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 Affine geometry , 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 affine geometry , 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 Affine geometry geometry stub ca Marc af fr Rep re affine ... 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
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 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 Affine geometry Category Closure operators de Affine H lle fr Sous espace affine engendr vi Bao afin ... 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
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
Italic title Taxobox name Bulbophyllum affine image regnum Plantae unranked divisio Angiosperms unranked classis Monocots ordo Asparagales familia Orchidaceae subfamilia Epidendroideae genus Bulbophyllum species Bulbophyllum affine binomial binomial authority synonyms Bulbophyllum affine is a species of orchid in the genus Bulbophyllum . References http bulbophyllum checklist.bulbophyllum.at The Bulbophyllum Checklist http www.orchidspecies.com indexbulb.htm The Internet Orchid Species Photo Encyclopedia DEFAULTSORT Bulbophyllum affine Category Bulbophyllum affine Bulbophyllum stub az Bulbophyllum affine vi C u di p g n ... more details
Italic title Taxobox name Eulithidium affine image image caption regnum Animal ia phylum Mollusca classis Gastropoda unranked superfamilia clade Vetigastropoda superfamilia Phasianelloidea familia Phasianellidae subfamilia genus Eulithidium species E. affine binomial Eulithidium affine binomial authority C. B. Adams, 1850 synonyms ref synonyms Eulithidium affine is a species of sea snail , a marine gastropod mollusk in the family biology family Phasianellidae . ref name WoRMS WRMS species 419483 Eulithidium affine C. B. Adams, 1850 19 April 2010 ref Description Empty section date April 2010 Distribution Empty section date April 2010 References reflist External links Use dmy dates date January 2011 DEFAULTSORT Eulithidium Affine Category Phasianellidae Phasianellidae stub vi Eulithidium affine ... 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 Affine geometry Affine space Affine geometry Affine 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 Affine geometry he hu Affin kombin ci nl Affiene combinatie pl Kombinacja afiniczna pt Combina o afim vi T h p afin ... more details
italic title Taxobox name Agonum affine image regnum Animal ia phylum Arthropod a classis Insect a ordo Beetle Coleoptera familia Carabidae genus Agonum species A. affine binomial Agonum affine binomial authority William Forsell Kirby Kirby , 1837 synonyms Agonum carbo small LeConte, 1850 small Agonum affine is a species of ground beetle s in the Carabidae family. It is found in the United States . ref http carabidae.pro carabidae affine kirby 1837.html Carabidae of the World ref References Reflist carabidae stub Category Animals described in 1837 Category Carabidae ... 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 ... 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 ... 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 ... 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 ... group contains an abelian subgroup of finite index. Important longstanding conjectures Geometry 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 affine crystallographic groups , Adv. Math. 47 1983 , 1 49. ref stating that any affine crystallographic ... part of a properly discontinuous group of affine transformations, J. Differential Geom., 60 ... more details
Image Affine logo.png right Affine is a commercial real estate company Soci t based in Paris , France. It was founded in 1990 and is currently a member of the CAC Small 90 . External links http www.affine group.com Corporate webpage en http biz.yahoo.com ic 132 132698.html Profile on Yahoo Finance. CAC Small 90 france company stub Category Real estate companies of France fr Affine soci t ... more details
This article was auto generated by User Polbot . taxobox image Flower 010.jpg image caption Exacum affine flower status LC status system IUCN3.1 regnum Plantae unranked divisio Angiosperms unranked classis Eudicots unranked ordo Asterids ordo Gentianales familia Gentianaceae genus Exacum species E. affine binomial Exacum affine binomial authority Isaac Bayley Balfour Balf.f. ex Regel Exacum affine is a species of plant in the Gentianaceae family. It is Endemism endemic to Socotra Socotra islands of Yemen . Its natural habitat is rocky areas. A small herbaceous biennial plant with dark green, ovate leaves. The small purple flowers have a yellow centre with fragrance. References Miller, A. 2004. http www.iucnredlist.org search details.php 44952 all Exacum affine . http www.iucnredlist.org 2006 IUCN Red List of Threatened Species. Downloaded on 21 August 2007. Category Flora of Yemen Category Exacum Category Least concern plants Category Biota of Socotra Gentianales stub vi Exacum affine ... more details
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine ... affine geometry, like projective geometry and Euclidean geometry , follows naturally from the Erlangen program of Felix Klein . Affine geometry is a form of geometry featuring the unique parallel ... postulates are ignored . First identified by Euler , many affine properties are familiar from Euclidean ... by parallel projection from one Plane mathematics plane to another are affine. In effect, affine geometry 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 affine geometry is the group mathematics .... Affine geometry can be developed on the basis of linear algebra . One can define an affine ... as obtaining the affine space from its associated vector space by forgetting the origin zero vector . History In 1748 Euler introduced the term affine ref cite web title Earliest Known Uses of Some ... see chapter XVII . In 1827 August M bius wrote on affine geometry in his Der barycentrische Calcul , chapter 3. Only after Felix Klein s Erlangen program was affine geometry recognized for being ... Systems of axioms Several axiomatic approaches to affine geometry have been put forward Pappus law As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria ... Coxeter 1955 The Affine Plane, 2 Affine geometry as an independent system ref If math A, B, C math ... Affine plane, p 8 ref The various types of affine geometry correspond to what interpretation is taken ... rotation. Ordered structure An axiomatic treatment of plane affine geometry can be built ... additional axioms. ref Coxeter, Introduction to Geometry, p. 192 ref Parallel postulate Affine ... to B C , then AC is parallel to A C . The affine concept of parallelism forms an equivalence relation ... more details
An affine representation of a topological group topological Lie group Lie group G on an affine space A is a continuity topology continuous smooth function smooth group homomorphism from G to the automorphism group of A , the affine group Aff A . Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff A of the affine group of A . An example is the action of the Euclidean group E n upon the Euclidean space E sup n sup . Since the affine group in dimension n is a matrix group in dimension n 1, an affine representation may be thought of as a particular kind of linear representation . We may ask whether a given affine representation has a fixed point mathematics fixed point in the given affine space A . If it does, we may take that as origin and regard A as a vector space in that case, we actually have a linear representation in dimension n . This reduction depends on a group cohomology question, in general. See also Group action Projective representation References citation first1 Elisabeth last1 Remm first2 Michel last2 Goze title Affine Structures on abelian Lie Groups arxiv math 0105023 journal Linear Algebra and its Applications volume 360 year 2003 pages 215&ndash 230 doi 10.1016 S0024 3795 02 00452 4 . Category Group theory Category Homological algebra Category Representation theory Category Representation theory of Lie algebras Category Representation theory of Lie groups algebra stub eo Afina prezento pt Representa o afim ... more details
distinguish affinity space In mathematics , an affine space is a geometric structure mathematics structure that generalizes the affine geometry affine properties of Euclidean space . In an affine space ... 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 ... 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 ..., 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 ... . The vector space math scriptstyle V math is said to underlie the affine space math scriptstyle ... A math into a vector space. Conversely, any vector space, math scriptstyle V math , is an affine ..., only such affine combinations of points have meaning. By noting that one can define subtraction of points of an affine space as follows math scriptstyle a , , b math is the unique vector in math scriptstyle ... an affine space as a point set math scriptstyle A math , together with a vector space math scriptstyle ... the number line as a one dimensional affine space. Any coset of a subspace math scriptstyle V math of a vector space is an affine space over math scriptstyle V math . If math scriptstyle A math ... more details
Italic title Taxobox name Medicorophium regnum Animal ia phylum Arthropod a subphylum Crustacean Crustacea classis Malacostraca ordo Amphipod a familia Corophiidae genus Medicorophium species M. affine binomial Medicorophium affine binomial authority Bruzelius, 1859 Medicorophium affine is a species of Amphipoda amphipod crustacean . ref cite WoRMS author Jim Lowry year 2011 title Medicorophium affine Brizelius, 1859 id 236501 accessdate November 27, 2011 ref It is a small up to 5 mm species which burrows in bottom sediments, between 10 and 60 metres deep. It occurs on coasts of Northern Europe. ref cite book url http ip30.eti.uva.nl bis crustacea.php?selected beschrijving&menuentry soorten&id 294 title Macrobenthos of the North Sea author M. J. de Kluijver & S. S. Ingalsuo year 1999 chapter Corophium affine ref References Reflist Category Corophiidea Category Crustaceans of the Atlantic Ocean Category Animals described in 1859 Amphipod stub ... more details
italictitle Taxobox image image width image caption image2 image2 caption regnum Animal ia phylum Arthropod a classis Insect a ordo Lepidoptera familia Sesiidae genus Pyropteron species P. affine binomial Pyropteron affine binomial authority Staudinger, 1856 ref http www.sesiidae.net Checklst.htm Checklist of the Sesiidae of the World ref synonyms Sesia affine small Staudinger, 1856 small Pyropteron affinis Synansphecia affinis Synansphecia affine Chamaesphecia erodiiphagum small Dumont 1922 small Pyropteron affine is a moth of the Sesiidae family. It is found in most of Europe , except Ireland , Great Britain , the Netherlands , Denmark , Fennoscandia , the Baltic region , Poland and Bulgaria . ref http www.faunaeur.org full results.php?id 440079 Fauna Europaea ref It is also found in Asia Minor , Georgia country Georgia , the Middle East and North Africa Tunisia , Algeria and Morocco . The wingspan is 15 18 mm. Adults are on wing from May to July. ref http ww.lepidoptera.pl show.php?ID 4133&country HR lepidoptera.pl ref The larvae of ssp. affine feed on Helianthemum chamaecistus , Helianthemum vulgare , Helianthemum nummularium and Fumana procumbens , while the larvae of ssp. erodiiphagum have been recorded on Erodium arborescens . ref http www.nic.funet.fi pub sci bio life insecta lepidoptera ditrysia sesioidea sesiidae sesiinae synansphecia index.html Synansphecia at funet ref Subspecies Pyropteron affine affine Pyropteron affine erodiiphagum small Dumont, 1922 small Tunisia, Algeria, Morocco References reflist Category Animals described in 1856 Category Sesiidae Sesioidea stub ... more details
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. Then the special affine curvature or equi affine curvature is given by math k s det begin bmatrix beta ... math t mapsto x t , y t , math the special affine curvature is math begin align k t & frac ... 2001a . Affine curvature Suppose as above that s is a curve parameterized by special affine arclength ... more details
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 group of V math operatorname Aff A V rtimes operatorname GL V math The action of GL V on V is the natural ... 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 .... Category Affine geometry Category Group theory Category Lie groups fr Groupe affine nl Affiene groep ... more details
taxobox image Stylidium affine flower1.jpg regnum Plantae unranked divisio Angiosperms unranked classis ... subg. Tolypangium Tolypangium sectio Stylidium sect. Squamosae Squamosae species S. affine binomial Stylidium affine binomial authority Sond. synonyms S. caricifolium subsp. affine Sond. Sherwin Carlquist Carlquist Stylidium affine is a species in the genus Stylidium family Stylidiaceae that is endemism endemic to Western Australia . Description Stylidium affine is a perennial plant that possesses ... like, and around 4 mm long. It has a diploid chromosome number of 2 n 14. Stylidium affine flowers ... , 12 1 43 57. ref Distribution and habitat Stylidium affine grows in laterite lateritic soils ... and west to Dunsborough, Western Australia Dunsborough . In the southern part of its range, S. affine ... illustration by Augusta Innes Withers The species now known as S. affine was first described ... valleys, which are within the bounds of the known distribution of S. affine . ref name Wege 2007 Wege, J. A. 2007. 1782 Proposal to conserve the name Stylidium affine against S. drummondii Stylidiaceae . Taxon , 56 2 613 614. ref In 1845, Otto Wilhelm Sonder published the name Stylidium affine ... with and antedates that of S. affine , thus making S. drummondii the proper name for the species ... Nomenclature ICBN . Citing the extensive use of S. affine over S. drummondii and the preference ... name conservation of the name S. affine over S. drummondii in 2007. ref name Wege 2007 The conservation of the name S. affine was recommended by a 17 to 1 vote by the Nomenclature ... varieties of S. affine have been described in the past, the first of which were S. affine var. minus and the autonym botany autonym S. affine var. affine described in Sonder s original 1845 publication. In 1904, Ernst Georg Pritzel published S. affine var. laxum ... , rendering the autonym var. affine unnecessary. ref APNI name Stylidium affine id 38502 ... more details
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 ... direction, because any such rounding would falsify the dependencies between affine forms that share ... more details
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 .... unref date December 2007 DEFAULTSORT Affine Involution Category Affine geometry ... more details
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