In mathematics affinegeometry is the study of geometric properties which remain unchanged by affine ... affinegeometry, like projective geometry and Euclidean geometry , follows naturally from the Erlangen program of Felix Klein . Affinegeometry is a form of geometry featuring the unique parallel ..., affinegeometry is a generalization of Euclidean geometry characterized by slant and scale distortions. Projective geometry is more general than affine since it can be derived from projective space ... of Klein s Erlangen program , the underlying symmetry in affinegeometry is the group mathematics .... Affinegeometry can be developed on the basis of linear algebra . One can define an affine ... see chapter XVII . In 1827 August M bius wrote on affinegeometry in his Der barycentrische Calcul , chapter 3. Only after Felix Klein s Erlangen program was affinegeometry recognized for being ... Systems of axioms Several axiomatic approaches to affinegeometry have been put forward Pappus law As affinegeometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria ... Coxeter 1955 The Affine Plane, 2 Affinegeometry as an independent system ref If math A, B, C math ... Affine plane, p 8 ref The various types of affinegeometry correspond to what interpretation is taken ... rotation. Ordered structure An axiomatic treatment of plane affinegeometry can be built ... additional axioms. ref Coxeter, Introduction to Geometry, p. 192 ref Parallel postulate Affine ... of affinegeometry over the field of real numbers. Ternary fields In 1984 Wanda Szmielew published ... chapter of From affine to Euclidean geometry . Affine transformations main Affine transformation ... programme this is its underlying group mathematics group of symmetry transformations for affinegeometry ... main Affine space Affinegeometry can be viewed as the geometry of affine space , of a given dimension ... generalization of coordinatized affine space, as developed in synthetic finite geometry . In projective ... more details
In geometry , an affine plane is a system of points and lines that satisfy the following axioms ref harvnb Cameron 1991 loc pg. 18 ref Any two distinct points lie on a unique line. Given a point and line ... three non collinear points points not on a single line . In an affine plane, two lines are called ... between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry . The familiar Euclidean plane is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields and division ring s , there are also .... The Moulton plane is an example of one of these. An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can ... affine planes. Finite affine planes If the number of points in an affine plane is finite, then if one ... n is called the order of the affine plane. All known finite affine planes have orders which are prime or prime power integers. The smallest affine plane of order 2 is obtained by removing a line and the three points on that line from the Fano plane . An affine plane of order n exists if and only ... of order in these cases is not the same . Thus, there is no affine plane of order 6 or order 10. The Bruck ..., the order of an affine plane. Affine spaces Affine space s can be defined in an analogous manner to the construction of affine planes from projective planes. It is also possible to provide a system of axioms for the higher dimensional affine spaces which does not refer to the corresponding projective ... year 1991 volume 13 Citation last Casse first Rey title Projective Geometry An Introduction publisher ... isbn 0 387 09614 0 Citation last1 Moulton first1 Forest Ray title A Simple Non Desarguesian Plane Geometry ... place San Francisco year 1972 isbn 0 7167 0443 9 Category Incidence geometry Category Geometry de Affine Ebene fr Plan affine structure d incidence ... more details
In the mathematics mathematical field of differential geometry , the affinegeometry of curves is the study of curve s in an affine space , and specifically the properties of such curves which are invariant mathematics invariant under the special affine group math mbox SL n, mathbb R ltimes mathbb R n. math In the classical Euclidean geometry of curves , the fundamental tool is the Frenet&ndash Serret frame . In affinegeometry, the Frenet&ndash Serret frame is no longer well defined, but it is possible ... geometry Volume 2 year 1999 publisher Publish or Perish isbn 0 914098 71 3 Category Curves Category Differential geometry Category Affinegeometry ... Favard . The affine frame Let x t be a curve in R sup n sup . Assume, as one does in the Euclidean ... not lie in any lower dimensional affine subspace of R sup n sup . Then the curve parameter t can ..., & mathbf x n end bmatrix pm 1. math Such a curve is said to be parametrized by its affine arclength ... determines a mapping into the special affine group, known as a special affine frame for the curve ... define a special affine frame for the affine space R sup n sup , consisting of a point x of the space ... differential geometry pullback of the Maurer&ndash Cartan form along this map gives a complete set of affine structural invariants of the curve. In the plane, this gives a single scalar invariant, the affine curvature of the curve. Discrete invariant The normalization of the curve parameter .... Curvature Suppose that the curve x in R sup n sup is parameterized by affine arclength. Then the affine ... affine arclength . Then, math dot A begin bmatrix 0&1&0&0& cdots&0&0 0&0&1&0& cdots&0&0 vdots& vdots ... n 1 &0 end bmatrix A CA. math In concrete terms, the matrix C is the pullback differential geometry ... n derivatives of x . See also Moving frame Affine sphere References reflist cite book first Heinrich last Guggenheimer title Differential Geometry year 1977 publisher Dover isbn 0 486 63433 7 cite ... more details
Affine differential geometry , is a type of differential geometry in which the differential invariants are invariant under volume preserving affine transformation s. The name affine differential geometry follows from Felix Klein Klein s Erlangen program . The basic difference between affine and Riemannian geometry Riemannian differential geometry is that in the affine case we introduce volume form .... last Nomizu first2 T. last2 Sasaki title Affine Differential GeometryGeometry of Affine Immersions ... last Su title Affine Differential Geometry publisher Harwood Academic year 1983 isbn 0 677 31060 9 ref From its dependence on volume forms for its definition we see that the affine normal vector field is invariant under volume preserving affine transformation s. These transformations are given by nowrap ... name Davis Davis, D. 2006 , Generic Affine Differential Geometry of Curves in R sup n sup , Proc. Royal ... the affine normal vector. References Reflist See also Affinegeometry of curves Affine sphere DEFAULTSORT Affine Differential Geometry Category Differential geometry ... Tangent space tangent to M and a transverse component, Parallel geometry parallel to . This gives ... 1,1 tensor nowrap 1 S &Psi M &rarr &Psi M is called the affine shape operator, the differential ... definition in differential geometry Euclidean differential geometry where is the surface ... are both satisfied. These two special transverse vector fields are called affine normal vector fields ... 1, R R sup n 1 sup forms a Lie group . The affine normal line The affine normal line at a point nowrap ... thumbnail Affine normal line for the curve nowrap 1 &gamma t t 2 t sup 2 sup , t sup 2 sup at nowrap 1 t 0. The affine normal vector field for a curve in the plane has a nice geometrical interpretation ... p is exactly the affine normal line, i.e. the line containing the affine normal vector to I at t sub 0 sub . Notice that this is an affine invariant construction since parallelism and midpoints ... more details
wiktionarypar affineAffine may refer to Prospective editors please keep this list alphabetical Affine cipher , a special case of the more general substitution cipher Affine combination , a certain kind of constrained linear combination Affine connection , a connection on the tangent bundle of a differentiable manifold Affinegeometry , a geometry not involving any notions of origin, length or angle Affine differential geometry , a geometry that studies differential invariants under the action of the special affine group Affine group , the group of all invertible affine transformations from any affine space over a field K into itself Affine representation , a continuous group homomorphism whose values are automorphisms of an affine space Affine scheme , the spectrum of prime ideals of a commutative ring Affine Soci t , A French Commercial real estate company. Affine space , an abstract structure that generalises the affine geometric properties of Euclidean space Affine transformation , a linear transformation followed by a translation between two vector spaces, or equivalently, a transformation that preserves all affine combinations See also Affinity disambiguation disambig Category Mathematical disambiguation fr Affine pt Afim ... more details
In affinegeometry , a branch of mathematics , an affine frame in an affine space A consists of a choice P of origin of A along with a basis of a vector space basis of the space of vectors based at P . Category Affinegeometrygeometry stub ca Marc af fr Rep re affine ... more details
In mathematics , an affine plane may refer to The Euclidean plane Affine plane incidence geometry , an abstract system of points and lines such that every pair of points has a line containing both of them A two dimensional affine space , an origin free generalization of a vector space The plane with two complex Cartesian coordinates , called the affine plane in algebraic geometry to emphasize the difference between it and its projective completion mathdab ... more details
In mathematics, and especially differential geometry , an affine sphere is a hypersurface for which the Affine differential geometry The affine normal line affine normal s all intersect in a single point. ref name spring cite web url http eom.springer.de a a011110.htm publisher Springer Online References title Affine Sphere author E. V. Shikin ref The term affine sphere is used because they play an analogous role in affine differential geometry to that of ordinary spheres in Euclidean differential geometry. An affine sphere is called improper if all of the affine normals are constant. ref name spring In that case, the intersection point mentioned above lies on the hyperplane at infinity . Affine spheres have been the subject of much investigation, with many hundreds of research article s devoted to their study. ref cite web url http scholar.google.co.uk scholar?hl en&q 22affine sphere 22&btnG Search&as sdt 1 2C5&as ylo &as vis 0 title Google Scholar Search publisher Google Inc ref Examples All quadric s are affine spheres the quadrics that are also improper affine spheres are the paraboloid s. ref cite book last1 Buchin first1 S. title Affine differential geometry year 1983 publisher Sci. Press and Gordon & Breach location language isbn 0 677 31060 9 ref If is a smooth function on the plane and the determinant of the Hessian matrix is 1 then the graph of in three space is an improper affine sphere. ref cite journal last1 Ishikawa first1 G. last2 Machida first2 Y. year 2005 title Singularities of improper affine spheres and surfaces of constant Gaussian curvature arxiv math 0502154 ref References reflist DEFAULTSORT Affine sphere Category Differential geometry differential geometry stub ... more details
In mathematics , an affine combination of vectors x sub 1 sub , ..., x sub n sub is a vector math sum i 1 n alpha i cdot x i alpha 1 x 1 alpha 2 x 2 cdots alpha n x n , math called a linear combination of x sub 1 sub , ..., x sub n sub , in which the sum of the coefficients is 1, thus math sum i 1 n alpha i 1. math Here the vectors are elements of a given vector space V over a field mathematics field K , and the coefficients math alpha i math are scalar mathematics scalars in K . This concept is important, for example, in Euclidean geometry . The act of taking an affine combination commutes with any affine transformation T in the sense that math T sum i 1 n alpha i cdot x i sum i 1 n alpha i cdot Tx i math In particular, any affine combination of the fixed point mathematics fixed point s of a given affine transformation math T math is also a fixed point of math T math , so the set of fixed points of math T math forms an affine subspace in 3D a line or a plane, and the trivial cases, a point or the whole space . When a stochastic matrix , A, acts on a column vector, B, the result is a column vector whose entries are affine combinations of B with coefficients from the rows in A. See also Related combinations details Linear combination Affine, conical, and convex combinations Convex combination Conical combination Linear combination AffinegeometryAffine space AffinegeometryAffine hull References Citation last1 Gallier first1 Jean title Geometric Methods and Applications publisher Springer Verlag location Berlin, New York isbn 978 0 387 95044 0 year 2001 . See chapter 2 . Category Affinegeometry he hu Affin kombin ci nl Affiene combinatie pl Kombinacja afiniczna pt Combina o afim vi T h p afin ... more details
In mathematics , the affine hull of a set mathematics set S in Euclidean space R sup n sup is the smallest affine set containing S , or equivalently, the intersection set theory intersection of all affine sets containing S . Here, an affine set may be defined as the translation mathematics translation of a vector subspace . The affine hull aff S of S is the set of all affine combination s of elements of S , that is, math operatorname aff S left sum i 1 k alpha i x i Bigg x i in S, , alpha i in mathbb R , , i 1,2, dots, k , sum i 1 k alpha i 1 k 1, 2, dots right . math Examples The affine hull of a set of two different points is the line through them. The affine hull of a set of three points not on one line is the plane going through them. The affine hull of a set of four points not in a plane in R sup 3 sup is the entire space R sup 3 sup . Properties math mathrm aff mathrm aff S mathrm aff S math math mathrm aff S math is a closed set Related sets If instead of an affine combination one uses a convex combination , that is one requires in the formula above that all math alpha i math be non negative, one obtains the convex hull of S , which cannot be larger than the affine hull of S as more restrictions are involved. The notion of conical combination gives rise to the notion of the conical hull If however one puts no restrictions at all on the numbers math alpha i math , instead of an affine combination one has a linear combination , and the resulting set is the linear span of S , which contains the affine hull of S . References R.J. Webster, Convexity , Oxford University Press, 1994. ISBN 0 19 853147 8. Category Affinegeometry Category Closure operators de Affine H lle fr Sous espace affine engendr vi Bao afin ... more details
Other uses Affine manifold disambiguation In differential geometry , an affine manifold is a manifold equipped with a flat connection flat , torsion tensor torsion free affine connection connection . Equivalently ... parts, so there is a unique connection associated with an affine structure. Note there is a link between linear connection mathematics connection also called affine connection and a web differential geometry web . Formal definition An affine manifold math M , math is a real manifold with charts ... group contains an abelian subgroup of finite index. Important longstanding conjectures Geometry of affine ... A Survey , Geom. Dedicata, 87,. 309 333 2001 ref Chern conjecture affinegeometry Chern conjecture ... theory Category Affinegeometry Category Structures on manifolds Category Differential geometry ... n math , with Covering space Monodromy action monodromy acting by affine transformation s. This equivalence ... with an atlas called the affine structure with all transition functions between chart s affine that is, have ... to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts . In each affine coordinate domain the coordinate ... math i, j , , math where math rm Aff Bbb R n math denotes the Lie group of affine transformations. An affine manifold is called complete if its universal covering is homeomorphism to math Bbb R n math . In the case of a compact affine manifold math M math , let math G math be the fundamental group ... affine manifold comes with a developing map math D colon tilde M to Bbb R n math , and a homomorphism ... complete flat affine manifold is called an affine crystallographic group . Classification of affine ... affine manifold is complete if and only if it has constant volume. ref Hirsch M. and Thurston ... affine manifolds, Topology 3 1964 , 131 139. ref ref Fried D. and Goldman W., Three dimensional ... more details
distinguish affinity space In mathematics , an affine space is a geometric structure mathematics structure that generalizes the affinegeometryaffine properties of Euclidean space . In an affine space ... from different origins. While Alice knows the linear structure , both Alice and Bob know the affine structure i.e. the values of affine combination s, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space. Definition An affine space ref cite book author Berger, Marcel chapter Affine spaces title Problems in Geometry ... sum i 1 n a i 0. math Axioms Affine space is usually studied as analytic geometry using coordinates ... space. harvtxt Coxeter 1969 loc p.192 axiomatizes affinegeometry over the reals as ordered geometry ... one may then projectivize, but this requires a choice. See also affinegeometryaffine transformation affine group affine hull equipollence geometry interval measurement , an affine observable in statistics ... first2 Robert J. title Metric AffineGeometry publisher Dover Publications year 1989 edition Dover ... S. title Affine Differential Geometry publisher Cambridge University Press year 1994 edition New isbn 978 0 521 44177 3 DEFAULTSORT Affine Space Category Affinegeometry Category Linear algebra ar ... set of an inhomogeneous linear equation is either empty or an affine space but note that a single point is also an affine space, over a zero dimensional vector space . Informal descriptions The following ... definition an affine space is what is left of a vector space after you ve forgotten which point is the origin or, in the words of the French mathematician Marcel Berger , An affine space is nothing ..., an affine space is a principal homogeneous space over the additive group of a vector space. cn date April 2012 Explicitly, an affine space is a point set math scriptstyle A math together with a map math ... Tarrida, Agusti R. chapter Affine spaces title Affine Maps, Euclidean Motions and Quadrics publisher ... more details
k parameterizes these leaves. See also Affinegeometry of curves Affine sphere References references ... year 2001a springer id a a010990 title Affine differential geometry first A.P. last Shirokov year ... 0 914098 71 3 year 1999 Category Differential geometry Category Curves Category Affinegeometry ar ...This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection . Special affine curvature , also known as the equi affine curvature or affine ... under a special affine group special affine transformation an affine transformation that preserves area . The curves of constant equi affine curvature k are precisely all non singular conic section ... point contact with the curve at the point. In the same way, the special affine curvature of a curve at a point P is the special affine curvature of its hyperosculating conic , which is the unique conic ... 1,P 2,P 3,P 4 to P. math In some contexts, the affine curvature refers to a differential invariant of the affine group general affine group , which may readily obtained from the special affine curvature k by k sup 3 2 sup d k d s , where s is the special affine arc length. Where the general affine group is not used, the special affine curvature k is sometimes also called the affine curvature harv Shirokov 2001b . Formal definition Special affine arclength To define the special affine curvature, it is necessary first to define the special affine arclength also called the equi affine arclength . Consider an affine plane curve math beta t math . Choose co ordinates for the affine plane such that the area ... invariant of the special affine group, and gives the signed area of the parallelogram spanned by the velocity ... 2 end bmatrix , ,dt. math This integral is called the special affine arclength , and a curve carrying this parameterization is said to be parameterized with respect to its special affine arclength. Special affine curvature Suppose that s is a curve parameterized with its special affine arclength ... more details
. Lyndon , Groups and Geometry , Cambridge University Press , 1985, ISBN 0 521 31694 4. Section VI.1. Category Affinegeometry Category Group theory Category Lie groups fr Groupe affine nl Affiene groep ...In mathematics , the affine group or general affine group of any affine space over a field mathematics field K is the group mathematics group of all invertible affine transformation s from the space into itself. It is a Lie group if K is the real or complex field or quaternions . Relation to general linear group Construction from general linear group Concretely, given a vector space V, it has an underlying affine space A obtained by forgetting the origin, with V acting by translations, and the affine group of A can be described concretely as the semidirect product of V by GL V , the general linear ... Given the affine group of an affine space A , the Group action Orbits and stabilizers stabilizer ... space math A,p math recall that if one fixes a point, an affine space becomes a vector space. All ... 1 to V to V rtimes operatorname GL V to operatorname GL V to 1 math . In the case that the affine ... space is the original GL V . Matrix representation Representing the affine group as a semidirect ... operatorname GL V oplus K math , with V embedded as the affine plane math v,1 v in V math , namely the stabilizer of this affine plane the above matrix formulation is the transpose of the realization ... ones. Each of these two classes of matrices is closed under matrix multiplication. Other affine ... an affine group, sometimes denoted math operatorname Aff G math analogously as math operatorname ... over R . ref an associated affine group math V rtimes rho G math one can say that the affine group ... sequence math 1 to V to V rtimes rho G to G to 1. math Special affine group main Special affine group The subset of all invertible affine transformations preserving a fixed volume form, or in terms of the semi ... as the special affine group . Poincar group main Poincar group The Poincar group is the affine ... more details
. unref date December 2007 DEFAULTSORT Affine Involution Category Affinegeometry ...In Euclidean geometry , of special interest are Involution mathematics involution s which are linear transformation linear or affine transformation s over the Euclidean space R sup n sup . Such involutions are easy to characterize and they can be described geometrically. Linear involutions To give a linear involution is the same as giving a square matrix A such that math A 2 I quad quad quad quad 1 math where I is the identity matrix . It is a quick check that a square matrix D that has zero off the main diagonal and 1 on the diagonal, that is, a signature matrix of the form math D begin pmatrix pm 1 & 0 & cdots & 0 & 0 0 & pm 1 & cdots & 0 & 0 vdots & vdots & ddots & vdots & vdots 0 & 0 & cdots & pm 1 & 0 0 & 0 & cdots & 0 & pm 1 end pmatrix math satisfies 1 , i.e. is the matrix of a linear involution. It turns out that all the matrices satisfying 1 are of the form A U sup 1 sup DU , where U is invertible and D is as above. That is to say, the matrix of any linear involution is of the form D up to a Similar matrix similarity . Geometrically this means that any linear involution can be obtained by taking oblique reflection s against any number from 0 through n hyperplane s going ... Projection linear algebra projection P . Affine involutions If A represents a linear involution, then x A x b b is an affine transformation affine involution. One can check that any affine involution in fact has this form. Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through a point b . Affine involutions can be categorized by the dimension of the affine space of Fixed point mathematics ... above , i.e., the dimension of the eigenspace for eigenvalue 1. The affine involutions in 3D are the identity ... 1 is orthogonal to every eigenvector with eigenvalue 1, such an affine involution is an isometry ... more details
Image Fractal fern explained.png thumb right 200px An image of a fern like fractal that exhibits affine self similarity In geometry , an affine transformation or affine map ref Citation last1 Berger first1 ... 6 sup y sup 5 sup y sup 3 sup y sup 2 sup 1 11101101 ED . Affine transformation in plane geometry ... is affine transformation in 1D 3D projection Flat geometry Notes refimprove date April 2012 Reflist ... first2 S. title Affine Differential Geometry publisher Cambridge University Press year 1994 edition ... software Category Affinegeometry Category Transformation function ar cs Afinn zobrazen ... not necessarily preserve angles or lengths. Translation geometry Translation , geometric contraction, Expansion geometry expansion , dilation, Reflection mathematics reflection , rotation , Shear mapping shear , Similarity geometry similarity transformations , and spiral similarities are all affine transformations, as are their combinations. An affine transformation is equivalent to a linear transformation followed by a translation. Mathematical Definition An affine map math f mathcal A to mathcal B math between two affine space s is a map that acts, on vectors defined by pairs of points, as a linear ..., this can be decomposed as an affine transformation math mathcal A , to , mathcal B math that sends ... Given two affine space s math mathcal A math and math mathcal B math , over the same field, a function math f , mathcal A to mathcal B math is an affine map if and only if for every family math a i, , lambda ... words, math f , math preserves barycenter s. In the finite dimensional case, an affine map can be specified in coordinates by a matrix A describing together with the vector math vec b math . An affine transformation preserves The Line geometry collinearity relation between points i.e., points ... is called affine transformation matrix , or projective transformation matrix as it can also be used ... invertible affine transformations as the semidirect product of K sup n sup and GL n , k . This is a group ... more details
the development . In the branch of mathematics called differential geometry , an affine connection ... programme . More generally, an n dimensional affine space is a Klein geometry for the affine ..., this approach does not explain the geometry behind affine connections nor how they acquired their name ... , and also Harvtxt Sharpe 1997 . ref in which a geometry is defined to be a homogeneous space . Affine ... affine manifold is viewed as curved deformation of the flat model geometry of affine space. Affine space as the flat model geometry Definition of an affine space Informally, an affine space is a vector ... bundle in this case P × aff n . The pair P , &eta defines the structure of an affinegeometry on M ... geometry, affine connections define a generalized notion of parametrized straight lines on any ...File Parallel transport sphere.svg right thumb An affine connection on the sphere rolls the affine tangent ... with values in a fixed vector space. The notion of an affine connection has its roots in 19th century geometry and tensor calculus, but was not fully developed until the early 1920s, by lie ... the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space . On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi Civita connection . The choice of an affine ... definition of an affine connection as a covariant derivative or linear connection vector bundle connection on the tangent bundle . A choice of affine connection is also equivalent to a notion of parallel ... yields another description of an affine connection, either as a Cartan connection for the affine group ... of an affine connection are its torsion of connection torsion and its curvature . The torsion measures how closely the Lie derivative Lie bracket of vector fields can be recovered from the affine connection ... more details
arithmetic Category Numerical analysis Category Affinegeometry it Aritmetica affine ...Affine arithmetic AA is a model for self validated computation self validated numerical analysis . In AA, the quantities of interest are represented as affine combination s affine forms of certain primitive .... Affine arithmetic is meant to be an improvement on interval arithmetic IA , and is similar ... approximations to general formulas. Affine arithmetic is potentially useful in every numeric ... control , worst case analysis of electric circuit s, and more. Definition In affine arithmetic ... X which is known to lie in the range 3,7 can be represented by the affine form math x 5 2 epsilon k ... affine forms math x math , math y math implies that the corresponding quantities X , Y are partially ... subset of the rectangle 2,18 13,27 . Affine arithmetic operations Affine forms can be combined ... to formulas. Affine operations For example, given affine forms math x,y math for X and Y , one can obtain an affine form math z math for Z X Y simply by adding the forms &mdash that is, setting math z j math math gets math math x j y j math for every j . Similarly, one can compute an affine form math ... math math gets math math alpha x j math for every j . This generalizes to arbitrary affine operations like Z math alpha math X math beta math Y math gamma math . br Non affine operations A non affine ... exactly, since the result would not be an affine form of the math epsilon i math . In that case, one should take a suitable affine function G that approximates F to first order, in the ranges ... previous form. The form math z math then gives a guaranteed enclosure for the quantity Z moreover, the affine ... on given quantities to be replaced by equivalent computations on their affine forms, while preserving .... For this reason, affine arithmetic will often yield much tighter bounds than standard interval ..., affine arithmetic operations must account for the roundoff errors in the computation of the resulting ... more details
Wiktionary Affine manifold Affine manifold may refer to See Algebraic variety Affine varieties for affine manifold, an affine variety which is smooth Affine manifold in differential geometry, differentiable manifold equipped with a flat, torsion free connection disambig ... more details
theorem , an important result in Euclidean geometry Euclidean and projective geometry . Image Oxyrhynchus ... fragment of Euclid s Elements Geometry lang grc wikt geo earth , wikt metria measurement ..., and the properties of space. Geometry arose independently in a number of early cultures as a body ... science emerging in the West as early as Thales 6th Century BC . By the 3rd century BC geometry was put into an axiomatic system axiomatic form by Euclid , whose treatment Euclidean geometry ... geometry in digital imaging . Academic Press . p.1. ISBN 0127039708 ref Archimedes developed ... works in the field of geometry is called a geometer. The introduction of coordinates by Ren Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curve s, could now be represented analytic geometry analytically , i.e., with functions ... century. Furthermore, the theory of perspective graphical perspective showed that there is more to geometry than just the metric properties of figures perspective is the origin of projective geometry . The subject of geometry was further enriched by the study of intrinsic structure of geometric objects ... geometry . In Euclid s time there was no clear distinction between physical space and geometrical space. Since the 19th century discovery of non Euclidean geometry , the concept of space ... geometry considers manifold s, spaces that are considerably more abstract than the familiar ... with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics , exemplified by the ties between pseudo Riemannian geometry and general relativity ... the visual nature of geometry makes it initially more accessible than other parts of mathematics ... traditional, Euclidean provenance for example, in fractal geometry and algebraic geometry . ref It is quite common in algebraic geometry to speak about geometry of algebraic variety algebraic varieties ... more details
Affine algebra may refer to affine Lie algebra , a type of Kac Moody algebras the Lie algebra of the affine group finitely generated algebra disambig ... more details
Affine logic is a substructural logic whose proof theory rejects the structural rule of Idempotency of entailment contraction . It can also be characterized as linear logic with weakening . The name affine logic is associated with linear logic , to which it differs by allowing the weakening rule. Jean Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterises linear logic in terms of linear algebra here he alludes to affine transformation s on vector spaces. ref Jean Yves Girard , 1997. http www.seas.upenn.edu sweirich types archive 1997 98 msg00134.html Affine . Message to the TYPES mailing list. ref The logic predated linear logic. V. N. Grishin used this logic in 1974, ref Grishin, 1974, and later, Grishin, 1981. ref after observing that Russell s paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom . ref Cf. Frederic Fitch s demonstrably consistent set theory ref Likewise, the logic formed the basis of a decidable subtheory of predicate logic , called Direct logic Ketonen & Wehrauch, 1984 Ketonen & Bellin, 1989 . Affine logic can be embedded into linear logic by rewriting the affine arrow math A rightarrow B math as the linear arrow math A circ B otimes top math . Whereas full linear logic i.e. propositional linear logic with multiplicatives, additives and exponentials is undecidable, full affine logic is decidable. Affine logic forms the foundation of ludics . Notes references References V.N. Grishin, 1974. A nonstandard logic and its application to set theory, Russian . Studies in Formalized Languages and Nonclassical Logics Russian , 135 171. Izdat, Nauka, Moskow. . V.N. Grishin, 1981. Predicate and set theoretic calculi based on logic without contraction rules, Russian . Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 45 1 47 68. 239. Math. USSR Izv., 18, no.1, Moscow. Ketonen and Weyhrauch, 1984, A decidable fragment of predicate calculus. Theoretical ... more details
italic title Unreferenced date March 2007 Taxobox name Sphagnum affine regnum Plant ae divisio Moss Bryophyta classis Sphagnopsida subclassis Sphagnidae ordo Sphagnales familia Sphagnaceae genus Sphagnum species S. affine binomial Sphagnum affine Sphagnum affine is a species of peat moss or sphagnum moss which is exploited to make commercial peat products. This moss has a yellowish coloring. DEFAULTSORT Sphagnum Affine Category Sphagnum Affine Bryophyte stub de Sphagnum affine es Sphagnum affine fr Sphagnum affine it Sphagnum affine fi Rannikkorahkasammal ... more details
In mathematics , the term affine Grassmannian has two distinct meanings. In one meaning the affine Grassmannian manifold is the variety of all k dimensional affine subspaces of a finite dimensional vector space this is a smooth finite dimensional variety over k . The concept treated in this article is the affine Grassmannian of an algebraic group G over a field k . It is an ind scheme a limit of finite dimensional scheme mathematics schemes which can be thought of as a flag variety for the loop group G k t and which describes the representation theory of the Langlands dual group sup L sup G through what is known as the geometric Satake correspondence. Definition of Gr via functor of points Let k be a field, and denote by math k mathrm Alg math and math mathrm Set math the category of commutative k algebras and the category of sets respectively. Through the Yoneda lemma , a scheme X over a field k is determined by its functor of points , which is the functor math X k mathrm Alg to mathrm Set math which takes A to the set X A of A points of X . We then say that this functor is representable functor representable by the scheme X . The affine Grassmannian is a functor from k algebras to sets which is not itself representable, but which has a filtration by representable functors. As such, although it is not a scheme, it may be thought of as a union of schemes, and this is enough to profitably apply geometric methods to study it. Let G be an algebraic group over k . The affine Grassmannian Gr sub G sub is the functor that associates to a k algebra A the set of isomorphism classes of pairs E , , where E is a principal homogeneous space for G over Spec A nowiki nowiki t nowiki nowiki and is an isomorphism, defined over Spec A t , of E with the trivial G bundle G Spec A t . By the Beauville Laszlo theorem , it is also possible to specify this data by fixing an algebraic curve ... O math . algebra stub Category Algebraic geometry ... more details
italic title Taxobox name Fusarium affine regnum Fungi phylum Ascomycota classis Sordariomycetes subclassis Hypocreomycetidae ordo Hypocreales familia Nectriaceae genus Fusarium species F. affine binomial Fusarium affine binomial authority Fautrey & Lambotte Fusarium affine is a fungus fungal plant pathogen. External links http www.speciesfungorum.org Names Names.asp Index Fungorum http nt.ars grin.gov fungaldatabases USDA ARS Fungal Database DEFAULTSORT Fusarium Affine Category Fusarium affine Category Plant pathogens and diseases Hypocreales stub plant disease stub ... more details
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