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See searches and references containing AFFINE INVOLUTION!AFFINE INVOLUTION
Linear or affine transformation which is its own inverse
In Euclidean geometry, an affine involution is an involution which is a linear or affine transformation over the Euclidean space R n {\displaystyle \mathbb
Affine_involution
Function that is its own inverse
a distinct example. These transformations are examples of affine involutions. An involution is a projectivity of period 2, that is, a projectivity that
Involution_(mathematics)
Geometric symmetry operation
but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where 1 ≤ k ≤ n − 1 {\displaystyle
Point_reflection
Invariance under a mathematical reflection
reflection symmetry. For example: with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc.) with respect to circle
Reflection_symmetry
Square matrix which is its own inverse
involutory. In fact, An will be equal to A if n is odd and I if n is even. Affine involution Higham, Nicholas J. (2008), "6.11 Involutory Matrices", Functions
Involutory_matrix
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
Number line and triangular tiling's symmetry mathematical structure
The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of
Affine_symmetric_group
rings is a representation of a group on coordinate rings of affine varieties. Let X be an affine algebraic variety over an algebraically closed field k of
Representation on coordinate rings
Representation_on_coordinate_rings
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)
(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
Deformation of the group algebra of a Coxeter group
objects. The generic multiparameter Hecke algebra, HA(W, S, q), has an involution bar that maps q1/2 to q−1/2 and acts as identity on Z. Then H admits a
Iwahori–Hecke_algebra
Generalized scaling operation in geometry
homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number k called
Homothety
Pictorial representation of symmetry
different ways, if quotienting by an involution) E 6 → F 4 {\displaystyle E_{6}\to F_{4}} Similar foldings exist for affine diagrams, including: A ~ 2 n − 1
Dynkin_diagram
Group theory
Kohl. RCWA – Residue-Class-Wise Affine Groups. GAP package. 2005. Stefan Kohl. A Simple Group Generated by Involutions Interchanging Residue Classes of
Residue-class-wise affine group
Residue-class-wise_affine_group
Mapping from a Euclidean space to itself
non-identity isometries that are involutions. The set of fixed points (the "mirror") of such an isometry is an affine subspace, but is possibly smaller
Reflection_(mathematics)
Geometric concept of a 2D space with "points at infinity" adjoined
is positive. The Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the previous example, to obtain the
Projective_plane
Integrable classical system
\Delta _{r}} 's and Δ ∞ {\displaystyle \Delta _{\infty }} are all in involution. It can be shown that the Δ r {\displaystyle \Delta _{r}} 's and Δ ∞ {\displaystyle
Garnier_integrable_system
Counterexample in algebraic geometry
points (0,0,0) and ( t , 0 , 0 ) {\displaystyle (t,0,0)} , and take the involution σ to be the one taking ( x , y , z ) {\displaystyle (x,y,z)} to ( x +
Hironaka's_example
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
parity under the Cartan involution, while h {\displaystyle {\mathfrak {h}}} has even parity. That is, denoting the Cartan involution at point p ∈ M {\displaystyle
Killing_vector_field
Isometry group of Euclidean space
Euclidean group of symmetries, is, therefore, a specialisation of affine geometry. All affine theorems apply. The origin of Euclidean geometry allows definition
Euclidean_group
Algebraic stack in mathematics
proper morphism of M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} to the affine line, the coarse moduli space of elliptic curves, given by the j-invariant
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Non-perpendicular Euclidean reflection
serving as a reference. An oblique reflection is an affine transformation, and it is an involution, meaning that the reflection of the reflection of a
Oblique_reflection
Space formed by the ''n''-tuples of real numbers
topological vector space. It is a Euclidean space and a real affine space, and every Euclidean or affine space is isomorphic to it. It is an analytic manifold
Real_coordinate_space
geometry. An inversion of the Möbius plane with respect to any circle is an involution which fixes the points on the circle and exchanges the points in the interior
Möbius_plane
Mathematics concept
(z_{2})\,} . A real structure on a complex vector space V is an antilinear involution σ : V → V {\displaystyle \sigma :V\to V} . A real structure defines a
Real_structure
Geometrical property
in three dimensions for more), one of the three types of order two (involutions), hence algebraically isomorphic to C2. The fundamental domain is a half-plane
Symmetry_(geometry)
Lie algebra with imaginary simple roots
degree zero piece (the Cartan subalgebra) is abelian. They have a (Cartan) involution w. (a, w(a)) is positive if a is nonzero. For example, for the algebras
Generalized_Kac–Moody_algebra
Sporadic simple group
conjugacy classes of involutions; these collapse to 2 in Co1, but there are 4-elements in Co0 that correspond to a third class of involutions in Co1. An image
Conway_group_Co1
or Steiner surface, a realization of the real projective plane in real affine space Tori, surfaces of revolution generated by a circle about a coplanar
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
Commutative monoid in simplicial abelian groups
− 1 ) | x | | y | y x {\displaystyle xy=(-1)^{|x||y|}yx} ) since the involution S 1 ∧ S 1 → S 1 ∧ S 1 {\displaystyle S^{1}\wedge S^{1}\to S^{1}\wedge
Simplicial_commutative_ring
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
Property of operations
itself is idempotent; the convex hull function from the power set of an affine space over the reals to itself is idempotent; the closure and interior functions
Idempotence
Locus of the zeros of a polynomial of degree two
quadric or a reducible quadric. A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in
Quadric
writes a semisimple real Lie algebra as the sum of eigenspaces of a Cartan involution. The Iwasawa decomposition G = K A N {\displaystyle G=KAN} of a semisimple
Lie_group_decomposition
Sporadic simple group
a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months, the order of M was found by Griess using the Thompson
Monster_group
Yangian is a degeneration of the quantum loop algebra (i.e. the quantum affine algebra at vanishing central charge). For any finite-dimensional semisimple
Yangian
variety is the spectrum of prime ideals of the ring of invariants (i.e., the affine GIT quotient). C [ Hom ( π , G ) ] G . {\displaystyle \mathbb {C} [\operatorname
Character_variety
Type of group in mathematics
unitary groups attached to nondegenerate Hermitian forms relative to an involution. Over C {\displaystyle \mathbb {C} } , the connected simple classical
Classical_group
Aspect of mathematical group theory
is the semi-direct product G of PGL2(K) = PGL2(9) against the Galois involution. This map carries the simple group A6 nontrivially into (hence onto) the
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Algebraic structure used in theoretical physics
group SU(3)×SU(2)×U(1) as their zero level algebra. Infinite-dimensional (affine) Lie superalgebras are important symmetries in superstring theory. Specifically
Lie_superalgebra
Term in mathematics
Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems that are relevant in several contexts. They were introduced
Satake_diagram
Set of n^3 + 1 points arranged into subsets of n + 1
ΓS,T is cyclic of order q - 1, and thus contains a unique involution, μ. Each such involution fixes exactly q + 1 points of P. Construct a block design
Unital_(geometry)
Algebra describing 2D conformal symmetry
cluster model. For any c , h ∈ C {\displaystyle c,h\in \mathbb {C} } , the involution L n ↦ L ∗ = L − n {\displaystyle L_{n}\mapsto L^{*}=L_{-n}} defines an
Virasoro_algebra
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
Universal construction of a complex Lie group from a real Lie group
}} Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (gt)−1 A−1 of SL(2n,C). It leaves the subgroups N±, TC and B
Complexification_(Lie_group)
Algebra used in 2D conformal field theories and string theory
(modeling lattice conformal field theories), VOAs given by representations of affine Kac–Moody algebras (from the WZW model), the Virasoro VOAs, which are VOAs
Vertex_operator_algebra
Curve in the dual projective plane made from all lines tangent to a given curve
coordinates, is known as the tangential equation of C. Duality is an involution: the dual of the dual of C is the original curve C. The construction of
Dual_curve
Simple encryption method
restore the original text (in mathematics, this is sometimes called an involution; in cryptography, a reciprocal cipher). The transformation can be done
ROT13
231) affine 1. Affine space is roughly a vector space where one has forgotten which point is the origin. 2. An affine variety is a variety in affine space
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Distance-preserving mathematical transformation
isometry of normed vector spaces over R {\displaystyle \mathbb {R} } is affine. A linear isometry also necessarily preserves angles, therefore a linear
Isometry
Sporadic simple group
orbits of sizes 4 and 8; centralizer of a quadruple transposition (an involution of class 2B) 10 42:(2 x S3) 192 = 26·3 495 = 32·5·11 imprimitive on 3
Mathieu_group_M12
Transformations induced by a mathematical group
of the vector in Zn. The affine group acts transitively on the points of an affine space, and the subgroup V of the affine group (that is, a vector space)
Group_action
Integral polynomial
{\displaystyle D(T_{w})=T_{w^{-1}}^{-1}} ; also D can be seen to be an involution. The Kazhdan–Lusztig polynomials Pyw(q) are indexed by a pair of elements
Kazhdan–Lusztig_polynomial
Binary relation over a set and itself
set X is the set 2X×X, which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition
Homogeneous_relation
Real numbers with an added point at infinity
and ∞. The projectively extended real number line is distinct from the affinely extended real number line, in which +∞ and −∞ are distinct. Unlike most
Projectively extended real line
Projectively_extended_real_line
Generalized manifold
by H and an involution τ subject to (τg)3 = 1 for each g in S. In fact, if Γ acts in this way, fixing an edge (v, w), there is an involution τ interchanging
Orbifold
Most widely known generalized inverse of a matrix
the field of complex numbers equipped with the identity involution (as opposed to the involution considered elsewhere in the article); do there exist matrices
Moore–Penrose_inverse
_{0}(T)\leq 1\}.}} Élie Cartan showed that it is a fundamental domain for the affine Weyl group. If G1 = G / Z and T1 = T / Z, it follows that the exponential
Borel–de_Siebenthal_theory
Type of group in abstract algebra
Sn is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps Sn → Cp are to S2 and all involutions are conjugate, hence
Symmetric_group
Four-dimensional number system
by q∗, qt, q ~ {\displaystyle {\tilde {q}}} , or q. Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns
Quaternion
System of resource-aware logic
proposition A in CLL has a dual A⊥, defined as follows: Observe that (-)⊥ is an involution, i.e., A⊥⊥ = A for all propositions. A⊥ is also called the linear negation
Linear_logic
Logical operation
intuitionistic negation. Algebraically, classical negation is called an involution of period two. However, in intuitionistic logic, the weaker equivalence
Negation
Mathematical transformation
Leibniz's notation. The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called
Legendre_transformation
Algebraic structure
we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting
Semigroup
American mathematician
40 (1934) 309–315. MR 1562842 The condition for a pfaffian system in involution. Bull. Amer. Math. Soc. 40 (1934) 316–320. MR 1562843 Sturm's theorem
Joseph_Miller_Thomas
Matrix operation which flips a matrix over its diagonal
^{\text{T}}\right)^{\text{T}}=\mathbf {A} .} The operation of taking the transpose is an involution (self-inverse). ( A + B ) T = A T + B T . {\displaystyle \left(\mathbf
Transpose
Construction in algebra
Hopf algebra is said to be involutive (and the underlying algebra with involution is a *-algebra). If H is finite-dimensional semisimple over a field of
Hopf_algebra
Mathematical group
Wonenburger, Anna (1966). "Transformations which are products of two involutions". Journal of Mathematics and Mechanics. 16 (4): 327–338. Symplectic Group
Symplectic_group
Relationship between elements of two sets
{\displaystyle 2^{X\times X}} which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition
Binary_relation
Open convex self-dual cones
Riemannian symmetric space isomorphic to G / K where G = Aut0 C. The Cartan involution is defined by σ(g)=(g*)−1, so that K = G ∩ O(V). In their classic paper
Symmetric_cone
Hypercomplex number system
_{3}-x_{4}e_{4}-x_{5}e_{5}-x_{6}e_{6}-x_{7}e_{7}} . Conjugation is an involution of O {\displaystyle \ \mathbb {O} \ } and satisfies (xy)* = y*x* (note
Octonion
Geometry founded on spheres
the Möbius transforms that fix the ideal point at infinity, namely the affine conformal maps. These groups also have a direct physical interpretation:
Lie_sphere_geometry
Study of angle-preserving transformations
the identity transformation which makes it a self-inversion (i.e. an involution). To make the inversion a total function that is also defined for O, it
Inversive_geometry
Operation in group theory
fixed (i.e., an orthogonal matrix with determinant –1 representing an involution), then φ : C2 → Aut(SO(n)) is given by φ(H)(N) = HNH−1 for all H in C2
Semidirect_product
Element of a unital algebra over the field of real numbers
hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real
Hypercomplex_number
adjunction See conjunction introduction. affine logics A subfield of linear logic focusing on the study of affine transformations and their implications
Glossary_of_logic
Group of unitary matrices
{\displaystyle K} a ↦ a ¯ {\displaystyle a\mapsto {\bar {a}}} which is an involution and fixes exactly k {\displaystyle k} ( a = a ¯ {\displaystyle a={\bar
Unitary_group
Formula relating the norm and the inner product in an inner product space
which are a narrower notion. More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes ε {\displaystyle \varepsilon
Polarization_identity
Concept in projective geometry
{x} )=(G(\mathbf {x} ^{\sigma }))^{\mathsf {T}}.} A duality that is an involution (has order two) is called a polarity. It is necessary to distinguish between
Duality_(projective_geometry)
Algebraic structure designed for geometry
group, although Lundholm deprecates this usage). We denote the grade involution as S ^ {\displaystyle {\widehat {S}}} and reversion as S ~ {\displaystyle
Geometric_algebra
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
Contents: Top 0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Affine connection Alexandrov space a generalization of Riemannian manifolds with
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Infinite sequence of differential equations
where I n [ u ] {\displaystyle I_{n}[u]} are conserved quantities in involution. This Hamiltonian point of view is closely related to the Gelfand–Dikii
Korteweg–De_Vries_hierarchy
Mathematical object
conjugacy class. For G = GL(n), the transposition can serve as such an involution. In this case, there is the following criterion for the pair (G, K) to
Gelfand_pair
Circle-like pointset in a geometric plane
common line). A set of points in an affine plane satisfying the above definition is called an affine oval. An affine oval is always a projective oval in
Oval_(projective_plane)
Normed vector space that is complete
C*-algebra is a complex Banach algebra A {\displaystyle A} with an antilinear involution a ↦ a ∗ {\displaystyle a\mapsto a^{*}} such that ‖ a ∗ a ‖ = ‖ a ‖ 2
Banach_space
Branch of mathematics
posets with a unique bottom element 0, as well as an order-reversing involution ∗ {\displaystyle *} such that a ≤ a ∗ ⟹ a = 0. {\displaystyle a\leq a^{*}\implies
Order_theory
Discrete dynamical system on polygons in the projective plane and on their moduli space
space, because it can be decomposed as the composition of two birational involutions. The corner invariants change in the following way: x k ′ = x k 1 − x
Pentagram_map
Manifold with inversion symmetry
be simple and K of maximal rank. From Borel-de Siebenthal theory, the involution σ is inner and K is the centralizer of its center, which is isomorphic
Hermitian_symmetric_space
Exterior algebraic map taking tensors from p forms to n-p forms
all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the
Hodge_star_operator
Direct sum of simple Lie algebras
Killing form is not all negative). Suppose, moreover, it has a Cartan involution θ {\displaystyle \theta } and let g = k ⊕ p {\displaystyle {\mathfrak
Semisimple_Lie_algebra
AFFINE INVOLUTION
AFFINE INVOLUTION
Girl/Female
Irish French
Beautiful.
Girl/Female
Latin
Red haired.
Male
English
English name, probably derived from the vocabulary word alpine, ALPINE means "of the Swiss Alps."
Female
English
Pet form of English Saffron, SAFFIE means "saffron (the spice)."
Male
English
Pet form of English Alfred, ALFIE means "elf counsel."
Girl/Female
Armenian
Valuable.
Female
English
 Variant spelling of English Aileen, ALINE means "little Eve." Compare with another form of Aline.
Girl/Female
Italian
Famous bearer: Alcine is mistress of alluring enchantments and sensual pleasures in the Orlando...
Girl/Female
English Latin
Warm.
Girl/Female
German
Soldier. Army Man. from the Old German Hariman.
Female
French
 Contracted form of French Adeline, ALINE means "little noble." Compare with another form of Aline.
Female
Hebrew
Variant spelling of Hebrew Amina, AMINE means "faithful, trusted."
Girl/Female
French
May Jehovah add. Addition (to the family). A feminine form of Joseph.
Female
English
Variant spelling of English Aline, ALLINE means "little Eve."Â
Female
Scandinavian
Scandinavian form of Hebrew Adiyna, ADINE means "slender."
Girl/Female
Irish
In charge.
Girl/Female
Irish American Celtic English French
Oath.
Male
English
Middle English form of Anglo-Saxon Ealdwine, ALDINE means "old friend."
Female
English
English pet form of Latin Euphemia, EFFIE means "Well I speak."
Girl/Female
French
Blond.
AFFINE INVOLUTION
AFFINE INVOLUTION
Boy/Male
American, Anglo, British, Christian, English
Lives in the Valley; Place Name; Valley of the Awesome One
Girl/Female
Hindu
Surname or Lastname
English (Kent)
English (Kent) : habitational name from Maxted Street in Kent.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Marathi, Oriya, Sindhi, Telugu
Full of Light; Lustrous
Male
Russian
(Гога) Russian Georgi, GOGA means "earth-worker, farmer."
Boy/Male
Arabic, Muslim, Sindhi
One who Collects Booty
Boy/Male
Indian, Punjabi, Sikh
Full of Flower Fragrance
Girl/Female
Arabic, Muslim
Pretty; Beautiful; Graceful
Male
Hebrew
Variant spelling of Hebrew Yediydeyah, Hebrew name YEDIDYA means "beloved of God," "delight of God," or "friend of God."Â
Male
African
I am given by God.
AFFINE INVOLUTION
AFFINE INVOLUTION
AFFINE INVOLUTION
AFFINE INVOLUTION
AFFINE INVOLUTION
a.
Of or pertaining to the Alps, or to any lofty mountain; as, Alpine snows; Alpine plants.
a.
Andean; as, Andine flora.
n.
That part of the sea at a good distance from the shore, or where there is deep water and no need of a pilot; also, distance from the shore; as, the ship had ten miles offing; we saw a ship in the offing.
v. t.
To determine or clearly exhibit the boundaries of; to mark the limits of; as, to define the extent of a kingdom or country.
a.
Of, from, in, or pertaining to, the belly or the intestines; as, alvine discharges; alvine concretions.
n.
A special duty, trust, charge, or position, conferred by authority and for a public purpose; a position of trust or authority; as, an executive or judical office; a municipal office.
v. t.
To define.
v. i.
To pay a fine. See Fine, n., 3 (b).
v. t.
To perform, as the duties of an office; to discharge.
v. t.
To reduce to a fine, unmixed, or pure state; to free from impurities; to free from dross or alloy; to separate from extraneous matter; to purify; to defecate; as, to refine gold or silver; to refine iron; to refine wine or sugar.
v. t.
To subjoin, annex, or add at the close or end; to append to; to fix to any part of; as, to affix a syllable to a word; to affix a seal to an instrument; to affix one's name to a writing.
v. t.
To fix or fasten figuratively; -- with on or upon; as, eyes affixed upon the ground.
n.
The place where a particular kind of business or service for others is transacted; a house or apartment in which public officers and others transact business; as, the register's office; a lawyer's office.
pl.
of Affix
a.
To make fine; to refine; to purify, to clarify; as, to fine gold.
n.
The company or corporation, or persons collectively, whose place of business is in an office; as, I have notified the office.
v. t.
To refine.
v. t.
To attach, unite, or connect with; as, names affixed to ideas, or ideas affixed to things; to affix a stigma to a person; to affix ridicule or blame to any one.
imp. & p. p.
of Affix