Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 10
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Foreach value of S, let t, n, and b be vector fieldsalong α defined byαsf (s),t (s) = αsf (s) , n (s) = αsf (s)andb (s) = t (s) × n (s) .The derivative bsf (s) = τ (s) n (s) yields thefunction τ : I → R , a geometric entity which isthe torsion of α in the neighborhood of s. It measures the arc-rate of turning of b. Generalizingto n-dimensional conformal space, the conformal torsion of a curve can be derived from theFrenet-Serret apparatus.
The concept of torsionassociated with a moving frame along a curvewas introduced by F. Frenet in 1847 and independently by J.A. Serret in 1851.congruence of linesRefers to a set of linesin projective, affine, or Euclidean space depending on a set of parameters. For example, let P 5be a 5-dimensional projective space, and Q ahyperquadrix defined by the equation p 01 p 23 −p02 p 13 + p03 p 12 = 0, where the p ij are homogeneous coordinates of P 5 .
Then we definea congruence of lines as a set of 2-parameterstraight lines corresponding to a surface of twodimensions on Q in P 5 . The theory of congruences is an important part of projective differential geometry. See also congruent objects inspace.congruence on a categoryAn equivalencerelation ∼ on the morphisms of a category Csuch that (i.) for every equivalence class E ofcongruent objects in planemorphisms in C, there exist objects A, B in Csuch that E is contained in the class of morphisms from A to B, and (ii.) for all morphismsf, f , g, g of C, if f ∼ f and g ∼ g , thenf ◦ g ∼ f ◦ g (assuming f ◦ g and f ◦ g exist).congruent objects in planeTwo objects Pand P ∗ in R2 are congruent if there exists a rigidmotion α : R2 → R2 such that α(P ) = P ∗ .conjugate anglesTwo angles whose sum is360 degrees (2π radians).conjugate arcs Two circular arcs whose unionis a complete circle.conjunctive normal formA propositional(sentential) formula of the formminAij ;i=1congruent objects in spaceTwo objects Sand S ∗ in R3 are congruent if there exists a rigidmotion α : R3 → R3 such that α(S) = S ∗ .conic section A geometric locus obtained bytaking planar sections of a conical surface (i.e.,of a circular cone formed from the rotation ofone line around another, provided the lines arenot parallel or orthogonal).
These sections donot pass through the intersection point of the twolines which produce the circular cone. The conicsection is thus a plane curve in R3 generated by apoint that moves so that the ratio of its distancefrom a fixed point to its distance from a fixedline is constant. It can be one of three types:an ellipse (where the intersecting plane meetsall generators of the cone in the points of onlyone convex half-cone), a parabola (where theintersecting plane is parallel to one of the tangentplanes to the cone going off to infinity), or ahyperbola (where the intersecting plane meetsboth half-cones).conical helixA space curve that lies on thesurface of a cone and cuts all the generators at aconstant angle.
See cone.conical surfaceA surface of revolution ofconstant curvature in R3 . It can be generatedby a straight line that connects a fixed point (thevertex) with each point of a fixed curve (the directrix). The conical surface consists of twoconcave pieces positioned symmetrically aboutthe vertex. It is quadric if the directrix is a conic.A circular conical surface is one whose directrixis a circle and whose vertex is on the line perpendicular to the plane of the circle and passingthrough the center of the circle.© 2001 by CRC Press LLCj =1i.e.,(A11 ∨ · · · ∨ A1m1 ) ∧ · · · ∧ (An1 ∨ · · · ∨ Anmn ),where each Aij , 1 ≤ i ≤ n, 1 ≤ j ≤ mi , iseither a sentence symbol or the negation of asentence symbol.
Every well-formed propositional formula is logically equivalent to one inconjunctive normal form.For example, if A, B, and C are sentencesymbols, then the well-formed formula ((A →B) → C) is logically equivalent to (¬A∨¬B ∨C) ∧ (A ∨ ¬B ∨ C) ∧ (A ∨ B ∨ C), which is aformula in conjunctive normal form.connected A subset A of a topological spaceX is connected if it is connected as a subspaceof X. In other words, A is connected if there donot exist nonempty disjoint sets U and V , whichare relatively open in A such that A = U ∪ V .That is, there are no disjoint open sets U and Vin X with U ∩ A = ∅, V ∩ A = ∅, and A =(U ∩A)∪(V ∩A) = (U ∪V )∩A.
For example,(0, 1) is connected in R, while [0, 0.5)∪(0.5, 1]is not, because [0, 0.5) and (0.5, 1] are relativelyopen in that space.connected component In a topological spaceX, the connected component of x ∈ X is thelargest connected A ⊆ X which contains x.Equivalently, A is the union of all connected setsin X which contain x. Any space can be partitioned into its connected components, whichmust be disjoint.connected im kleinenA topological spaceX is connected im kleinen if, for any x ∈ X andopen set U containing x, there is a connectedA ⊆ U and an open V ⊆ A with x ∈ V .
Thiscontact formform of connectedness is stronger than beinglocally connected.connected setSee connected.connected sum An n-dimensional manifoldformed from n-dimensional manifolds M1 andM2 (and denoted M1 #M2 ) as follows: Let B1and B2 be closed n-dimensional balls in M1and M2 , respectively. Let h : S1 −→ S2 be ahomeomorphism of the boundary sphere of B1to the boundary of B2 . Then M1 #M2 is the unionof M1 minus the interior of B1 and M2 minus theinterior of B2 , with each x in S1 identified withh(x) in S2 .
The resulting space is a manifold;different choices of balls and homeomorphismsmay give rise to inequivalent manifolds.connected topological spaceA topologicalspace X such that there do not exist nonemptydisjoint open sets U and V such that X = U ∪V .Equivalently, the only subsets of X that are bothopen and closed are ∅ and X itself.connective fiber spaceA fiber space thatcannot be represented as the sum of two nonempty disjoint open-closed subsets.
Connectivity is preserved under homeomophisms.conservativeLet L1 and L2 be first orderlanguages with L1 ⊆ L2 ; let T1 be a theory ofL1 and T2 be a theory of L2 . The theory T2 is aconservative extension of T1 if(i.) T2 is an extension of T1 ; i.e., T1 ⊆ T2 ,and(ii.) for every sentence σ of L1 , if σ is atheorem of T2 (T2 σ ), then σ is a theorem ofT1 (T1 σ ).consistent Let L be a first order language andlet be a set of well-formed formulas of L.
Theset is consistent if it is not inconsistent; i.e., ifthere does not exist a formula α such that both αand (¬α) are theorems of . If is consistent,then there must exist a formula α such that α isnot a theorem of . Note that is consistentif and only if is satisfiable, by soundness andcompleteness of first order logic.If is a set of sentences and ϕ is a wellformed formula, then ϕ is consistent with if has a model that is also a model of ϕ.© 2001 by CRC Press LLCconsistent axioms A set of axioms such thatthere is no statement A such that both A and itsnegation are provable from the set of axioms.Informally, a collection of axioms is consistentif there is a model for the axioms; the axioms ofgroup theory are consistent as G = {e}, where ·is defined on G by e · e = e, is a model of theseaxioms.In the case of formal systems, Gödel’s Second Incompleteness Theorem states, roughly,that the consistency of any sufficiently strongtheory cannot be proved in that theory; for example, it is not possible to prove the consistencyof Zermelo-Fraenkel (ZF ) set theory from theaxioms of ZF .
Consequently, only a relativenotion of consistency can be considered; i.e.,given a set of axioms of a formal system anda statement A in the language of that system,one asks whether ∪ {A} is consistent, assuming that is consistent. For example, Gödelproved in 1936 that, assuming ZF is consistent,so is ZF C, where ZF C is ZF set theory together with the Axiom of Choice. In addition,Cohen proved in 1963 that, assuming ZF is consistent, so is ZF + ¬AC, where ZF + ¬AC isZF set theory together with the negation of theAxiom of Choice.constructible set An element of the class L,defined below in the constructible hierarchy:(i.) L0 = Ø;(ii.) Lα = β<α Lβ , if α is a limit ordinal;(iii.) Lα+1 = the set of all subsets definableover Lα ;(iv.) L = α∈Ord Lα .contact elementIn the Euclidean plane R2 ,an ordered pair (p, ) consisting of a point p anda line containing the point p.
More generally,a contact element in a smooth n-dimensionalmanifold M is a pair (p, H ) consisting of a pointp in M and an n − 1-dimensional plane H in thetangent space at p. The contact elements in Mform a (2n−1)-dimensional manifold which hasa special structure on it called a contact structure.contact formA one-form ω on a smooth(2n + 1)-dimensional manifold M such that ω ∧(dω)n = 0 everywhere on M. At each point pin M, the kernel K(p) of ω(p) is a plane ofdimension 2n in the tangent space at p. Thecontact manifoldcondition on ω is equivalent to the statement:for any vector V ∈ K(p) there is a W such thatdω(V , W ) = 0. Darboux’s Theorem says thatit is always possible to find local coordinates(x 1 , .
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