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Theorem in differential topology
differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: The strong Whitney embedding theorem states that any smooth real
Whitney_embedding_theorem
Conditions under which a chaotic system can be reconstructed by observation
M} with box counting dimension dA. Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with k > 2 d A . {\displaystyle
Takens's_theorem
Every Riemannian manifold can be isometrically embedded into some Euclidean space
Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into
Nash_embedding_theorems
Equivariant version of the Whitney embedding theorem
In mathematics, the Mostow–Palais theorem is an equivariant version of the Whitney embedding theorem. It states that if a manifold is acted on by a compact
Mostow–Palais_theorem
American mathematician (1907–1989)
01016. Loomis–Whitney inequality Whitney extension theorem Stiefel–Whitney class Whitney's conditions A and B Whitney embedding theorem Whitney graph isomorphism
Hassler_Whitney
Theorem in differential topology
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m > 1 {\displaystyle m>1} , any smooth m {\displaystyle
Whitney_immersion_theorem
Inclusion of one mathematical structure in another, preserving properties of interest
n} must be for an embedding, in terms of the dimension m {\displaystyle m} of M {\displaystyle M} . The Whitney embedding theorem states that n = 2 m
Embedding
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
the modern study of both fields. Embed M in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) Take a small neighborhood of M
Poincaré–Hopf_theorem
Waldhausen's theorem (geometric topology) Whitney embedding theorem (differential manifolds) Whitney immersion theorem (differential topology) Radó's theorem (harmonic
List_of_theorems
Graph that can be embedded in the plane
planar graph. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is k-outerplanar if removing the
Planar_graph
Projection of data onto lower-dimensional manifolds
GitHub) Manifold hypothesis Spectral submanifold Taken's theorem Whitney embedding theorem Discriminant analysis Elastic map Feature learning Growing
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Proof that every structure with certain properties is isomorphic to another structure
full and faithful limit-preserving embedding of any category into a category of presheaves. Mitchell's embedding theorem for abelian categories realises
Representation_theorem
Branch of mathematics
Famous theorems in differential topology include the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's
Differential_topology
Topological space that locally resembles Euclidean space
ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. In Riemannian geometry, one may ask for maps to
Manifold
Subset of a manifold that is a manifold itself; an injective immersion into a manifold
because, by the Whitney embedding theorem, any second-countable smooth (abstract) m {\displaystyle m} -manifold can be smoothly embedded in R 2 m {\displaystyle
Submanifold
Branch of mathematics studying (smooth) functions of manifolds
because the Whitney embedding theorem, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows
Geometric_topology
Two-dimensional manifold
surfaces in the extrinsic sense. However, the Whitney embedding theorem asserts every surface can in fact be embedded homeomorphically into Euclidean space,
Surface_(topology)
Type of mathematical set
be embedded in a ( 2 d + 1 ) {\displaystyle (2d+1)} -dimensional space. This result is piecewise linear counterpart of the (weak) Whitney embedding theorem
Simplicial_complex
ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. In complex geometry, ramified covering spaces
Maps_of_manifolds
Manifold
not the same. For example, the Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R2n
Complex_manifold
Differentiable function whose derivative is everywhere injective
immersion, and in fact to an embedding for 2m < n; these are the Whitney immersion theorem and Whitney embedding theorem, respectively. Stephen Smale
Immersion_(mathematics)
American mathematician
fixed-point theorem Chern's conjecture (affine geometry) Differential structure Homotopy principle Immersion (mathematics) Whitney embedding theorem Cr section
Morris_Hirsch
Concept in topology
a cancelling pair as desired, so long as we can embed this disk into the boundary of W. This embedding exists if dim ∂ W − 1 = n − 1 ≥ 2 ( k + 1 ) {\displaystyle
H-cobordism
Graph representing edges of another graph
properties of the underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Line
Line_graph
Embedding Whitney embedding theorem Critical value Sard's theorem Saddle point Morse theory Lie derivative Hairy ball theorem Poincaré–Hopf theorem Stokes'
List of differential geometry topics
List_of_differential_geometry_topics
On the intersection form of a smooth, closed 4-manifold with a spin structure
Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (equivalently, if the second Stiefel–Whitney class w 2
Rokhlin's_theorem
Concept in mathematics
the Whitney embedding theorem, every manifold admits a normal bundle, given such an embedding. There is in general no natural choice of embedding, but
Normal_bundle
Smooth manifold with an inner product on each tangent space
use of a partition of unity. An alternative proof uses the Whitney embedding theorem to embed M {\displaystyle M} into Euclidean space and then pulls back
Riemannian_manifold
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
the Nash embedding theorem can be assumed. However, this theorem was not available then, as John Nash published his famous embedding theorem for Riemannian
Chern–Gauss–Bonnet_theorem
are equivalent by regular homotopy, though not by isotopy. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane;
Regular_homotopy
American mathematician and Nobel Laureate (1928–2015)
various sciences. In the 1950s, Nash discovered and proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising
John_Forbes_Nash_Jr.
Statistical model used in machine learning
{R} ^{2n+1}} , proved by combining Whitney embedding theorem for manifolds and the universal approximation theorem for neural networks. To regularize
Flow-based_generative_model
Type of mathematical functions
into the complex plane. The Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R
Function of several complex variables
Function_of_several_complex_variables
Basic question in geometry and topology
embeddings and immersions include: Whitney embedding theorem Whitney immersion theorem Nash embedding theorem Smale-Hirsch theorem Key tools in studying these
Classification_of_manifolds
Topological mapping
h-cobordism theorem, where it is used to cancel the intersection points; and its failure in low dimensions corresponds to not being able to embed a Whitney disc
Whitney_disk
German mathematician (1930–2011)
approximation theorem, due to Grauert, is an analog of Whitney’s approximation theorem for real-analytic maps. It states: with respect to the Whitney topology
Hans_Grauert
Topological space associated to a vector bundle
available, we can use them and the isomorphism of the theorem to construct the Stiefel–Whitney classes. Recall that the Steenrod operations (mod 2) are
Thom_space
Partial differential equation technique
appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric C 1 {\displaystyle C^{1}} embedding theorem and the Smale–Hirsch
Homotopy_principle
Abstract simplicial complex describing a graph's cliques
known as Whitney complexes, after Hassler Whitney. A Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph
Clique_complex
Darboux-Moser-Weinstein theorem, taking X = L {\displaystyle X=L} a Lagrangian submanifold, together with a version of the Whitney Extension Theorem for smooth manifolds
Weinstein's neighbourhood theorem
Weinstein's_neighbourhood_theorem
Graph representing faces of another graph
6. Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique. By Steinitz's theorem, these graphs
Dual_graph
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
Lie group theory. The Whitney embedding theorem showed that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as
History of manifolds and varieties
History_of_manifolds_and_varieties
Branch of the mathematical field of graph theory
the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph
Topological_graph_theory
Measure theory
_{n\in \mathbb {N} }a_{n}} diverges. Prevalence version of the Whitney embedding theorem: Let M {\displaystyle M} be a compact manifold of class C 1 {\displaystyle
Prevalent_and_shy_sets
Branch of mathematics
Vietoris Hassler Whitney J. H. C. Whitehead Gordon Thomas Whyburn Blakers–Massey theorem Borsuk–Ulam theorem Brouwer fixed point theorem Cellular approximation
Algebraic_topology
Space formed by the ''n''-tuples of real numbers
geometry. On the other hand, Whitney embedding theorems state that any real differentiable m-dimensional manifold can be embedded into R2m. Other structures
Real_coordinate_space
Spivak. Given an embedding of a manifold in Euclidean space (provided by the theorem of Hassler Whitney), it has a normal bundle. The embedding is not unique
Stable_normal_bundle
Area of discrete mathematics
embedding (or imbedding) of a graph in surface and linkless embedding, graph minors, crossing number, map coloring, and voltage graph. The embedding of
Graph_theory
Set of topological invariants
particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that
Stiefel–Whitney_class
Undirected graph named after S. S. Shrikhande
toroidal graph. The embedding forms a regular map in the torus, with 32 triangular faces. The skeleton of the dual of this map (as embedded in the torus) is
Shrikhande_graph
identifying their boundaries. As the result we get a manifold without boundary. Embedding Exotic structure – See exotic sphere and exotic R 4 {\textstyle \mathbb
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
British-Canadian codebreaker and mathematician (1917–2002)
graph theory have been influential to modern graph theory and many of his theorems have been used to keep making advances in the field, most of his terminology
W._T._Tutte
Calculus of functions generalization
differential is injective. An embedding is an immersion that is homeomorphic (thus diffeomorphic) to the image. Whitney's embedding theorem—Each k {\displaystyle
Calculus_on_Euclidean_space
Algorithmic problem of finding non-crossing drawings
graphs to incrementally build planar embeddings of every 3-connected component of G (and hence a planar embedding of G itself). The construction starts
Planarity_testing
Topological operation of turning a sphere inside-out without creasing
S^{n}} embedded in euclidean space R n + 1 {\displaystyle \mathbb {R} ^{n+1}} admits eversion. Nylon string open model Whitney–Graustein theorem Bednorz
Sphere_eversion
Non-orientable surface with one edge
that force an embedding to be developable versus the assumptions under which the Nash–Kuiper theorem allows arbitrarily flexible embeddings, see remarks
Möbius_strip
Subgraph of planar graph with Hamiltonian cycle
MR 2749626. For instance in a 2003 technical report "Book embeddings of graphs and a theorem of Whitney", Paul Kainen defines subhamiltonian graphs to be subgraphs
Subhamiltonian_graph
Graph cycle which does not separate remaining elements
graph G {\displaystyle G} , and every planar embedding of G {\displaystyle G} , the faces of the embedding that are induced cycles must be peripheral cycles
Peripheral_cycle
American mathematician
received his Ph.D. from the University of California, Berkeley in 1982 (Embedding 4-manifolds in S5). He then returned to MIT as a C.L.E. Moore Postdoctoral
Tim_Cochran
h-cobordism theorem, the following construction is used. Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold
Casson_handle
Concept in algebraic geometry
resolve the singularities of a variety X embedded into a larger variety W. Suppose we have a closed embedding of X into a regular variety W. A strong desingularization
Resolution_of_singularities
Special tangential structure
Mellor 1995, Theorem 5 Albanese & Milivojević 2021, p. 6 Mellor 1995, Theorem 2 Nicolaescu, Example 1.3.16 Lawson & Michelson 90, Theorem D.2 und Corollary
Spinc_structure
Averages of repeated trials converge to the expected value
Conjecturing) in 1713. He named this his "golden theorem" but it became generally known as "Bernoulli's theorem". This should not be confused with Bernoulli's
Law_of_large_numbers
Grassmannian of all lines of projective space. The Birkhoff–Grothendieck theorem classifies the n-dimensional vector bundles over a projective line as corresponding
Jumping_line
Partially ordered set equipped with a rank function
having rank i . The Whitney numbers are connected with a lot of important combinatorial theorems. The classic example is Sperner's theorem, which can be formulated
Graded_poset
Unirational surfaces of characteristic 0 Veronese surface, the Veronese embedding of the projective plane into projective 5-space White surfaces, the blow-up
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
Tree graph with one central node and leaves of length 1
subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomorphic line graphs are themselves
Star_(graph_theory)
Family of probability distributions related to the normal distribution
distributed data using a fixed number of values. (Pitman–Koopman–Darmois theorem) Exponential families have conjugate priors, an important property in Bayesian
Exponential_family
local Hasse invariants and the signatures coming from real embeddings. Hasse–Minkowski theorem Lam (2005) p.118 Milnor & Husemoller (1973) p.79 Serre (1973)
Hasse invariant of a quadratic form
Hasse_invariant_of_a_quadratic_form
Graph of numbers differing by a square
(2001) finds embeddings of the Paley graphs of order q ≡ 1 (mod 8) that are highly symmetric and self-dual, generalizing a natural embedding of the Paley
Paley_graph
Continuous deformation between two continuous functions
t = 0 giving the K1 embedding, ending at t = 1 giving the K2 embedding, with all intermediate values corresponding to embeddings. However, this definition
Homotopy
Characteristic class in algebraic topology
Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem. It is named for
Todd_class
Distribution function associated with the empirical measure of a sample
uniform rate of convergence in Donsker’s theorem can be quantified by the result known as the Hungarian embedding: lim sup n → ∞ n ln 2 n ‖ n ( F ^ n −
Empirical distribution function
Empirical_distribution_function
Problem in algebraic geometry
applications, one combines Bézout's theorem. Let X i ↪ Y {\displaystyle X_{i}\hookrightarrow Y} be regular embeddings of schemes, separated and of finite
Residual_intersection
et plongements riemanniens, d'après Nash et Moser (Nash embedding theorem, Nash–Moser theorem) Laurent Schwartz, Sous-espaces hilbertiens et antinoyaux
Séminaire Nicolas Bourbaki (1960–1969)
Séminaire_Nicolas_Bourbaki_(1960–1969)
Concept in geometric topology
is homotopic to an embedding by a theorem of Whitney. On the other hand, every stably trivial normal bundle of such an embedding is automatically trivial
Normal_invariant
Statistical test that compares goodness of fit
maxima and the allowed ranges they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) λ LR {\displaystyle \lambda _{\text{LR}}}
Likelihood-ratio_test
Point on a curve not given by a smooth embedding of a parameter
singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on
Singular_point_of_a_curve
Characteristic classes of vector bundles
information about this through, for instance, the Riemann–Roch theorem and the Atiyah–Singer index theorem. Chern classes are also feasible to calculate in practice
Chern_class
Central limit theorem Central limit theorem (illustration) – redirects to Illustration of the central limit theorem Central limit theorem for directional
List_of_statistics_articles
Topological spaces whose union is a boundary
the disjoint union M ⊔ M ′ {\displaystyle M\sqcup M'} by surgery on an embedding of S 0 × D n {\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{n}} in
Cobordism
Algebraic structure used in topology
Section IV.3. Hopf 1933. van Kampen 1932. Whitney 1937. May 1999, p. 95. Switzer 1975, p. 117, 331, Theorem 9.27; Corollary 14.36; Remarks. "Are spectra
Cohomology
Two pentagonal pyramids fused base-to-base
Within this structure, the graph forms a topological surface called a Whitney triangulation. The pentagonal bipyramid has applications in many fields
Pentagonal_bipyramid
Branch of mathematics
properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments
Algebraic_geometry
Mathematical space
on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman (1982) implies that the homeomorphism type of the manifold
4-manifold
Tool to track locally defined data attached to the open sets of a topological space
These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, the decomposition theorem, paving the path
Sheaf_(mathematics)
Type of numerical analysis
scaling, where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity
Isotonic_regression
Awarded every year by the American Mathematical Society
Academic Press. ISBN 9780080873732. Edwards, Harold M. (1977). Fermat's Last Theorem. Graduate Texts in Mathematics. Vol. 50. Springer New York. ISBN 978-0-387-90230-2
Leroy_P._Steele_Prize
Discrete (i.e., incremental) version of infinitesimal calculus
These two points of view are related to each other by the fundamental theorem of discrete calculus.[definition needed] The study of the concepts of change
Discrete_calculus
Concept in stochastic analysis
can recover classical results—such as the Wong–Zakai theorem, the Stroock–Varadhan support theorem, and the construction of stochastic flows—without relying
Rough_path
Mathematics timeline
2018. Gallier, Jean; Xu, Dianna (2013). A Guide to the Classification Theorem for Compact Surfaces. Springer Science & Business Media. p. 156. ISBN 9783642343643
Timeline_of_manifolds
Value that appears most often in a set of data
one-dimensional vector space) and the integers (which can be considered embedded in the reals). For example, a distribution of points in the plane will
Mode_(statistics)
Text in Hinduism, Buddhism, or Jainism, often a collection of aphorisms
compilation of short aphoristic statements. Each sutra is any short rule, like a theorem distilled into few words or syllables, around which teachings of ritual
Sutra
Concept in differential geometry
an embedding (of concrete categories) if it is injective on objects and faithful, and D ∘ E = U {\displaystyle D\circ E=U} . To specify an embedding, we
Diffeology
Manifold upon which it is possible to perform calculus
onto its image, then f is an embedding. Embeddings formalize the notion of M being a submanifold of N. In general, an embedding is an immersion without self-intersections
Differentiable_manifold
Vector bundle of rank 1
bundle is sufficiently ample this construction verifies the Kodaira embedding theorem. In general if V {\displaystyle V} is a vector bundle on a space X
Line_bundle
P. Hochschild (local class field theory) Laurent Schwartz, Les théorèmes de Whitney sur les fonctions différentiables (singularity theory) Jean-Pierre
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
Variable capable of taking on a limited number of possible values
not recommended as it will lead to uninterpretable statistical results. Embeddings are codings of categorical values into low-dimensional real-valued (sometimes
Categorical_variable
Symplectic topology tool
solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the counts of flow lines defining the differential
Floer_homology
travel, tourism, insurance
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
Girl/Female
Anglo, Australian, British, English
Place Name; White Island
Girl/Female
Christian & English(British/American/Australian)
Island
Boy/Male
English
From the white meadow.
Surname or Lastname
English
English : patronymic from White.
Surname or Lastname
English
English : variant spelling of Whitley.
Female
English
Variant spelling of English Brittany, BRITNEY means "Little Britain."
Surname or Lastname
English
English : variant of Whinery.
Girl/Female
American, Australian, British, Dutch, English
From the White Meadow; White Wood
Surname or Lastname
English
English : variant of Whitley.
Surname or Lastname
English
English : habitational name from any of various places named with Old English hwīt ‘white’ + lēah ‘wood’, ‘clearing’.
Girl/Female
Anglo, Australian, British, English
Form of Whitney
Surname or Lastname
English
English : habitational name from a place in Herefordshire, the etymology of which is uncertain. The second element is Old English ēg ‘island’, ‘piece of higher ground in a low-lying area’; the first appears to be hwītan, which is either the genitive singular of an Old English byname Hwīta (meaning ‘white’), or the weak dative case (originally used after a preposition and article) of the adjective hwīt ‘white’.John Whitney came from London, England, to Watertown, MA, in 1635, and had numerous prominent descendents.
Boy/Male
English Scandinavian
From the white farm.
Boy/Male
British, English
From the White Farm
Girl/Female
Anglo, Australian, British, English
Place Name; White Island
Boy/Male
American, Anglo, Australian, British, English, Irish, Jamaican
From the White Island
Girl/Female
Anglo, British, English
Place Name; White Island
Boy/Male
British, English
A Small Field; From the White Meadow
Surname or Lastname
English (chiefly Yorkshire)
English (chiefly Yorkshire) : variant of Whitley.
Girl/Female
English
White meadow.
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
WHITNEY EMBEDDING-THEOREM
v. i.
To whinny.
n.
One who, or that which, whines.
v. t.
To make white; to whiten; to whitewash; to bleach.
a.
Like, or coming near to, white.
n. pl.
Cloth or garments of a plain white color.
p. pr. & vb. n.
of Imbed
v. i.
To grow white; to turn or become white or whiter; as, the hair whitens with age; the sea whitens with foam; the trees in spring whiten with blossoms.
n.
The act of embedding, or the state of being embedded.
adv.
To what place; -- used interrogatively; as, whither goest thou?
n.
A person with a white skin; a member of the white, or Caucasian, races of men.
n.
A white pigment; as, Venice white.
v. t.
To make white; to bleach; to blanch; to whitewash; as, to whiten a wall; to whiten cloth.
p. pr. & vb. n.
of Embed
imp. & p. p.
of Whine
n.
A kind of carriage. See Whiskey.
imp. & p. p.
of Whiten
a.
Whitened; make white.
n. pl.
The finest flour made from white wheat.
imp. & p. p.
of White
n.
One who, or that which, whitens; a bleacher; a blancher; a whitewasher.
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