Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 29
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. . , xn ) in Rn such that xi ≥ 0, 1 ≤ i ≤ n. Inthe case of the plane, this is the positive quadrant.postulates of Euclid Euclid based his geometry on five basic assumptions, known as ThePostulates. The first four postulates assert thatone can join any two points by a straight line,extend straight lines continuously, and draw acircle with any given center and radius, and thatall right angles are equal.
The fifth postulate asserts the uniqueness of a straight line, through apoint, parallel to a given line.power set (of a set) The set of all subsets ofa given set. The power set of S is denoted byP(S) or 2S . For example, if S = {3, 5} thenP(S) = {Ø, {3}, {5}, {3, 5}}. The power set ofany set S, together with the operations of union,intersection, and complementation with respectto S, forms a Boolean algebra. The unit andzero element of this Boolean algebra are S andØ, respectively.
See also Cantor’s Theorem.predicate calculusThe syntactical part offirst order logic. A predicate calculus is a formalsystem consisting of a first order language, theset of all well-formed formulas, a particular set of well-formed formulas, which are calledlogical axioms, and a list of rules of deduction.The well-formed formulas that are logical axioms should be valid formulas. A typical axiomthat might occur in a predicate calculus is∀x(α → β) → (∀xα → ∀xβ),where α and β are any well-formed formulas,or, if the language contains equality,x = y → (α → α ),where α is an atomic formula, and α is obtainedfrom α by replacing the variable x in α in zeroor more places by the variable y. A typical ruleof deduction in a predicate calculus is modusponens.
See modus ponens.A predicate calculus is used to prove theorems. See also proof, theorem. While the actualchoice of logical axioms and rules of deductionis not important, it is important that a predicatecalculus be both sound (i.e., any well-formedpredicate logicformula which is provable in the formal systemshould be a logical consequence of the logicalaxioms) and complete (i.e., any logical consequence of the logical axioms should be provablein the formal system).Predicate calculus is sometimes called firstorder predicate calculus.predicate logicSee first order logic.primary cohomology operationtransformation of functorsA naturalto the positive real number x, thenlim x→∞π(x)xlog(x) =1.xThat is, if x is large, π(x) ≈ log(x).
An equivalent formulation of the theorem is that x dxπ(x)= 1, where Li(x) = 2 log(x)(thelimx→∞ Li(x)so-called logarithmic integral). The Prime Number Theorem was first proved, independently,by Jacques Hadamard and Charles de la ValéePoussin in 1896.H i (X, A; M) −→ H i+j (X, A; N ) ;an operation may be defined for many choicesof i and j and many choices of Abelian groupsM and N , or only for specific choices.
Operations are often additive. The squaring operation,which takes u to u2 , is not additive; the Steenrodsquare operations (also called reduced squares)are additive.Cohomology operations also exist on generalized cohomology theories, for example Ktheory and cobordism theories.
Adams operations on K-theory are cohomology operations.See also secondary cohomology operation.prime factorA prime p that is a divisor ofan integer n. For example, the prime factors of24 are 2 and 3. See divisor.prime idealLet S be a set. An ideal I on Sis a prime ideal if, for all X ⊆ S, either X ∈ Ior S\X ∈ I .prime number(1) An integer with exactlytwo positive integer factors (including itself and1). For example, 5 is prime because its positiveinteger factors are 1 and 5, while 6 is not primebecause the positive integer factors of 6 are 1,2, 3, and 6.
Note that the integer 1 is not primesince it has only one positive integer factor, itself.(2) More generally, an element p of a ringis prime (or irreducible) if it is not a unit andall of its factors (in the ring) are associates (unitmultiples) of p.Prime Number TheoremIf π(x) denotesthe number of prime numbers less than or equal© 2001 by CRC Press LLCprimitive recursive functionAll functionsmentioned are functions on the natural numbers.An n-ary function f is primitive recursive if itcan be derived from a certain set of initial functions by finitely many applications of composition and recursion; i.e., there is a finite sequencef0 , f1 , .
. . , fk = fof functions such that for all i, 0 ≤ i ≤ k,(i.) fi is an initial function or(ii.) fi can be obtained from {fj : 0 ≤ j <i} by composition or recursion.The following functions are initial functions.• S(x) = x + 1 (the successor function)• Cin (x1 , . . . , xn ) = i, for all natural numbers i, n ≥ 0 (the constant functions)• Pin (x1 , .
. . , xn ) = xi , for all natural numbers n ≥ 1 and 1 ≤ i ≤ n (the projectionfunctions).Let g1 , . . . , gk , f be n-ary functions and leth be a k-ary function; let x denote an n-tuplex1 , . . . , xn . The function f is obtained fromg1 , . . . , gk and h by composition if for all natural numbers x1 , . . . , xn , f (x) = h(g1 (x), .
. . ,gk (x)).Let f be an n-ary function, n ≥ 1, g be an(n−1)-ary function, h be an (n+1)-ary function,and y denote the (n−1)-tuple y1 , . . . , yn−1 . Thefunction f is obtained from g and h by recursionif for all natural numbers x, y1 , . . . , yn−1 ,f (0, y)=g(y)f (x + 1, y)=h(x, f (x, y), y).product categoryFor example, the function f (x, y) = x + yis primitive recursive. Informally, the recursionequations for f areprincipal ultrafilterAn ultrafilter U over aBoolean algebra B such that there is a b ∈ Bsuch that U = {x ∈ B : b ≤ x}.==principle of dependent choicesSuppose Ris a binary relation on a nonempty set S, and that,for every x ∈ S, there exists y ∈ S such that(x, y) ∈ R. Then there exists a countable sequence x0 , x1 , .
. . , xn , . . . (n ∈ N) of elementsin S such that (xn , xn+1 ) ∈ R, for all n ∈ N.This principle is also known as the Axiom ofDependent Choice. It is a consequence of, butis weaker than, the Axiom of Choice, and it isthe usual replacement of the Axiom of Choiceif the Axiom of Determinacy is assumed.f (0, y)f (x + 1, y)yS(f (x, y)).More formally,f (0, y)=P11 (y)f (x + 1, y)=h(x, f (x, y), y),where h(x, y, z) = S(P23 (x, y, z)).principal curvature If P is a point in a surface S in R3 , then the principal curvatures at Pare the minimum and maximum values of thecurvatures of the curves formed by intersectingS with a plane through P containing the normal vector to the surface at P .
Equivalently, theprincipal curvatures are the eigenvalues of theWeingarten map at P .principal fiber bundle A fiber bundle whosefiber is a topological group G and whose structure group is also G, acting on itself by (left)multiplication. See fiber bundle. It consists ofa base space B, a total space E, and a projection map π : E −→ B.
There is a coveringof B by open sets Ui and homeomorphisms φi :Ui ×G −→ π −1 (Ui ) such that π ◦φi (x, q) = x.This identifies π −1 (x) with G as a topological space. Examples of principal fiber bundles are constructed by taking the quotient mapπ : L −→ (L/G) from a Lie group L to thequotient space of L by a closed subgroup G. Auniversal covering map π : E −→ B is a principal bundle with the fundamental group of B(with the discrete topology) as fiber and group.principal ideal Let S be a nonempty set andlet P(S) be the power set of S. An ideal I on Sis a principal ideal if there exists a set A ⊆ Ssuch that I = {X ∈ P(S) : X ⊆ A}.principal typeA type (x̄) of a theory Tin a language L such that there is an L-formulaθ (x̄) in (x̄) such that T ∀x̄ θ (x̄) → φ(x̄)for every φ(x̄) ∈ (x̄).
That is, under T , thesingle formula θ generates the entire set .© 2001 by CRC Press LLCprinciple of inclusion-exclusionA combinatorial formula for the cardinality of the unionof a finite collection of finite sets. In the case oftwo sets, the formula is |A ∪ B| = |A| + |B| −|A ∩ B|. For r arbitrary finite sets A1 , . . . , Ar ,the formula isr|A1 ∪ · · · ∪ Ar | =|Ak1 ∩ · · · ∩ Akn | ,n+1n=1 (−1)where the second sum ranges over all n-tuplesof natural numbers (k1 , . . . , kn ) such that 1 ≤k1 < · · · < kn ≤ r.productThe general term used for the result obtained by applying some operation, usually called multiplication.
For example, productof natural numbers, product of complex numbers, product of real-valued functions, Cartesianproduct of sets, product of matrices, product ofcardinal numbers, product of ordinal numbers,product of elements of a group, product of objects in a category.product bundle Formed by taking the tensorproduct of the fibers (of two vector bundles Eand E over B) over each point of B.
Thus, thetensor product of two line bundles is again a linebundle. Line bundles over a space form a groupwith respect to this product; the group identityis the trivial bundle B × R −→ B.product categoryLet C1 , . . . , Cn be categories. The product C1 ×· · ·×Cn is the categorywhose objects are n-tuples (A1 , . . . , An ), whereeach Ai is an object of Ci , and the morphismsproduct metricare n-tuples (f1 , . .
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