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On the preimage of points in a manifold under the action of a smooth map
of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold
Preimage_theorem
Theorem in topology
every point of the preimage of p {\displaystyle p} . In particular, by the inverse function theorem, every point of the preimage of f {\displaystyle
Brouwer_fixed-point_theorem
Group of mathematical theorems
specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients
Isomorphism_theorems
(topology) Preimage theorem (differential topology) Reeb sphere theorem (foliations) Reidemeister–Singer Theorem (geometric topology) Riemann–Roch theorem for
List_of_theorems
Set of the values of a function
produce, that is, the image of X {\displaystyle X} . The preimage of f {\displaystyle f} is the preimage of the codomain Y {\displaystyle Y} . Because it always
Image_(mathematics)
Sufficient criterion for uniform convergence
In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space
Dini's_theorem
Theorem about right triangles
In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle
Geometric_mean_theorem
Association of one output to each input
preimage under f of an element y of the codomain Y is the set of all elements of the domain X whose images under f equal y. In symbols, the preimage of
Function_(mathematics)
Movement with a fixed point is rotation
containing A is the axis of rotation and the theorem is proved. Otherwise we label A’s image as a and its preimage as α, and connect these two points to A
Euler's_rotation_theorem
Theorem in statistics
In statistics, Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statistic. This is a 1955 result
Basu's_theorem
Degree of differentiability of a function or map
point p ∈ F − 1 ( q ) {\displaystyle p\in F^{-1}(q)} . This is the preimage theorem. Similarly, the image of an embedding is an embedded submanifold. Smoothness
Smoothness
On the approximate structure of sets whose sumset is small
In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate structure of sets whose
Freiman's_theorem
Algebraic geometry theorem
In algebraic geometry, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties
Theorem_of_Bertini
Set of all points in a function's domain that all map to some single given point
field at p . {\displaystyle p.} Fibration Fiber bundle Fiber product Preimage theorem Zero set Lee, John M. (2011). Introduction to Topological Manifolds
Fiber_(mathematics)
High-area shapes can shift to hold many grid points
Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area A {\displaystyle
Blichfeldt's_theorem
Two continuous functions can be glued together to create another continuous function
∩ Y {\displaystyle f^{-1}(U)\cap Y} are both closed since each is the preimage of f {\displaystyle f} when restricted to X {\displaystyle X} and Y {\displaystyle
Pasting_lemma
Mathematical concept
surjective. If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is defined to be the set of all
Inverse_function
Kind of mathematical function
two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition
Measurable_function
Elements taken to zero by a homomorphism
involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the
Kernel_(algebra)
Poisson integrals of homeomorphisms are diffeomorphisms
In mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism
Radó–Kneser–Choquet_theorem
Area formula from geometric measure theory
\mathbb {R} ^{m}} , is the (possibly infinite) number of points in the preimage f − 1 ( y ) ∩ A {\displaystyle f^{-1}(y)\cap A} . The multiplicity function
Area formula (geometric measure theory)
Area_formula_(geometric_measure_theory)
Differential map between manifolds whose differential is everywhere surjective
{\displaystyle q\in N} is a regular value of f if all points p in the preimage f − 1 ( q ) {\displaystyle f^{-1}(q)} are regular points. A differentiable
Submersion_(mathematics)
In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion
Myhill_isomorphism_theorem
Formal language that can be expressed using a regular expression
logic (Büchi–Elgot–Trakhtenbrot theorem) it is recognized by some finite syntactic monoid M, meaning it is the preimage {w ∈ Σ* | f(w) ∈ S} of a subset
Regular_language
Most general completion of a commutative square given two morphisms with same domain
and g is the disjoint union of X and Y, where elements sharing a common preimage (in Z) are identified, together with the morphisms i1, i2 from X and Y
Pushout_(category_theory)
Study of space and shapes locally given by a convergent power series
analytic functions. A fundamental result in the theory is the Riemann mapping theorem. The following are some of the most important topics in geometric function
Geometric_function_theory
Most general completion of a commutative square given two morphisms with same codomain
pullback of f and g (in Set) is given by the preimage f−1[B0] together with the inclusion of the preimage in A f−1[B0] ↪ A and the restriction of f to
Pullback_(category_theory)
bimeasurable" means that, first, f {\displaystyle f} is measurable (that is, the preimage f − 1 ( B ) {\displaystyle f^{-1}(B)} is measurable for every measurable
Schröder–Bernstein theorem for measurable spaces
Schröder–Bernstein_theorem_for_measurable_spaces
Mathematical formula of two surfaces
n>1} . Equivalently, the point P {\displaystyle P} has exactly one nearby preimage π − 1 ( P ) ∩ U ′ = { P ′ } {\displaystyle \pi ^{-1}(P)\cap U'=\{P'\}}
Riemann–Hurwitz_formula
Subspace preserved by a linear mapping
subspace V ⊂ A/M is an invariant under {Φ'(a) | a ∈ A} if and only if its preimage under the quotient map, V + M, is a left ideal in A. The invariant subspace
Invariant_subspace
Mathematical function with no sudden changes
topologies used on X and Y. This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in
Continuous_function
Analysis of datasets using techniques from topology
) {\textstyle M(\mathbb {U} ,f):=N(f^{-1}(\mathbb {U} ))} , where each preimage is split into its connected components. This is a very general concept
Topological_data_analysis
Integer
specified inside the function f, its inverse will yield an inverse image, or preimage, of that subset under the function. Exponentiation to negative integers
−1
Type of module over a ring
maximal right ideal: If M is a non-zero proper submodule of R/I, then the preimage of M under the quotient map R → R/I is a right ideal which is not equal
Simple_module
On the homotopy groups of the infinite symmetric product of a connected CW complex
In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same
Dold–Thom_theorem
Continuous surjection satisfying a local triviality condition
trivialization of the bundle. Thus for any p ∈ B {\displaystyle p\in B} , the preimage π − 1 ( { p } ) {\displaystyle \pi ^{-1}(\{p\})} is homeomorphic to F {\displaystyle
Fiber_bundle
Four-dimensional number system
versors is a point group, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same
Quaternion
Mathematical set of all subsets of a set
functor is different from the covariant version in that it sends f to the preimage morphism, so that if f (A) = B ⊆ T, Pf (B) = A. This is because a general
Power_set
Concept in topology
about Polish groups that Baire-measurable mappings (i.e., for which the preimage of any open set has the property of Baire) that are homomorphisms between
Polish_space
Type of function in complex analysis
{\displaystyle f:\;M\mapsto {\mathbb {R} }} is called exhaustive if the preimage f − 1 ( ( − ∞ , c ] ) {\displaystyle f^{-1}((-\infty ,c])} is compact for
Plurisubharmonic_function
Type of mathematical functions
complex manifolds with boundary called Stein domain. A Stein domain is the preimage { z ∣ − ∞ ≤ ψ ( z ) ≤ c } {\displaystyle \{z\mid -\infty \leq \psi (z)\leq
Function of several complex variables
Function_of_several_complex_variables
Measurable set whose measure is zero
preimage of a Borel set by a continuous function is measurable; g ( F ) = ( g − 1 ) − 1 ( F ) {\displaystyle g(F)=(g^{-1})^{-1}(F)} is the preimage of
Null_set
Expression that may be integrated over a region
allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general
Differential_form
Term in mathematics
complex manifolds with boundary called Stein domains. A Stein domain is the preimage { z ∣ − ∞ ≤ ψ ( z ) ≤ c } {\displaystyle \{z\mid -\infty \leq \psi (z)\leq
Stein_manifold
preimage: f − 1 ( y ) := f − 1 ( { y } ) = { x ∈ X : f ( x ) = y } . {\displaystyle f^{-1}(y):=f^{-1}(\{y\})=\{x\in X~:~f(x)=y\}.} Any preimage of
Saturated_set
Finite extension of the rationals
maps (that is, maps f : X → Y {\displaystyle f:X\to Y} such that the preimages of all points y in Y consist only of finitely many points): the cardinality
Algebraic_number_field
Structure in group theory (in mathematics)
\cap \operatorname {dom} \beta ]\alpha ^{-1}\,} where α−1 denotes the preimage under α. Partial transformations had already been studied in the context
Inverse_semigroup
Nonabelian group of order 120
group I or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism Spin ( 3
Binary_icosahedral_group
Property of functions which is weaker than continuity
{\displaystyle F} defines a set F ( x ) ⊂ B . {\displaystyle F(x)\subset B.} The preimage of a set S ⊂ B {\displaystyle S\subset B} under F {\displaystyle F} is
Semi-continuity
Mathematical concept for comparing objects
surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain. Thus an equivalence relation over X , {\displaystyle
Equivalence_relation
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
I)^{b-1}p_{b}} is an ordinary eigenvector corresponding to λ. In general, pi is a preimage of pi−1 under A − λ I {\displaystyle A-\lambda I} . So the lead vector
Jordan_normal_form
structure-preserving function between measurable spaces in the sense that the preimage of any measurable set is measurable. measurable set A measurable set is
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Maximal proper filter
The property of being ultra is preserved under bijections. However, the preimage of an ultrafilter is not necessarily ultra, not even if the map is surjective
Ultrafilter_on_a_set
Method of mathematical integration
intervals in the domain of f, which, taken together, is defined to be the preimage of the lower bound of that layer, under the simple function. In this way
Lebesgue_integral
Principle in geometry
forms a conic. Note that X and Y are on this conic by considering the preimage and image of the line XY (which is respectively a line through X and a
Five_points_determine_a_conic
f: X → Y and a closed subscheme Y′ ⊆ Y defined by an ideal sheaf J, the preimage Y′ ×Y X is defined by the ideal sheaf f*(J)OX = im(f*J → OX). The pull-back
Ideal_sheaf
Construction in functional analysis, useful to solve differential equations
complex number λ is in the essential range of h if for all ε > 0, the preimage of the open ball Bε(λ) under h has strictly positive measure. We will show
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
about the preimage c from the image f(x). For instance, while RSA is conjectured to be a one-way function, the Jacobi symbol of the preimage can be easily
Hard-core_predicate
Graph drawing used to study Riemann surfaces
as a term for polygon); he used a white circle for the preimage of 0 and a '+' for the preimage of 1, rather than a black circle for 0 and white circle
Dessin_d'enfant
If there are more items than boxes holding them, one box must contain at least two items
exists an element b of B such that there exists a bijection between the preimage of b and A. This is a quite different statement, and is absurd for large
Pigeonhole_principle
Functions that send open (resp. closed) subsets to open (resp. closed) subsets
only if the preimage of every open set of Y {\displaystyle Y} is open in X . {\displaystyle X.} (Equivalently, if and only if the preimage of every closed
Open_and_closed_maps
Economic Model
McKenzie, however, did not receive the award. The contents of both theorems [fundamental theorems of welfare economics] are old beliefs in economics. Arrow and
Arrow–Debreu_model
Topological space which is a generalization of certain compact spaces
S}f_{U}\upharpoonright N\,} , which is a continuous function; hence the preimage under f {\displaystyle f\,} of a neighbourhood of f ( x ) {\displaystyle
Paracompact_space
Concept in model theory
onto M such that the f {\displaystyle f} -preimage (more precisely the f k {\displaystyle f^{k}} -preimage) of every set X ⊆ Mk definable in M by a first-order
Interpretation_(model_theory)
Group-theoretic concept
^{-1}\left(H''\right)\right)=H'',} because every h ∈ H {\displaystyle h\in H} has a preimage g ∈ G ′ {\displaystyle g\in G'} with φ ( g ) = h . {\displaystyle \varphi
Subquotient
Subgroup of the Clifford algebra associated to a quadratic space
has a ±1 ambiguity. The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker (Spin(V) → SO(V)), and the two pin
Pin_group
Ideal of a ring contained in no other ideal except the ring itself
field, the preimage of a maximal ideal of a finitely generated k-algebra under a k-algebra homomorphism is a maximal ideal. However, the preimage of a maximal
Maximal_ideal
Generalization of the Legendre transformation
X , A x = y } {\displaystyle (Af)(y)=\inf\{f(x):x\in X,Ax=y\}} is the preimage of f {\displaystyle f} with respect to A {\displaystyle A} and A ∗ {\displaystyle
Convex_conjugate
Part of mathematics that addresses the stability of solutions
span ker A {\displaystyle \ker A} , and let w {\displaystyle w} be a preimage of v {\displaystyle v} , then in { v , w } {\displaystyle \{v,w\}} basis
Stability_theory
Mathematical theory
covering. In particular, a) Each point has the same (finite) number of preimages, counting multiplicity. This number is the degree of the covering. b)
Ahlfors_theory
Natural basic set in product spaces
{\displaystyle Y} component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form
Cylinder_set
A to B. If eB is the neutral element of B, then the kernel of f is the preimage of the singleton set {eB}; that is, the subset of A consisting of all those
Malcev_algebra
Mathematical function that outputs real values
real numbers. If X has its σ-algebra and a function f is such that the preimage f −1(B) of any Borel set B belongs to that σ-algebra, then f is said to
Real-valued_function
Group in which the order of every element is a power of p
a p-group, then so is G/Z, and so it too has a non-trivial center. The preimage in G of the center of G/Z is called the second center and these groups
P-group
Collection of open sets used to define a topology
open set of Y {\displaystyle Y} , it is an open map. Similarly, if every preimage of a basic open set of Y {\displaystyle Y} is open in X {\displaystyle
Base_(topology)
Matrix factorisation in mathematics
before, T would have an eigenspace, say Wμ ⊂ Cn modulo Vλ. Notice the preimage of Wμ under the quotient map is an invariant subspace of A that contains
Schur_decomposition
(Mathematical) ring with a unique maximal ideal
y]/(x^{3},x^{2}y,y^{4})} sending x ↦ x {\displaystyle x\mapsto x} . The preimage of ( x , y ) {\displaystyle (x,y)} is ( x ) {\displaystyle (x)} . Another
Local_ring
Set with algorithmic membership test
if A and the complement of A are both computably enumerable(c.e.). The preimage of a computable set under a total computable function is computable. The
Computable_set
Ideal in a ring which has properties similar to prime elements
x3, x4, ...}, of all positive powers of a non-nilpotent element. The preimage of a prime ideal under a ring homomorphism is a prime ideal. The analogous
Prime_ideal
{\displaystyle [y]} whose preimage under E is exactly this collection. In particular, there will be an isomorphism type [v] whose preimage under E is the collection
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Subfield of mathematical logic
Y are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This
Descriptive_set_theory
Inputs at which function values are highest
minima Mode (statistics) Mathematical optimization Kernel (linear algebra) Preimage Softmax function For clarity, we refer to the input (x) as points and the
Arg_max
Tool to track locally defined data attached to the open sets of a topological space
{\displaystyle V} is an open subset of Y {\displaystyle Y} , so that its preimage is open in X {\displaystyle X} by the continuity of f {\displaystyle f}
Sheaf_(mathematics)
Type of group in mathematics
minus signs. The Weyl group of SO(2n) is represented in SO(2n) by the preimages under the standard injection SO(2n) → SO(2n + 1) of the representatives
Orthogonal_group
Algebraic structure of set algebra
function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces
Σ-algebra
{\textstyle \cup _{k\in \mathbb {Z} }T^{k}A} , so the preimages of A {\textstyle A} are disjoint from the preimages of E ∩ I {\textstyle E\cap I} . Since A ∈ S
Rokhlin_lemma
Mathematical function such that every output has at least one input
= g o f. To see this, define Y to be the set of preimages h−1(z) where z is in h(X). These preimages are disjoint and partition X. Then f carries each
Surjective_function
Map from algebra to geometric transforms
{\displaystyle \rho } is faithful, then H {\displaystyle H} is isomorphic to the preimage in G L ( V ) {\displaystyle \mathrm {GL} (V)} of ρ ( G ) ⊆ P G L ( V )
Projective_representation
Concept in algebra
{\displaystyle R} . Equivalently, I {\displaystyle {\sqrt {I}}} is the preimage of the ideal of nilpotent elements (the nilradical of the ring) of the
Radical_of_an_ideal
Linear map over a ring
replaced with f ( x r ) = f ( x ) r . {\displaystyle f(xr)=f(x)r.} The preimage of the zero element under f is called the kernel of f. The set of all module
Module_homomorphism
Subgroup invariant under conjugation
{\displaystyle H.} Also, the preimage of any subgroup of H {\displaystyle H} is a subgroup of G . {\displaystyle G.} We call the preimage of the trivial group
Normal_subgroup
{n(n+1)}{2}}} . The Chinese monoid equivalence class of a permutation is the preimage of an involution under the map w ↦ w ∘ w − 1 {\displaystyle w\mapsto w\circ
Chinese_monoid
Graph theory concept
deleted edge or vertex in H, delete its preimage in C, and for each contracted edge or vertex in H, contract its preimage in C. The result of applying these
Planar_cover
Short exact sequence of sheaves on projective space
open set. Pulled-back in V, this is equivalent to a derivation on the preimage of U that preserves 0-homogeneous functions. Any vector field on P V {\displaystyle
Euler_sequence
Description of how spaces intersect in mathematics
manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space
Transversality
Special type of principal bundle
{\displaystyle E\times \operatorname {SU} (2)\rightarrow E} , which preserves all preimages of points, hence p ( e g ) = p ( e ) {\displaystyle p(eg)=p(e)} for all
Principal_SU(2)-bundle
In mathematics, a mapping between categories
to sheaves on topoi, such as étale sheaves. There, instead of the above preimage f−1(U), one uses the fiber product of U and X over Y. Forming sheaf categories
Direct_image_functor
2019 Mathematics book
function, each point on the unit circle has three preimages, also on the unit circle. These triples of preimages form triangles inscribed in the unit circle
Finding_Ellipses
Information theory
assigning each Borel set the P {\displaystyle {\mathfrak {P}}} -measure of its preimage in F {\displaystyle {\mathcal {F}}} . This is called the pushforward measure
Conditional mutual information
Conditional_mutual_information
Mathematical logic concept
with the Cantor pairing function) are computably enumerable sets. The preimage of a computably enumerable set under a partial computable function is a
Computably_enumerable_set
PREIMAGE THEOREM
PREIMAGE THEOREM
Boy/Male
Indian
Love
Surname or Lastname
English
English : from a Middle English personal name or nickname. The personal name existed in Old English, and is probably derived from Old English prim ‘early morning’ (from Latin primus ‘first’, used as the name of one of the canonical hours). The surname may be derived from this word as a Middle English nickname in the sense ‘fine’, ‘excellent’.French : feminine form of Prim 3.Dutch : variant of Priem.Probably an Americanized spelling of German Preim, a topographic name (of Slavic origin), perhaps from a river near Hannover; or of Preime, a variant of Primus.
Surname or Lastname
English
English : probably a variant of Bromage (see Brumage).
Surname or Lastname
English
English : variant of Prestwich, reflecting the old local pronunciation of the place name.
PREIMAGE THEOREM
PREIMAGE THEOREM
Boy/Male
Hindu
Affectionate
Girl/Female
Tamil
Prapya | பà¯à®°à®¾à®ªà¯à®¯
Achieving
Boy/Male
Celebrity, Hindu, Indian, Tamil
King of the Serpents
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Rapid; A River
Boy/Male
Celtic Gaelic Irish
Handsome.
Boy/Male
Arabic, Muslim
The Place Where Earth and Sky Meet
Boy/Male
Indian, Sanskrit
Inflamed
Girl/Female
American, Australian, British, Christian, English, German, Indian, Latin
Act of Kindness; Charity; Brotherly Love; Affection
Boy/Male
Tamil
Lord Shiva
Boy/Male
Indian
King, Ruler, Emperor, Royal
PREIMAGE THEOREM
PREIMAGE THEOREM
PREIMAGE THEOREM
PREIMAGE THEOREM
PREIMAGE THEOREM
n.
A right belonging to the crown of England, of taking two tuns of wine from every ship importing twenty tuns or more, -- one before and one behind the mast. By charter of Edward I. butlerage was substituted for this.
v. i.
To fortell; to presage; to augur.
n.
Presage of coming ill; expectation of misfortune.
v. t.
To have a presentiment of; to feel beforehand; to foreknow.
n.
To prophesy; to presage.
n.
The chief primate.
imp. & p. p.
of Presage
v. t.
To intimate or signify beforehand; to presage.
v. t.
To foretell; to predict; to foreshow; to indicate.
p. pr. & vb. n.
of Presage
a.
Of or pertaining to a primate.
v. t.
To foretell; to predict; to presage.
a.
One of the Primates.
n.
A charge in addition to the freight; originally, a gratuity to the captain for his particular care of the goods (sometimes called hat money), but now belonging to the owners or freighters of the vessel, unless by special agreement the whole or part is assigned to the captain.
v. i.
To form or utter a prediction; -- sometimes used with of.
n.
Prognostication; presage.
a.
The chief ecclesiastic in a national church; one who presides over other bishops in a province; an archbishop.
n.
To presage; to tell the fortune of.
n.
The share of merchandise taken as lawful prize at sea which belongs to the king or admiral.
v. t. & i.
To presage; to foreshow; to foretoken.