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Concept in geometric topology
defined, but inequivalent, concepts of normal maps and normal invariants. It is possible to perform surgery on normal maps, meaning surgery on the domain
Normal_invariant
Invariant of a quadratic form over a field of characteristic 2
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941)
Arf_invariant
Topics referred to by the same term
functions Normal function, in set theory Normal invariants, in geometric topology Normal matrix, a matrix that commutes with its conjugate transpose Normal measure
Normal
Subgroup invariant under conjugation
abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by
Normal_subgroup
Topological space that locally resembles Euclidean space
orientability (a normal invariant, also detected by homology) and genus (a homological invariant). Smooth closed manifolds have no local invariants (other than
Manifold
Property that is not changed by mathematical transformations
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations
Invariant_(mathematics)
Topics referred to by the same term
Normal map may refer to: Normal mapping in 3D computer graphics Normal invariants in mathematical surgery theory Normal matrix in linear algebra Normal
Normal_map
Tool to classify manifolds within a homotopy type in dim > 4
which are usually easier to determine. These are on one hand the normal invariants which form generalized cohomology groups, and hence one can use standard
Surgery_exact_sequence
Class of topological space
lens spaces are determined by simple homotopy type, and there are no normal invariants (like characteristic classes) or surgery obstruction. A knot-theoretic
Lens_space
Techniques in topology used to produce one finite-dimensional manifold from another
(called normal invariants) are classified by the set of homotopy classes [ X , G / O ] {\displaystyle [X,G/O]} . Each of these normal invariants has a surgery
Surgery_theory
Matrix normal form
So the Smith normal form is ( 2 0 0 0 2 0 0 0 156 ) {\displaystyle {\begin{pmatrix}2&0&0\\0&2&0\\0&0&156\end{pmatrix}}} and the invariant factors are 2
Smith_normal_form
Locally compact topological group with an invariant averaging operation
G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms
Amenable_group
addition each Ai is normal in G, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series. A series
Subgroup_series
Probability distribution
discrete log-normal distribution. City sizes (population) satisfy Gibrat's Law. The growth process of city sizes is proportionate and invariant with respect
Log-normal_distribution
Canonical form of matrices over a field
divisors used in the construction of the Jordan normal form do not exist over F[X], so the invariant factors fi as given above must be used instead. The
Frobenius_normal_form
American mathematician (1934–2025)
Invariant of Framed Manifolds and Its Generalization", Annals of Mathematics 90, 157–186 (1969) Assembly map Exotic sphere Kervaire invariant Normal invariant
William Browder (mathematician)
William_Browder_(mathematician)
American mathematician (born 1941)
conjecture Flexible polyhedron Formal manifold Loch Ness monster surface Normal invariant Ring lemma Rummler–Sullivan theorem Ruziewicz problem Holden, Helge;
Dennis_Sullivan
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
matrix A may be put in Jordan normal form. Since the underlying vector space can be shown to be the direct sum of invariant subspaces associated with the
Jordan_normal_form
Subgroup mapped to itself under every automorphism of the parent group
characteristic in G. A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. ∀φ ∈ Inn(G)： φ(H) ≤ H Since
Characteristic_subgroup
Concept in differential topology
In mathematics, the Kervaire invariant is an invariant of a framed ( 4 k + 2 ) {\displaystyle (4k+2)} -dimensional manifold that measures whether the
Kervaire_invariant
Generalization of the one-dimensional normal distribution to higher dimensions
tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives. The
Multivariate normal distribution
Multivariate_normal_distribution
Mathematical property of a space
mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological
Topological_property
Type of mathematical curve
Weierstrass normal form with coefficients in the field of definition of the cubic. Invariant theory is mainly concerned with the study of invariants of homogeneous
Cubic_plane_curve
Partially unsolved problem in mathematics
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded
Invariant_subspace_problem
invariant factors of M {\displaystyle M} and are unique up to associatedness. The invariant factors of a matrix over a PID occur in the Smith normal form
Invariant_factor
Smooth manifold with an inner product on each tangent space
nontrivial normal subgroup of G, fix any complemented subspace W of the Lie algebra of K within the Lie algebra of G. If this subspace is invariant under the
Riemannian_manifold
Left-invariant (or right-invariant) measure on locally compact topological group
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral
Haar_measure
Topological quantum field theory
s(M) is the section of the normal orthogonal bundle P. Moreover, the Chern–Simons term is described as the eta invariant defined by Atiyah, Patodi and
Chern–Simons_theory
Features that do not change if length or energy scales are multiplied by a common factor
closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the
Scale_invariance
Indication of topological symmetry groups to topological condensed matter
matter physics. It indicates the mathematical group for the topological invariant of the topological insulators and topological superconductors, given a
Periodic table of topological insulators and topological superconductors
Periodic_table_of_topological_insulators_and_topological_superconductors
Feature detection algorithm in computer vision
The scale-invariant feature transform (SIFT) is a computer vision algorithm to detect, describe, and match local features in images, invented by David
Scale-invariant feature transform
Scale-invariant_feature_transform
Type of knot in knot theory
number associated to a given link is called the Schubert normal form of the link (as this invariant was first defined by Schubert), and is precisely the fraction
2-bridge_knot
Representation of a matrix as a product
such as the SVD, that are invariant with respect to diagonal scaling. Applicable to: m-by-n matrix A. Unit-Scale-Invariant Singular-Value Decomposition:
Matrix_decomposition
Result about when a matrix can be diagonalized
v1. By Hermiticity, K n − 1 {\displaystyle {\mathcal {K}}^{n-1}} is an invariant subspace of A. To see that, consider any k ∈ K n − 1 {\displaystyle k\in
Spectral_theorem
Basic question in geometry and topology
characteristic Fundamental group Cohomology ring Geometric topology normal invariants (orientability, characteristic classes, and characteristic numbers)
Classification_of_manifolds
Are certain algebras finitely generated
for X normal. (See also: Zariski's finiteness theorem.) Éfendiev F.F. (Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary
Hilbert's_fourteenth_problem
Representation of mechanical stress at every point within a deformed 3D object
non-invariant fluids, such as polymers. At every point in a stressed body there are at least three planes, called principal planes, with normal vectors
Cauchy_stress_tensor
Mathematics
In mathematics, the Nevanlinna invariant of an ample divisor D on a normal projective variety X is a real number connected with the rate of growth of
Nevanlinna_invariant
Map from the normal invariants to the L-groups
{\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))} from the normal invariants to the L-groups which is in the first instance a set-theoretic map
Surgery_obstruction
Set of quantities in probability theory
{\textstyle \kappa _{n}(X+c)=\kappa _{n}(X),} i.e. the cumulant is translation invariant. (If n = 1 {\textstyle n=1} then we have κ 1 ( X + c ) = κ 1 ( X ) + c
Cumulant
Observation that in many real-life datasets, the leading digit is likely to be small
agreement with Benford's law, the distribution has to be approximately invariant when scaled up by any factor up to 10; a log-normally distributed data
Benford's_law
Topological structure in number theory
series. The μ-invariant of a finitely-generated torsion module is the number of times the module Zp[[T]]/(p) occurs in it. This invariant is additive on
Iwasawa_algebra
(on a complex Hilbert space) continuous linear operator
{\displaystyle \ell ^{2}(\mathbb {Z} )} , which is normal, but has no eigenvalues. The invariant subspaces of a shift acting on Hardy space are characterized
Normal_operator
In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same
Invariant_estimator
German mathematician (1882–1935)
associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under
Emmy_Noether
Meanings of mass in special relativity
"mass" has two meanings in special relativity: invariant mass (also called rest mass) is an invariant quantity which is the same for all observers in
Mass_in_special_relativity
Group with series of normal subgroups where all factors are cyclic
mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is
Supersolvable_group
and orientability are a complete set of invariants. The Jordan normal form of a matrix is a complete invariant for matrices over a field up to conjugation
Complete_set_of_invariants
Group that is also a differentiable manifold with group operations that are smooth
to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of
Lie_group
definition, K' is invariant under B1* and contains H. The normality of B1 and the assumption that H is invariant under B1 imply K' is invariant under B1. Therefore
Subnormal_operator
Statement in abstract algebra
it in Smith normal form. This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors. Another
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Unique knot with a crossing number of four
The figure-eight knot is a prime knot. The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Algebro-geometric stability condition
Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability
K-stability
Group that is a topological space with continuous group operations
d {\displaystyle d} on G {\displaystyle G} is called left-invariant (resp. right-invariant) if and only if d ( a x 1 , a x 2 ) = d ( x 1 , x 2 ) {\displaystyle
Topological_group
Statistical procedure
errors, residuals, means and standard deviations, which are hence scale invariant – some of which may be summarized as follows. Note that in terms of levels
Normalization_(statistics)
Invariant of framed knots
In knot theory, the self-linking number is an invariant of framed knots. It is related to the linking number of curves. A framing of a knot is a choice
Self-linking_number
Mathematical model combining space and time
hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate hyperboloids of one sheet, while the invariant hyperbolae displaced
Spacetime
Probability distribution
directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution)
Projected_normal_distribution
Group whose operation is a composition of braids
to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry. In this introduction let n = 4; the generalization
Braid_group
Integral of the Gaussian function, equal to sqrt(π)
polynomial in n variables may depend only on SL(n)-invariants of the polynomial. One such invariant is the discriminant, zeros of which mark the singularities
Gaussian_integral
Model of set theory constructed using permutations
closed under taking finite unions and subsets, and is called invariant if it is invariant under the action of the group G. For each element S of the ideal
Permutation_model
Stress case in finite deformations
strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure
Piola–Kirchhoff stress tensors
Piola–Kirchhoff_stress_tensors
Standard representation of a mathematical object
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical
Canonical_form
Cell type in the immune system
Mucosal-associated invariant T cells (MAIT cells) make up a subset of T cells in the immune system that display innate, effector-like qualities. In humans
Mucosal-associated invariant T cell
Mucosal-associated_invariant_T_cell
Relative deformation of a physical body
volume, as arising from dilation or compression; it is the first strain invariant or trace of the tensor: δ = Δ V V 0 = I 1 = ε 11 + ε 22 + ε 33 {\displaystyle
Strain_(mechanics)
System whose output depends on moment of observation and input signal application
true for time invariant systems (TIV). There are many well developed techniques for dealing with the response of linear time invariant systems, such as
Time-variant_system
differential geometry, affine differential geometry is the study of differential invariants of curves, surfaces, and higher-dimensional submanifolds under affine
Affine_differential_geometry
central fibre is a normal variety. In this case there exists an intersection-theoretic formula for the Donaldson–Futaki invariant of a normal test configuration
K-stability_of_Fano_varieties
Theory of response to surprise events
the beholder and warns that objectively defining a black swan in a way "invariant in the eyes of all observers" would be erroneous. Taleb provides the example
Black_swan_theory
Describes the objects of a given type, up to some equivalence
in solving it. (A combination of invariant values is realizable if there in fact exists an object whose invariants take on the specified set of values)
Classification_theorem
Statistical distance measure
the transformed space. The Mahalanobis distance is thus unitless, scale-invariant, and takes into account the correlations of the data set. Given a probability
Mahalanobis_distance
Mathematical concept
phrase "singular values of the j {\displaystyle j} -invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication
Supersingular_elliptic_curve
Type of polytope in mathematics
normality property is invariant under affine-lattice isomorphisms of lattice polytopes and the integrally closed property is invariant under an affine change
Normal_polytope
Map from tangent space to the manifold
{\displaystyle T_{p}M} . In the case of Lie groups with a bi-invariant metric—a pseudo-Riemannian metric invariant under both left and right translation—the exponential
Exponential map (Riemannian geometry)
Exponential_map_(Riemannian_geometry)
Probability distribution
{\displaystyle z} has now become a ratio of uncorrelated non-central normal samples with an invariant z {\displaystyle z} -offset.. Finally, to be explicit, the
Ratio_distribution
Formulas in differential geometry
{r} ^{(n)}} ) from the usual torsion. The Frenet–Serret formulas are invariant under flipping the sign of both χn−1 and en, and this change of sign makes
Frenet–Serret_formulas
Algebraic formula
give the final invariant factor; as long as M is non-empty, repeat to find the invariant factors before it. Invariant factors Smith normal form B. Hartley;
Elementary_divisors
Group of transformations under which the object is invariant
object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation
Symmetry_group
Mod 2 invariant of (4k+1)-dimensional manifold
In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of Z / 2 {\displaystyle \mathbf
De_Rham_invariant
has a nontrivial invariant subspace. However, when A is normal, an affirmative answer is given by the spectral theorem. Every normal operator A is obtained
Quasinormal_operator
type II1 if A has a faithful finite normal G-invariant trace. This corresponds to X having a finite G invariant measure, absolutely continuous with respect
Crossed_product
Protective layer that lines the interior of hollow organs
clearance Mucocutaneous boundary Mucosal immunology Mucosal-associated invariant T cell Mucosal melanoma Rete pegs Epimysium "Modes of locomotion in protists:
Mucous_membrane
dynamical systems, a spectral submanifold (SSM) is the unique smoothest invariant manifold serving as the nonlinear extension of a spectral subspace of
Spectral_submanifold
Concept in probability theory
the normal distribution. The only rotationally invariant probability distributions on Rn that have independent components are multivariate normal distributions
Maxwell's_theorem
Concept in statistics and wave theory
355\;\sigma .} The FWHM does not depend on the expected value x0; it is invariant under translations. The area within this FWHM is approximately 76% of
Full_width_at_half_maximum
Mathematical concept
modelling via invariant manifolds for non-autonomous dynamical systems". arXiv:1804.06998 [math.DS]. Hochs, Peter; Roberts, A.J. (2019). "Normal forms and
Center_manifold
Measure of linear correlation
mathematical property of the Pearson correlation coefficient is that it is invariant under separate changes in location and scale in the two variables. That
Pearson correlation coefficient
Pearson_correlation_coefficient
Special orthogonal group
which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
number Perfect digital invariant Happy number Perfect digit-to-digit invariant Factorion Emirp Palindromic prime Home prime Normal number Stoneham number
List of recreational number theory topics
List_of_recreational_number_theory_topics
Matrix factorisation in mathematics
Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V0 ⊂ V1 ⊂ ⋯ ⊂ Vn = Cn, and that there exists an ordered
Schur_decomposition
Set of isolated points in the spectrum of an operator with finite-rank Riesz projectors
{N}}_{\lambda }} , where N λ {\displaystyle {\mathfrak {N}}_{\lambda }} is an invariant subspace of A {\displaystyle A} in which A − λ I B {\displaystyle A-\lambda
Discrete spectrum (mathematics)
Discrete_spectrum_(mathematics)
Mathematical study of linear operators
questions in function theory. For example, Beurling's theorem describes the invariant subspaces of the unilateral shift in terms of inner functions, which are
Operator_theory
branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There
Stable_normal_bundle
Theory of interwoven space and time by Albert Einstein
presented as being based on just two postulates: The laws of physics are invariant (identical) in all inertial frames of reference (that is, frames of reference
Special_relativity
Process of word formation, by alteration to express grammatical categories
never subject to inflection are said to be invariant; for example, the English verb must is an invariant item: it never takes a suffix or changes form
Inflection
Property of algebraic varieties and complex manifolds
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds. The geometric genus can be
Geometric_genus
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
the kth moment about the mean, which are also dimensionless and scale invariant. The variance-to-mean ratio, σ 2 / μ {\displaystyle \sigma ^{2}/\mu }
Coefficient_of_variation
Topics referred to by the same term
mathematician Claude Chevalley bear his name. Chevalley–Shephard–Todd theorem in invariant theory of finite groups. Chevalley–Warning theorem concerning solvability
Chevalley_theorem
Programming language concept
datatypes. By making type constructors covariant or contravariant instead of invariant, more programs will be accepted as well-typed. On the other hand, programmers
Type_variance
Type of feedforward neural network
interfaces, and financial time series. CNNs are also known as shift invariant or space invariant artificial neural networks, based on the shared-weight architecture
Convolutional_neural_network
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