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Function that is its own inverse
In mathematics, an involution, involutory function, or self-inverse function is a function f that is its own inverse, f(f(x)) = x for all x in the domain
Involution_(mathematics)
Topics referred to by the same term
up involution in Wiktionary, the free dictionary. Involution may refer to: Involution (mathematics), a function that is its own inverse Involution algebra
Involution
General concept and operation in mathematics
structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. In other cases
Duality_(mathematics)
Property of operations
generalization of idempotence to binary relations Idempotent (ring theory) Involution (mathematics) Iterated function List of matrices Nilpotent Pure function Referential
Idempotence
types: a de Jonquières involution, a Geiser involution, or a Bertini involution. The normalized fixed curve of a Geiser involution is a non-hyperelliptic
Cremona_group
Mathematical structure in abstract algebra
may happen that an algebra admits no involution. Look up * or star in Wiktionary, the free dictionary. In mathematics, a *-ring is a ring A with a map * :
*-algebra
Natural number
The Book of Involutions. American Mathematical Society Colloquium Publications. Vol. 44. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-0904-4
2
Semigroup in abstract algebra
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism
Semigroup_with_involution
Number that, when added to the original number, yields the additive identity
|x|). Inverse element Inverse function Involution (mathematics) Monoid Multiplicative inverse Reflection (mathematics) Reflection symmetry Semigroup Gallian
Additive_inverse
Number of ways to pair up n objects
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person
Telephone number (mathematics)
Telephone_number_(mathematics)
In mathematics, a Fricke involution is the involution of the modular curve X0(N) given by τ → –1/Nτ. It is named after Robert Fricke. The Fricke involution
Fricke_involution
Length in a vector space
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance
Norm_(mathematics)
Category equipped with involution
category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with
Dagger_category
Topics referred to by the same term
inverse Involution (mathematics), a function that is its own inverse (when applied twice, the starting value is obtained) Inversion (discrete mathematics),
Inversion
Family of linear transformations
matrix. These are both symmetric, they are their own inverses (see Involution (mathematics)), and each have determinant −1. This latter property makes them
Lorentz_transformation
Mathematical finite group theory
In mathematical finite group theory, the classical involution theorem of Aschbacher (1977a, 1977b, 1980) classifies simple groups with a classical involution
Classical_involution_theorem
Generalized matrix decomposition for Lie groups and Lie algebras
semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent. A Cartan involution on s l n ( R ) {\displaystyle {\mathfrak
Cartan_decomposition
Element mapped to itself by a mathematical function
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation
Fixed_point_(mathematics)
Part of the theory of modular forms
identity; for this reason, the resulting operator is called an Atkin–Lehner involution. If e and f are both Hall divisors of N, then We and Wf commute modulo
Atkin–Lehner_theory
Mapping from a Euclidean space to itself
axis (a horizontal reflection) would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original
Reflection_(mathematics)
Chinese term for social competition
inwards' IPA: [nei̯˥˩tɕɥɛn˩˧]) is the Chinese calque of the English word involution. Neijuan is written with two characters which mean 'inside' and 'rolling'
Neijuan
Method for producing composition algebras
Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension. Hurwitz's theorem states that
Cayley–Dickson_construction
Square matrix which is its own inverse
by the matrix A n × n {\displaystyle {\mathbf {A}}_{n\times n}} is an involution if and only if A 2 = I , {\displaystyle {\mathbf {A}}^{2}={\mathbf {I}}
Involutory_matrix
Condition for a mathematical function to map some value to itself
first place. Every involution on a finite set with an odd number of elements has a fixed point; more generally, for every involution on a finite set of
Fixed-point_theorem
Theorem classifying finite simple groups
group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component (where O(C) is the core of C, the maximal normal
Classification of finite simple groups
Classification_of_finite_simple_groups
Natural number
26 is the number of letters in the Latin alphabet. "Sloane's A000085 : Involution numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
26_(number)
British-American mathematician (1946–2024)
Walter D. Neumann, Department of Mathematics, Columbia University. Accessed October 2, 2024 Walter Neumann at the Mathematics Genealogy Project Home page at
Walter_Neumann
Homomorphism reversing the order of something
Semigroup with involution Jacobson, Nathan (1943). The Theory of Rings. Mathematical Surveys and Monographs. Vol. 2. American Mathematical Society. p. 16
Antihomomorphism
System of logic lacking the excluded middle law
distributive lattice, and ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies De Morgan's laws)
De_Morgan_algebra
Family of groups in group theory
2E6(L) (thinking of the group as a subgroup of E6(L) fixed by an outer involution). Over finite fields these groups form one of the 18 infinite families
2E6_(mathematics)
Natural number
form and the seventh of the form (22.q). a Lucas number. a telephone or involution number, the number of different ways of connecting 6 points with pairwise
76_(number)
Distance from zero to a number
In mathematics, the absolute value or modulus of a real number x {\displaystyle x} , denoted | x | {\displaystyle |x|} , is the (non-negative) magnitude
Absolute_value
Range of related ideas and movements that have developed in the Western world
education Philosophy of information Philosophy of language Philosophy of mathematics Philosophy of religion Philosophy of science Political philosophy Practical
Western_esotericism
Algebraic structure used in theoretical physics
In mathematics and theoretical physics, a superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. That is, it is an algebra over a commutative
Superalgebra
Belgian mathematician
study of involution algebras. In 1996, he was invited by the European Congress of Mathematics in Budapest to speak on "Algebras with involution and classical
Jean-Pierre_Tignol
Topological complex vector space
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the
C*-algebra
Manx mathematician (1944–2021)
UCD, he completed his PhD thesis, Hermitian Forms over Algebras with Involution, under the supervision of Professor Wall and was awarded a doctorate by
David W. Lewis (mathematician)
David_W._Lewis_(mathematician)
Algebra where division is always defined
group but respectively a commutative monoid and a commutative monoid with involution. A wheel is an algebraic structure ( W , 0 , 1 , + , ⋅ , / ) {\displaystyle
Wheel_theory
says that after a finite number of prolongations the system is either in involution (admits at least one 'large' integral manifold), or is impossible. The
Cartan–Kuranishi prolongation theorem
Cartan–Kuranishi_prolongation_theorem
Theorem in group theory
The theorem states that if C {\displaystyle C} is the centralizer of an involution of a finite group, then every component of C / O ( C ) {\displaystyle
B-theorem
Monster and modular connection
the –1 involution of the Leech lattice, there is an involution h of VL, and an irreducible h-twisted VL-module, which inherits an involution lifting
Monstrous_moonshine
Algebraic structure with a ternary operation
considered an involuted semigroup with operation given by ab = [a, e, b] and involution by a–1 = [e, a, e]. When the above construction is applied to a heap,
Heap_(mathematics)
Mathematics concept
In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by Atiyah (1966), motivated by
KR-theory
German mathematician (1832–1903)
condition) and differential geometry, as well as number theory, algebras with involution and classical mechanics. Rudolf Lipschitz was born on 14 May 1832 in Königsberg
Rudolf_Lipschitz
Topics referred to by the same term
the classical involution theorem The infinite Thompson groups F, T and V studied by the logician Richard Thompson. Outside of mathematics, it may also
Thompson_group
True when either but not both inputs are true
The function is linear. Involution: Exclusive or with one specified input, as a function of the other input, is an involution or self-inverse function;
Exclusive_or
Composition and Triality". The book of involutions. Vol. v44. Providence (R. I.): Colloquium Publications, American mathematical society. pp. 451–511. ISBN 0-8218-0904-0
Okubo_algebra
Straight line that only contains one real point
of the double points (imaginary) of the overlapping involutions in which an overlapping involution pencil (real) is cut by real transversals is a pair
Imaginary_line_(mathematics)
(pseudo-)Riemannian manifold whose geodesics are reversible
subgroup H that is (a connected component of) the invariant group of an involution of G. This definition includes more than the Riemannian definition, and
Symmetric_space
Group theoretic operation
In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarisation
Rosati_involution
generates contains a unique involution x. Aschbacher, Michael (2000), Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10 (2nd ed.), Cambridge
Thompson_order_formula
Mathematical formula
μ by adding r elements, no two in the same column. By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule for
Pieri's_formula
Low-rank isomorphisms in mathematics
(1998). The Book of Involutions. American Mathematical Society Colloquium Publications. Vol. 44. Providence, RI: American Mathematical Society. ISBN 978-0-8218-0904-4
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
morphism R : X → Y {\displaystyle R\colon X\to Y} is associated with an anti-involution, i.e. a morphism R ∘ : Y → X {\displaystyle R^{\circ }\colon Y\to X} with
Allegory_(mathematics)
Theorem about finite groups
count involutions (elements of order 2) in G. Perhaps more important is another result that the authors derive from the same count of involutions, namely
Brauer–Fowler_theorem
Mathematical method in functional analysis
automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a
Tomita–Takesaki_theory
Algebraic surface with special triviality properties
quotient by the involution taking (u:v:w:x:y:z) to (–x:–y:–z:u:v:w). For generic quadrics this involution is a fixed-point-free involution of a K3 surface
Enriques_surface
structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Alexander duality
List_of_dualities
American composer
the 2018 Magic Rectangle, Infinity, Four-Momentum, Euler's Identity; Involution (2020); and the full length Bohlen-Pierce scale mini-movie album Macrochip
Elaine_Walker_(composer)
Reversal of the order of elements of a binary relation
relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more
Converse_relation
Fundamental operation on complex numbers
{\displaystyle \left|{\overline {z}}\right|=|z|.} Conjugation is an involution, that is, the conjugate of the conjugate of a complex number z {\displaystyle
Complex_conjugate
In the mathematical field of combinatorics, jeu de taquin is a construction due to Marcel-Paul Schützenberger (1977) which defines an equivalence relation
Jeu_de_taquin
In mathematics, the argument shift method is a method for constructing functions in involution with respect to Poisson–Lie brackets, introduced by Mishchenko
Argument_shift_method
Swiss mathematician born 1942
write The Book of Involutions published by the American Mathematical Society. This book is about "central simple algebras with involution, in relation to
Max-Albert_Knus
Product of a number by itself
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same
Square_(algebra)
Relationship between elements of two sets
In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the
Binary_relation
Arithmetic operation
+ cx3 + d. Samuel Jeake introduced the term indices in 1696. The term involution was used synonymously with the term indices, but had declined in usage
Exponentiation
Type of residuated Boolean algebra with extra structure
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation
Relation_algebra
Geometric symmetry operation
preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation. An object that is invariant
Point_reflection
Group that admits a formal description in terms of reflections
for all i {\displaystyle i} ; as such the generators are involutions. If m i j = 2 {\displaystyle m_{ij}=2} , then the generators r i {\displaystyle
Coxeter_group
Type of physical or mathematical property
one-to-one, so that for every state there exists a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution
Time_reversibility
Algebraic structure
we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting
Semigroup
Sporadic simple group
the Monster group is S3 × Th, so Th centralizes 3 involutions alongside the 3-cycle. These involutions are centralized by the Baby monster group, which
Thompson_sporadic_group
Quasisimple subnormal subgroup of a finite group
centralizer has even order, it is normal in the centralizer of every involution centralizing it, and it commutes with none of its conjugates. This concept
Component_(group_theory)
Number which when multiplied by x equals 1
one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and
Multiplicative_inverse
Gamma matrices for arbitrary Clifford algebras
correspond to those actions on matrices), and in physics, where the "main involution" α {\displaystyle \alpha } corresponds to a combined P-symmetry and T-symmetry
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Type of group in mathematics
of Mathematics. Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The Book of Involutions. American Mathematical Society
Classical_group
Algebraic structure modeling logical operations
also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa,
Boolean_algebra_(structure)
Topics referred to by the same term
notion of adjoints C*-algebra, a Banach algebra equipped with a unary involution operation Von Neumann algebra (or W*-algebra) Coalgebra is the dual notion
Algebra_(disambiguation)
1955 book by Meher Baba
of the atma (soul) through its imagined evolution, reincarnation, and involution, to its goal, its origin, of Paramatma (Over-soul). The journey winds
God_Speaks
German mathematician
finite simple group having a centralizer of an involution isomorphic to that of the centralizer of an involution in the center of a Sylow 2-subgroup of the
Dieter_Held
Term in mathematics
In the mathematical study of Lie algebras and Lie groups, Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems
Satake_diagram
representation can be decomposed using the characters of SO(E) and the involution coming from the nontrivial coset of O(E). In this decomposition, one of
Θ10
Four finite groups derived from the Leech lattice
other than 2. Any involution in Co0 can be shown to be conjugate to an element of the Golay code. Co0 has 4 conjugacy classes of involutions. A permutation
Conway_group
Sporadic simple group
Ivanov, A.A. (2009). The Monster group and Majorana involutions. Cambridge tracts in mathematics. Vol. 176. Cambridge University Press. doi:10.1017/CBO9780511576812
Monster_group
Russian mathematician (1937–2010)
published "On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral
Vladimir_Arnold
Sporadic simple group
outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group. HS is one of the 26 sporadic groups
Higman–Sims_group
American mathematician (born 1943)
received her B.A. in 1965 from the University of Michigan and her Ph.D. in Mathematics from the University of Chicago in 1969 under the supervision of I. N
Susan_Montgomery
suitable boundary conditions to the Chern-Simons action. In this scheme, the involution of conserved charges of the AKNS system yields an infinite-dimensional
AKNS_system
group of characteristic 2 type, where involutions resemble unipotent elements, and other groups, where involutions resemble semisimple elements. Groups
Characteristic_2_type
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
In mathematics and theoretical physics, a Killing vector field or Killing field (named after Wilhelm Killing) is a vector field on a Riemannian manifold
Killing_vector_field
Classification of finite simple groups
various other assumptions are satisfied, then G has a centralizer of an involution with a "standard component" with small centralizer. Aschbacher, Michael
Component_theorem
Type of permutation
of modules. Guibert, Pergola & Pinzani (2001) showed that vexillary involutions are enumerated by Motzkin numbers. Riffle shuffle permutation, a subclass
Vexillary_permutation
Anticommutating number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior
Grassmann_number
American computer scientist and mathematician (born 1938)
notation Knuth–Morris–Pratt algorithm Davis–Knuth dragon Bender–Knuth involution TPK algorithm Fisher–Yates shuffle Robinson–Schensted–Knuth correspondence
Donald_Knuth
Bound lattice in which every element has a complement
complemented lattice. An orthocomplementation on a complemented lattice is an involution that is order-reversing and maps each element to a complement. An orthocomplemented
Complemented_lattice
Set of the elements not in a given subset
follows from the equivalence of a conditional with its contrapositive). Involution or double complement law: ( A c ) c = A . {\displaystyle \left(A^{c}\right)^{c}=A
Complement_(set_theory)
Integral transform closely related to the Fourier transform
Hartley transform has the convenient property of being its own inverse (an involution): f = { H { H f } } . {\displaystyle f=\{{\mathcal {H}}\{{\mathcal {H}}f\}\}\
Hartley_transform
Four-dimensional number system
In mathematics, the quaternions form a number system similar to the complex numbers, with the usual arithmetical operations of addition, subtraction,
Quaternion
Natural number
Mathematics portal 1,000,000,000,000 (one trillion on the short scale; one billion on the long scale; one thousand billion; one million million) is the
1,000,000,000,000
INVOLUTION MATHEMATICS
INVOLUTION MATHEMATICS
Boy/Male
Hindu
Revolution
Boy/Male
Hindu, Indian, Telugu
Revolution
Boy/Male
Hindu
Floating, Revolution
Girl/Female
Hindu, Indian, Traditional
Inspired Invocation
Boy/Male
Indian, Sanskrit
Invocation
Male
Chinese
revolution.
Girl/Female
Indian, Sanskrit
Invocation
Girl/Female
Biblical
Passage, revolution.
Boy/Male
Hindu
Light, Revolution
Boy/Male
Gujarati, Hindu, Indian, Kannada, Marathi
Floating; Revolution
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit
Revolution
Boy/Male
Tamil
Kranthi | கà¯à®°à®¾à®‚தி
Light, Revolution
Kranthi | கà¯à®°à®¾à®‚தி
Girl/Female
Biblical
Wheel, revolution.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu, Traditional
Revolution
Boy/Male
Tamil
Revolution
Boy/Male
Indian, Tamil
Revolution
Boy/Male
Tamil
Floating, Revolution
Boy/Male
Tamil
Floating, Revolution
Boy/Male
Bengali, Indian
Revolution
Boy/Male
Hindu, Indian, Punjabi, Sikh, Telugu
Revolution
INVOLUTION MATHEMATICS
INVOLUTION MATHEMATICS
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Perpetrator of the Kuru Race
Boy/Male
Indian, Sanskrit, Sindhi
As Hard as Diamond
Male
Greek
(ΣτÎφανος) Greek name derived from the word stephanos, STEPHANOS means "crown." In the bible, this is the name of one of the seven deacons of the church at Jerusalem who was stoned to death by the Jews.Â
Girl/Female
Arabic, Muslim
Virtues; Plural of Shamila
Boy/Male
Australian, British, Czechoslovakian, Danish, English, Finnish, German, Norse, Polish, Scandinavian, Swedish
Divine Spear; Gentle Friend; Spear of the Gods
Boy/Male
British, English
Old Leader
Female
Egyptian
, a hippo goddess.
Boy/Male
Hindu, Indian
Virtuous
Girl/Female
Gujarati, Hindu, Indian
River Ganga
Girl/Female
Latin
Laughter.
INVOLUTION MATHEMATICS
INVOLUTION MATHEMATICS
INVOLUTION MATHEMATICS
INVOLUTION MATHEMATICS
INVOLUTION MATHEMATICS
n.
The extraction of roots; -- the reverse of involution.
n.
A total or radical change; as, a revolution in one's circumstances or way of living.
n.
The act or process of raising a quantity to any power assigned; the multiplication of a quantity into itself a given number of times; -- the reverse of evolution.
n.
Involution in one's self; hence, abstraction of thought; reverie.
n.
Act of involving, or state of being involved; involution.
n.
The state of being entangled or involved; complication; entanglement.
n.
Return to a point before occupied, or to a point relatively the same; a rolling back; return; as, revolution in an ellipse or spiral.
a.
Relating to evolution.
n.
The motion of any body, as a planet or satellite, in a curved line or orbit, until it returns to the same point again, or to a point relatively the same; -- designated as the annual, anomalistic, nodical, sidereal, or tropical revolution, according as the point of return or completion has a fixed relation to the year, the anomaly, the nodes, the stars, or the tropics; as, the revolution of the earth about the sun; the revolution of the moon about the earth.
n.
The act of revolving, or turning round on an axis or a center; the motion of a body round a fixed point or line; rotation; as, the revolution of a wheel, of a top, of the earth on its axis, etc.
n.
Partial or incomplete involution; as, subinvolution of the uterus.
n.
A call or summons; especially, a judicial call, demand, or order; as, the invocation of papers or evidence into court.
n.
That in which anything is involved, folded, or wrapped; envelope.
n.
The insertion of one or more clauses between the subject and the verb, in a way that involves or complicates the construction.
n.
The relation which exists between three or more sets of points, a.a', b.b', c.c', so related to a point O on the line, that the product Oa.Oa' = Ob.Ob' = Oc.Oc' is constant. Sets of lines or surfaces possessing corresponding properties may be in involution.
n.
The motion of a point, line, or surface about a point or line as its center or axis, in such a manner that a moving point generates a curve, a moving line a surface (called a surface of revolution), and a moving surface a solid (called a solid of revolution); as, the revolution of a right-angled triangle about one of its sides generates a cone; the revolution of a semicircle about the diameter generates a sphere.
n.
The return of an enlarged part or organ to its normal size, as of the uterus after pregnancy.
n.
The act of unfolding or unrolling; hence, in the process of growth; development; as, the evolution of a flower from a bud, or an animal from the egg.
n.
The act of involving or infolding.
n.
Evolution of one's self; development by inherent quality or power.