Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 27

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. . , vn }. A second basis{w1 , . . . , wn } gives the same orientation precisely when the determinant of the change of basis matrix from {v1 , . . . , vn } to {w1 , . . . , wn } ispositive.For example, the ordered basis{(1, 0, 0), (0, 1, 0), (0, 0, 1)} gives the orientation of R3 known as the right-handed orientation.(2) An orientation of a simplex is a specificordering of its vertices; two orderings are equivalent if one is an even permutation of the other.(3) An orientation of a cell en (homeomorphic to an n-dimensional ball) is a choice of generator for the infinite cyclic relative homologygroup Hn (en , Bd(en )).(4) Given a manifold M, each point is contained in an open set that is homeomorphic toRn . Thus, using the definition of orientationfor vector spaces above, M can be covered byopen sets such that each open set has an orientation.

If two such open sets U and V are notdisjoint, we may ask if the orientation on U ∩ Vinherited from U is the same or opposite to thatinherited from V . If the orientation on U ∩ V isthe same either way, we say U and V are coherently oriented. An orientation on a manifold Mis defined to be a choice of coherently orientedopen sets that cover M.(5) Similarly, in a simplicial or cell complex,each point is contained in a simplex or cell whichcan be given an orientation. An orientation ofthe complex is a coherent choice of orientationfor each cell or simplex.orientation preserving mappingNote that if a manifold or complex can begiven one orientation, then it can also be giventhe reverse orientation.

Many manifolds andcomplexes cannot be oriented. For example,an open Möbius band cannot be oriented. SeeMöbius band.orientation preserving mappingAny mapbetween oriented bundles or oriented manifoldswhich maps the orientation of the domain to theorientation of the codomain. (Since homology isnatural, a map between manifolds induces a mapfrom the homology of one bundle or manifoldto the other.)orientation reversing mappingAny mapbetween oriented bundles or oriented manifoldswhich maps the orientation of the domain to thenegative orientation of the codomain. Example:Any reflection reverses orientation: the mapS 2 −→ S 2 given by reflection in the equator(horizontal plane) sends the generator of H2 (S 2 )to its negative generator.oriented complexA simplicial or cell complex with an orientation. See orientation.orthocenter The point of intersection of thethree altitudes of a triangle.orthogonalAt right angles.orthogonal complementGiven a subspaceW of a vector space V , the unique subspace Uof V such that V = U ⊕ W and every vector inU is perpendicular to every vector in W .orthogonal coordinate system A coordinatesystem in which, whenever i = j , the vectorwith a 1 in the ith position and zeros in everyother is orthogonal to the vector with a 1 in thej th position and zeros in every other.© 2001 by CRC Press LLCorthogonal frameIn differential geometry,the ordered set (x, v1 , .

. . , vn ) consisting of apoint x and orthonormal vectors v1 , . . . , vn .orthogonal groupThe group of all n × northogonal matrices under multiplication. Anorthogonal matrix is one whose inverse equalsits transpose.orthogonal projection A linear transformation T : V → V from an inner product spaceV to itself such that T = T 2 = T ∗ , where T ∗denotes the adjoint of T .orthogonal transformationA linear transformation whose matrix A is an orthogonal matrix, i.e., A−1 = At .orthogonal vectorspairwise orthogonal.A set of vectors that areorthonormalA set of vectors that are orthogonal and have magnitude 1.orthonormalizationA process by which aset of independent vectors may be transformedinto an orthonormal set of equal size while spanning the same space.osculating circle Given a point P on a curve,the circle that is the limit (if this exists) as apoint Q approaches P along the curve of circlespassing through Q and tangent to C at P .osculating processA method, due to P.Koebe (1912), of proving the existence of Green’sfunction on any simply or multiply connecteddomain in the complex plane.oval Any egg-shaped curve.

More generally,the boundary of a convex body in R2 .ovaloidR3 .The boundary of a convex body inpartial orderingparallelizable manifoldtangent bundle is trivial.A manifold whosePparallelogram A four-sided polygon havingopposite sides parallel.pair of relatively prime integersTwo integers whose greatest common divisor is 1. Forexample, 6 and 25 are relatively prime sincegcd(6,25)= 1.

See greatest common divisor.Parallel Postulate The fifth postulate of Euclid’s Elements, which requires that, if two linesare cut by a third, and if the sum of the interiorangles on one side of the third is less than 180◦ ,then the two lines will meet on that side of thethird.pairwise disjoint A collection of sets in whichany two distinct sets are disjoint is a pairwisedisjoint family.

For example, {[2n, 2n + 1) :n ∈ N} is a collection of pairwise disjoint sets(where each interval [2n, 2n + 1) is the set ofreal numbers x such that 2n ≤ x < 2n + 1).pairwise relatively prime numbersA setof integers with the property that no two share acommon divisor greater than 1.parabola The set of points in the plane equidistant from a given point and a given line. Alternatively, a conic section formed by the intersection of a circular cone with a plane such thatthe intersection is connected but unbounded.paraboloidThe surface given by the set of2solutions in R3 to an equation of the form xa 2 +y2b2= z (elliptic paraboloid) or(hyperbolic paraboloid).x2a2−y2b2= zparacompact topological spaceA Hausdorff space with the property that each opencover has a locally finite open refinement thatcovers X.

See refinement of a cover. Metricspaces are paracompact but, of course, not generally compact. Paracompactness is importantin the theory of manifolds because it is a sufficient condition on a space to construct a partitionof unity.parallelEquidistant, in some sense. In Euclidean space, two lines are parallel if they donot intersect and there is a plane in which theyboth lie.parallelepipedA polyhedron whose facesare parallelograms.© 2001 by CRC Press LLCpartially ordered setA set with a partialordering. See also partial ordering. A partiallyordered set is sometimes called a poset.partial orderingA binary relation on a setA (i.e., a subset of A × A), often denoted by ≤,which is reflexive (for all x ∈ A, x ≤ x), antisymmetric (for all x, y ∈ A, if x ≤ y and y ≤ x,then x = y), and transitive (for all x, y, z ∈ A,if x ≤ y and y ≤ z, then x ≤ z).

Given a partialordering ≤ on A and x, y ∈ A, x < y is definedto mean x ≤ y and x = y. Note that if ≤ is apartial ordering on A, then it is not necessarilythe case that every two distinct elements of Aare comparable; i.e., there may exist x, y ∈ Awith x = y and x ≤ y and y ≤ x.Sometimes one sees an alternative definitionof partial ordering, where < is defined first and≤ is defined in terms of <.A partial ordering on a set A is a binary relation on A, often denoted by <, which is antireflexive (for all x ∈ A, x < x) and transitive.Given a partial ordering < on A and x, y ∈ A,x ≤ y is defined to mean x < y or x = y.

Using this definition of partial ordering, one canprove that ≤ is antisymmetric. Note that if < isa partial ordering on A, then it is not necessarilythe case that every pair of distinct elements arecomparable; i.e., there may exist x, y ∈ A withx = y and x < y and y < x.Regardless of the choice of defining < from≤ or ≤ from <, the respective notions of ≤ and< are the same.One example of a partial ordering is the usualordering ≤ on the natural numbers N. This partial ordering is in fact a total, or linear ordering.See linear ordering. Let P(N) be the power setof the set of natural numbers; that is, the set ofpartial recursive functionall subsets of N.

Then set containment ⊆ is apartial ordering on P(N) which is not a linear ordering, as {1, 2} and {3} are not ⊆-comparable;i.e., {1, 2} ⊆ {3} and {3} ⊆ {1, 2}.partial recursive function All functions mentioned are functions on the natural numbers N;an n-ary function is partial if its domain is somesubset of Nn (i.e., the function may not be defined on all inputs). The notion of a partialrecursive function is a formalization (Kleene,1936) of the notion of an intuitively computablepartial function.

An n-ary partial function ϕis partial recursive if it can be derived from acertain set of initial functions by finitely manyapplications of composition, recursion, or theµ-operator; i.e., there is a finite sequenceϕ0 , ϕ1 , . . . , ϕk = ϕof functions such that for all i, 0 ≤ i ≤ k,(i.) ϕi is an initial function or(ii.) ϕi can be obtained from {ϕj : 0 ≤ j <i} by composition, recursion, or the µ-operator.The following functions are initial functions.• S(x) = x + 1 (the successor function)• Cin (x1 , . .

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