Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 23
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Note that ifa set has a least upper bound, then it is unique.Compare with greatest lower bound.left adjoint functorLet C and D be categories with functors F : C −→ D and G :D −→ C such that if X is an object of C and Yis an object of D, we have a bijection of hom-setshomC (X, G(Y )) = homD (F (X), Y )which is natural in both X and Y .
Then F is aleft adjoint for G and G is a right adjoint for F .Example: The forgetful functor from Abelian groups to sets which forgets the group structure has left adjoint given by taking the freeAbelian group on the elements of the set. Notethat this is not an “inverse” functor.leg In a right triangle, either of the two sidesincident to the right angle.level (of a tree)The αth level of a tree T ,Levα (T ), is the set of all elements of T whosepredecessors have order type α. That is, for anyt ∈ T , the set of predecessors of t, {s ∈ T :s < t}, must be well ordered, and the level of tis given by their order type.Lev0 (T ) is the set of elements in T with nopredecessors, while Lev1 (T ) is the set of elements with exactly one predecessor (which mustcome from level 0).Lie groupA group that is also a differentiable manifold and for which the product andinverse maps are infinitely differentiable (and isLie line-sphere transformationtherefore a topological group).
See topologicalgroup.Example: Consider the unit circle in the complex numbers, all points of the form eix for xreal. This is a Lie group.Lie line-sphere transformationA correspondence between lines in space R, say, andspheres in a corresponding space S, named afterthe Norwegian mathematician Marius SophusLie. A point (X, Y, Z) ∈ S determines a line inR by the two equations:(X + iY ) − zZ − x = 0z(X − iY ) + Z − y = 0For any fixed line l in R, the set of such linesthat meet l corresponds to a sphere in S (whosecenter and/or radius may be complex numbers).limit cardinal A cardinal ℵα whose index αis a limit ordinal. See limit ordinal.limit ordinalAn ordinal α that is not a successor ordinal.Therefore,α has the form sup{β :β < α} = β<α β.
(It should be noted that 0 isalso a limit ordinal; we define sup ∅ = 0.) Seesuccessor ordinal, ordinal.line bundle A term used in the theory of vector bundles. A vector bundle over a topologicalspace X consists of a space E called the totalspace, a vector space F called the fiber, and amap π : E −→ X.
The space X has a covering by open sets Ui with homeomorphisms φifrom Ei = π −1 (Ui ) to Ui × F . The projectionmap π respects these product structures; i.e.,π ◦φi−1 (x, V ) = x. When x ∈ Ui ∩Uj , the mapgij (x) : F −→ F defined by (x, gij (x)(V )) =φj (φi−1 (x, V )) is required to be linear. Thisimplies that π −1 (x) has the structure of a vector space. A line bundle is a vector bundlewith one-dimensional fiber.
Usually the field iseither the real numbers or the complex numbers.line of curvatureA curve C on a surface,having the property that, at each point C(t) onthe curve, the tangent vector C (t) is a principalvector of the surface at C(t), that is, an eigenvector of the Weingarten map.© 2001 by CRC Press LLCline segmentAll points P on the line determined by two given points A and B, lyingbetween A and B in the plane. For such a pointP , the points A and B lie on different rays fromP . This definition also makes sense in hyperbolic plane geometry but not in elliptic geometry, where betweenness is not a well-definedconcept.linear fractional function A bijection of theextended complex plane C ∪ {∞} defined byz → az+bcz+d for given complex numbers a, b, c, dsuch that ad − bc = 0.
The inverse is given byz → dz−bcz−a . Also called Möbius transformation,linear fractional transformation, linear transformation.linearly ordered setA set A with a linearordering on A. See linear ordering.linear ordering A partial ordering ≤ on a setA in which every pair of distinct elements of Ais comparable; i.e., for all x, y ∈ A, if x = y,then x ≤ y or y ≤ x.
If the partial ordering isof the < type (see partial ordering), then < is alinear ordering if, for all x, y ∈ A, if x = y,then x < y or y < x.The usual ordering ≤ on Q, the set of rationalnumbers, is a linear ordering.A linear ordering is also called a total ordering.linkA link of n components in R3 is a subset of R3 which is homeomorphic to n distinctcopies of S 1 . Individual components may beknotted with themselves or with other components.More generally, a link of n components inRm+2 is an m embedding of a finite numberof copies of S m in Rm+2 or S m+2 . Two linksL1 , L2 are equivalent if there is a homeomorphism h : Rm+2 → Rm+2 (or h : S m+2 →S m+2 ) such that h(L1 ) = L2 .linking numberA numerical invariant oflinks in 3 space which measures the number oftimes pairs of components of a link wrap abouteach other.
See link.For a link of 2 components L = α ∪ β in R3 ,the linking number of α with β, lk(α, β), is thesum of the signed undercrossings of α with β inlogical consequencea regular projection of α ∪ β. The sign (±1) ofan undercrossing is determined by a choice ofan orientation at the undercrossings.For example, the linking of a two-componentlink (the Hopf link) is given in the figure with achoice of undercrossing orientations.αβββlogical connective Used to build new propositional (sentential) or first-order formulas fromexisting ones.
The usual logical connectivesare ∧ (and), ∨ (or), ¬ (not), → (implies), and↔ (if and only if). For example, if A and Bare well-formed propositional formulas, then soare (A ∧ B), (A ∨ B), (¬A), (A → B) and(A ↔ B).The truth tables for these logical connectivesare as follows, where T is interpreted as true andF is interpreted as false.ATF(¬A)FTATTFFBTFTF(A ∧ B)TFFFATTFFBTFTF(A ∨ B)TTTFATTFFBTFTF(A → B)TFTTATTFFBTFTF(A ↔ B)TFFT+1αα-1Left: Oriented undercrossings; Right:−1.lk(α, β) =Liouville’s functionThe arithmetic function, denoted λ, which, for any positive integer n = p1i1 .
. . pkik , returns the number λ(n) =(−1)i1 +...+ik . (See arithmetic function.) For example, λ(540) = λ(22 · 33 · 5) = (−1)6 = 1. Itis completely multiplicative.locally n-connected topological spaceAtopological space such that, for every point p,every neighborhood of p contains a smallerneighborhood of p which is n-connected. Aconnected topological space X is n-connected if for every k ≤ n, every map of the kdimensional sphere into X is homotopic to aconstant map.
For example, any manifold islocally n-connected for every n, as is every locally finite simplicial complex. The one pointunion of infinitely many n-spheres, with theweak topology, is locally n − 1-connected butnot locally n-connected.© 2001 by CRC Press LLClogical consequenceIn propositional (sentential) logic, a well-formed formula β is a logical consequence of a well-formed formula αif α logically implies β; i.e., if every truth assignment that satisfies α also satisfies β.
Forexample, if A and B are sentence symbols, thenA is a logical consequence of (A ∧ B). In addition, β is a logical consequence of a set ofwell-formed formulas if logically implies β;i.e., if every truth assignment that satisfies everymember of also satisfies β.logically equivalentFor first order logic, let L be a first order language, and let α and β be well-formed formulasof L. Then β is a logical consequence of α ifα logically implies β; i.e., if, for every structureA for L and for every s : V → A, wheneverA satisfies α with s, A also satisfies β with s.(Here, V is the set of variables of L and A isthe universe of A.) In addition, β is a logicalconsequence of a set of well-formed formulasof L if logically implies β; i.e., if, for everystructure A for L and for every s : V → A,whenever A satisfies every member of with s,A also satisfies β.logically equivalentIn propositional (sentential) logic, well-formed formulas α and βare logically equivalent if α logically impliesβ and β logically implies α; that is, if everytruth assignment either satisfies both α and β,or both ¬α and ¬β.
For example, if A andB are sentence symbols, then (¬(A ∨ B)) and((¬A) ∧ (¬B)) are logically equivalent.For first order logic, let L be a first order language, and let α and β be well-formed formulasof L. Then α and β are logically equivalent ifα logically implies β and β logically implies α;that is, if for every structure A for L and for every s : V → A, A satisfies α with s if and onlyif A satisfies β with s. (Here, V is the set ofvariables of L and A is the universe of A.)logically impliesIn propositional (sentential) logic, a well-formed formula α logically implies another well-formed formula β (notation:α |= β) if every truth assignment that satisfiesα also satisfies β. A set of well-formed formulas logically implies a well-formed formulaβ (notation: |= β) if every truth assignmentthat satisfies every member of also satisfies β.For example, if A, B, and C are sentence symbols, = {A, (A → B)}, and β = B, then logically implies β.
This notion in propositional© 2001 by CRC Press LLClogic is called “tautologically implies” by someauthors.For first order logic, let L be a first order language, and let α and β be well-formed formulas of L. Then α logically implies β (notation:α |= β) if, for every structure A for L and forevery s : V → A such that A satisfies α withs, A also satisfies β with s. (Here, V is the setof variables of L and A is the universe of A.)A set of well-formed formulas of L logicallyimplies a well-formed formula β of L (notation: |= β) if, for every structure A for L and forevery s : V → A such that A satisfies everymember of with s, A also satisfies β with s.Loop TheoremA theorem addressing theconditions in which a loop in the boundary of athree-dimensional manifold which is contractiblewithin the manifold has an equivalent embedding which bounds a disk.Specifically, let M be a compact threedimensional manifold and let N be a component of its boundary. If the kernel of the homomorphism π1 (N ) → π1 (M) is non-trivial, thenthere exists a disk D 2 ⊂ M such that ∂D 2 ⊂ Nis a simple loop that is not homotopic to zero inM.lower bound Let S be a subset of a partiallyordered set (P , ≤).
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