connected submanifolds of codimension 1. Note that two submanifolds of codimension 1 are orthogonal iff their normal vectors are orthogonal
Web_(differential_geometry)
Completion of the usual space with "points at infinity"
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Projective_space
Theorem about mass-minimizing surfaces
2000), states that the singular set of a mass-minimizing surface has codimension at least 2. Almgren's proof of this was 955 pages long. Within the proof
Almgren_regularity_theorem
(assuming that some technical nondegeneracy conditions are satisfied). Three codimension-one bifurcations occur nearby: a saddle-node bifurcation, an Andronov–Hopf
Bogdanov–Takens_bifurcation
Theoretical (d–1)-dimensional singularity
(d−1)-dimensional singularity. A domain wall is meant to represent an object of codimension one embedded into space (a defect in space localized in one spatial dimension)
Domain_wall_(string_theory)
Invariant measure of fractal dimension
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Hausdorff_dimension
Discontinuous change of a quantity in algebraic geometry or string theory
integer geometric invariant, an index or a space of BPS state, across a codimension-one wall in a space of stability conditions, a so-called wall of marginal
Wall-crossing
Type of commutative ring in mathematics
some p × q matrix of elements of S. If the codimension (or height) of I is equal to the "expected" codimension (p−r+1)(q−r+1), R is called a determinantal
Cohen–Macaulay_ring
Concept in algebraic geometry
(which kills codimension 1 singularities) with blowing up points (which makes codimension 2 singularities better, but may introduce new codimension 1 singularities)
Resolution_of_singularities
Multi-dimensional generalization of triangle
\mathbf {R} ^{n}} (maximal dimension, codimension 0) rather than of R n + 1 {\displaystyle \mathbf {R} ^{n+1}} (codimension 1). The facets, which on the standard
Simplex
Mathematical theory
stability theorem, named after Georges Reeb, asserts that if one leaf of a codimension-one foliation is closed and has finite fundamental group, then all the
Reeb_stability_theorem
Conjecture in algebraic geometry
of V to ks; these groups are representations of G. For any i ≥ 0, a codimension-i subvariety of V (understood to be defined over k) determines an element
Tate_conjecture
Number used in algebraic geometry
theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees
Degree of an algebraic variety
Degree_of_an_algebraic_variety
Singularities of algebraic varieties
exceptional divisors of f (the codimension-1 subvarieties of Y, these being irreducible by definition, whose image in X has codimension at least 2). The ai are
Canonical_singularity
Result about foliation of compact 3-manifolds
Novikov's compact leaf theorem, named after Sergei Novikov, states that A codimension-one foliation of a compact 3-manifold whose universal covering space
Novikov's compact leaf theorem
Novikov's_compact_leaf_theorem
Faster-than-light travel in science fiction
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Hyperspace
point, (0, 0, 0), which is singular. The same considerations apply to codimension. For example a smooth complex hypersurface in complex projective space
Complex_dimension
Algebraic structure used in topology
isomorphism HiX ≅ Hn−iX. As a result, a closed oriented submanifold S of codimension i in X determines a cohomology class in HiX, called [S]. In these terms
Cohomology
Way to join two given mathematical manifolds together
element from H 1 ( V ) {\displaystyle H^{1}(V)} . A connected sum along a codimension-two V {\displaystyle V} can also be carried out in the category of symplectic
Connected_sum
Skeletonized version of algebraic geometry
dimension d that satisfies the zero-tension condition and is connected in codimension one. When d is one, the zero-tension condition means that around each
Tropical_geometry
Causal relationships between points in a manifold
spacelike infinity has codimension 2. Anti-de Sitter space: there's no timelike or null infinity, and spacelike infinity has codimension 1. de Sitter space:
Causal_structure
Four-dimensional number system
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Quaternion
Convex polytope, the n-dimensional analogue of a square and a cube
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Hypercube
Geometric model of the physical space
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Three-dimensional_space
Local ring in commutative algebra
embedding codimension c, meaning that c = dimk(m/m2) − dim(R). In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular
Gorenstein_ring
Concept in algebraic geometry
of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. More
Nef_line_bundle
effective algebraic cycle in P n − 1 {\displaystyle \mathbb {P} ^{n-1}} of codimension 1 and degree d can be defined by the vanishing of a single degree d polynomial
Chow_variety
Property of a mathematical space
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Dimension
Surgery operation in minimal model program
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative
Flip_(algebraic_geometry)
Real-valued number of spatial dimensions
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Fractal_dimension
Branch of mathematics
X^{(p)}} the set of codimension p {\displaystyle p} points, meaning the set of subschemes x : Y → X {\displaystyle x:Y\to X} of codimension p {\displaystyle
K-theory
Topological space that locally resembles Euclidean space
dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney
Manifold
Study of mathematical knots
{\displaystyle \mathbb {R} ^{6}} (Haefliger 1962) (Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension
Knot_theory
{\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that
Regular_embedding
Riemannian manifold. In the special case of the space of null-homologous codimension 1 cycles with mod 2 coefficients on a closed Riemannian manifold Almgren
Almgren's_isomorphism_theorem
Mathematical formula
Riemann curvature tensor. In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental
Simons'_formula
Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf. By Novikov's compact
Reeb_foliation
Geometric space with four dimensions
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Four-dimensional_space
Method for producing composition algebras
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Cayley–Dickson_construction
Vector space consisting of affine subsets
V/U\to 0.\,} If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
In mathematics, a module that has a basis
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Free_module
Concept in mathematics
tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets
Taut_foliation
Diffeomorphism that has a hyperbolic structure on the tangent bundle
for codimension-one Anosov diffeomorphisms (i.e., those for which the contracting or the expanding subbundle is one-dimensional) and for codimension one
Anosov_diffeomorphism
F(U-Y)} is bijective for every open subset U and a closed subset Y of U of codimension at least 2. With this terminology, a coherent sheaf on an integral normal
Reflexive_sheaf
American mathematician and Nobel Laureate (1928–2015)
continuously differentiable mapping. Nash's construction allows the codimension of the embedding to be very small, with the effect that in many cases
John_Forbes_Nash_Jr.
Element of a unital algebra over the field of real numbers
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Hypercomplex_number
In mathematics, vector subspace
linear functional F specifies its kernel subspace F = 0 of codimension 1. Subspaces of codimension 1 specified by two linear functionals are equal, if and
Linear_subspace
Theorem about the dual of a Hilbert space
} The vector subspace ker φ {\displaystyle \ker \varphi } has real codimension 1 {\displaystyle 1} in ker φ R , {\displaystyle \ker \varphi _{\mathbb
Riesz_representation_theorem
Measure of a mathematical object studied in the field of algebraic geometry
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Low energy theories not compatible with string theory
{\mathcal {O}}_{g}(\Sigma )} to any symmetry element g {\displaystyle g} and codimension-1 hypersurface Σ {\displaystyle \Sigma } such that O g ( Σ ) {\displaystyle
Swampland_(physics)
American mathematician (born 1942)
Ph.D. in 1969 at Columbia University. His dissertation, A Theory of Codimension One Phenomena with an Application to the Theory of Purely Inseparable
William_Haboush
Mathematical theorem
{\displaystyle i:Z\to X} a closed immersion of a regular scheme of pure codimension r, an integer n that is invertible on the base scheme, F {\displaystyle
Theorem_of_absolute_purity
Geometric model of the planar projection of the physical universe
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Euclidean_plane
Romanian American mathematician
al.), and the proof of Getzler's conjecture, asserting vanishing in codimension at least g of the tautological ring of the moduli space of genus-g curves
Eleny_Ionel
Mathematical model combining space and time
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Spacetime
Mathematical space with two coordinates
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Two-dimensional_space
American mathematician
singular sets of codimension seven. In 1970, Federer proved that this codimension is optimal: all such singular sets have codimension of at least seven
Herbert_Federer
Mathematical element
singularities since it gives a process for resolving singularities of codimension 1. For example, the integral closure of C [ x , y , z ] / ( x y ) {\displaystyle
Integral_element
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz–Killing curvature (the average of
Chern–Gauss–Bonnet_theorem
Number of independent parameters of a system
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Degrees_of_freedom
Class of mathematical function
with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two (in the given example this set consists of the origin ( 0 , 0 ) {\displaystyle
Meromorphic_function
Theorem on extension of bounded linear functionals
{\displaystyle M} to a larger vector space in which M {\displaystyle M} has codimension 1. {\displaystyle 1.} Lemma (One–dimensional dominated extension theorem)—Let
Hahn–Banach_theorem
Description of how spaces intersect in mathematics
then the intersection will be a submanifold whose codimension is equal to the sums of the codimensions of the two manifolds. In the absence of the transversality
Transversality
American mathematician
Lawson, Osserman studied the minimal surface problem in the case that the codimension is larger than one. They considered the case of a graphical minimal submanifold
Robert_Osserman
Discrete group type in group theory
/ c i j {\displaystyle 2\pi /c_{ij}} fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter
Reflection_group
Method of determining fractal dimension
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Minkowski–Bouligand_dimension
Extremely small quantity in calculus; thing so small that there is no way to measure it
indivisibles related to geometrical figures as being composed of entities of codimension 1.[clarification needed] John Wallis's infinitesimals differed from indivisibles
Infinitesimal
Generalization of vector bundles
{\mathcal {E}}} on a smooth projective variety X {\displaystyle X} and codimension 2 subvarieties Y {\displaystyle Y} using a certain Ext 1 {\displaystyle
Coherent_sheaf
Class of partial differential equations
the initial value problem is well-posed for initial data given on a codimension-one hypersurface. And later, in 2022, a research team at the University
Ultrahyperbolic_equation
Topological space of dimension zero
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Zero-dimensional_space
Theorem in topology
diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2. Hilbert–Smith conjecture Eilenberg, Samuel (1949), "On the Problems
Smith_conjecture
Geometric object with flat sides
numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category
Polytope
Algebraic structure
2 {\displaystyle \geq 2} . Item (i) is often phrased as "regular in codimension 1". Note that (i) implies that the set of associated primes A s s ( A
Integrally_closed_domain
CODIMENSION
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Girl/Female
American, Australian, British, Christian, English, French, German, Jamaican, Portuguese, Spanish
Precious Green Gem Stone; Jewel Name; Emerald; Praise; The Prized Green Emerald Gemstone
Boy/Male
Hindu, Indian, Kannada, Telugu
One who has Red-colored Body
Boy/Male
Arabic, Muslim
Ruler over Heroes
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
An Idol
Boy/Male
Arabic, Hindu, Indian, Muslim
Irritable; Impatient
Boy/Male
African, American, Anglo, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Gujarati, Indian, Irish, Italian, Jamaican, Netherlands, Norwegian, Polish, Portuguese, Swedish, Swiss, Tamil
Ruler of the Enclosure; Estate Ruler; House Owner; Lord of the Manor; Home Ruler
Surname or Lastname
English (Cornwall and Wales)
English (Cornwall and Wales) : variant of Jack.Czech (JaÄka), Polish, and German (of Slavic origin) : from a pet form (Czech JaÄ, Polish Jacz) of any of the various Slavic personal names beginning with Ja-, for example Jakub, Jan, Jacenty (see Jacek).
Boy/Male
Indian
Lord Perumal
Female
Danish
, pure.
Boy/Male
American, British, English
Son of Charles; A Man; Variant of Carl
CODIMENSION
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CODIMENSION