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# Dictionary of-algebra-,arithmetic-and-trigonometry-(p)

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### Dictionary of-algebra-,arithmetic-and-trigonometry-(p)

1. 1. such that ∂p−1,q ◦ ∂p,q = ∂p,q−1 ◦ ∂p,q P = ∂p,q−1 ◦ ∂p,q + ∂p−1,q ◦ ∂p,q = 0 . We deﬁne the associated chain complex (Xn , ∂)p-adic numbers The completion of the ratio- by settingnal ﬁeld Q with respect to the p-adic valuation Xn = Xp,q , ∂n = ∂p,q + ∂p,q .|·|p . See p-adic valuation. See also completion. p+q=n p+q=np-adic valuation For a ﬁxed prime integer p, We call ∂ the total boundary operator, and ∂ ,the valuation |·|p , deﬁned on the ﬁeld of rational ∂ the partial boundary operators.numbers as follows. Write a rational number inthe form pr m/n where r is an integer, and m, n partial derived functor Suppose F is a func-are non-zero integers, not divisible by p. Then tor of n variables. If S is a subset of {1, . . . , n},|p r m/n|p = 1/p r . See valuation. we consider the variables whose indices are in S as active and those whose indices are inparabolic subalgebra A subalgebra of a Lie {1, . . . n}S as passive. By ﬁxing all the passivealgebra g that contains a maximal solvable sub- variables, we obtain a functor FS in the activealgebra of g. variables. The partial derived functors are then deﬁned as the derived functors R k FS . See alsoparabolic subgroup A subgroup of a Lie functor, derived functor.group G that contains a maximal connectedsolvable Lie subgroup of G. An example is the partial differential The rate of change of asubgroup of invertible upper triangular matrices function of more than one variable with respectin the group GLn (C) of invertible n×n matrices to one of the variables while holding all of thewith complex entries. other variables constant.parabolic transformation A transformation partial fraction An algebraic expression ofof the Riemann sphere whose ﬁxed points are ∞ the formand another point. nj aj mparaholic subgroup A subgroup of a Lie m .group containing a Borel subgroup. j m=1 z − αjparametric equations The name given toequations which specify a curve or surface by partially ordered space Let X be a set. Aexpressing the coordinates of a point in terms of relation on X that satisﬁes the conditions:a third variable (the parameter), in contrast with (i.) x ≤ x for all x ∈ Xa relation connecting x, y, and z, the cartesian (ii.) x ≤ y and y ≤ x implies x = ycoordinates. (iii.) x ≤ y and y ≤ z implies x ≤ z is called a partial ordering.partial boundary operator We call (Xp,q ,∂ , ∂ ) over A a double chain complex if it is partial pivoting An iterative strategy, usinga family of left A-modules Xp,q for p, q ∈ Z pivots, for solving the equation Ax = b, wheretogether with A-automorphisms A is an n × n matrix and b is an n × 1 matrix. In the method of partial pivoting, to obtain the ∂p,q : Xp,q → Xp−1,q matrix Ak (where A0 = A), the pivot is chosen to be the entry in the kth column of Ak−1 atand or below the diagonal with the largest absolute ∂p,q : Xp,q → Xp,q−1 value.c 2001 by CRC Press LLC
2. 2. partial product Let {αn }∞ be a given se- n=1 1 − e and e are orthogonal idempotent elements,quence of numbers (or functions deﬁned on a andcommon domain in Rn or Cn ) with terms R = eR + (1 − e)Rαn = 0 for all n ∈ N. The formal inﬁnite prod- is the direct sum of left ideals. This is calleduct α1 · α2 · · · is denoted by ∞ αj . We call j =1 Peirce’s right decomposition. n Pell’s equation The Diophantine equations Pn = αj x 2 − ay 2 = ±4 and ±1, where a is a positive j =1 integer, not a perfect square, are called Pell’sits nth partial product. equations. The solutions of such equations can be found by continued fractions and are used inpeak point See peak set. the determination of the units of rings such as √ Z[ a]. This equation was studied extensivelypeak set Let A be an algebra of functions by Gauss. It can be regarded as a starting pointon a domain ⊂ Cn . We call p ∈ a peak of modern algebraic number theory.point for A if there is a function f ∈ A such that When a < 0, then Pell’s equation has onlyf (p) = 1 and |f (z)| < 1 for all z ∈ {p}. ﬁnitely many solutions. If a > 0, then all solu-The set P(A) of all peak points for the algebra tions xn , yn of Pell’s equation are given byA is called the peak set of A. √ n √ x1 + ay1 xn + ayn ± = ,Peirce decomposition Let A be a semisimple 2 2Jordan algebra over a ﬁeld F of characteristic 0 provided that the pair x1 , y1 is a solution with √and let e be an idempotent of A. For λ ∈ F , let the smallest x1 + ay1 > 1. Using continuedAe (λ) = {a ∈ A : ea = λa}. Then fractions, we can determine x1 , y1 explicitly. A = Ae (1) ⊕ Ae (1/2) ⊕ Ae (0) . penalty method of solving non-linear pro- gramming problem A method to modify aThis is called the Peirce decomposition of A, constrained problem to an unconstrained prob-relative to E. If 1 is the sum of idempotents ej , lem. In order to minimize (or maximize) a func-let Aj,k = Aej (1) when j = k and Aej ∩ Aek tion φ(x) on a set which has constraints (suchwhen j = k. These are called Peirce spaces, as f1 (x) ≥ 0, f2 (x) ≥ 0, . . . fm (x) ≥ 0), aand A = ⊕j ≤k Aj,k . See also Peirce space. penalty or penalty function, ψ(x, a), is intro- duced (where a is a number), where ψ(x, a) = 0Peirce’s left decomposition Let e be an idem- if x ∈ X or ψ(x, a) > 0 if x ∈ X and ψ in- /potent element of a ring R with identity 1. Then volves f1 (x) ≥ 0, f2 (x) ≥ 0, . . . fm (x) ≥ 0. Then, one minimizes (or maximizes) φa (x) = R = Re ⊕ R(1 − e) φ(x) + ψ(x, a) without the constraints.expresses R as a direct sum of left ideals. Thisis called Peirce’s left decomposition. percent Percent means hundredths. The symbol % stands for 100 . We may write a per-Peirce space Suppose that the unity element cent as a fraction with denominator 100. For ex-1 ∈ K can be represented as a sum of the mutu- ample, 31% = 100 , 55% = 100 , . . . etc. Simi- 31 55ally orthogonal idempotents ej . Then, putting larly, we may write a fraction with denominator 100 as a percent.Aj,j = Aej (1), Aj,k = Aej (1/2)∩Aek (1/2) , perfect ﬁeld A ﬁeld such that every algebraicwe have A = j ≤k ⊕Aj,k . Then Aj,k are extension is separable. Equivalently, a ﬁeld F iscalled Peirce spaces. perfect if each irreducible polynomial with co- efﬁcients in F has no multiple roots (in an alge-Peirce’s right decomposition Let e be an braic closure of F ). Every ﬁeld of characteristicidempotent element of a unitary ring R, then 0 is perfect and so is every ﬁnite ﬁeld.c 2001 by CRC Press LLC
3. 3. perfect power An integer or polynomial corresponding to the sign of the permutation iswhich can be written as the nth power of another missing from each summand.integer or polynomial, where n is a positive in-teger. For example, 8 is a perfect cube, because permutation group Let A be a ﬁnite set with8 = 23 , and x 2 + 4x + 4 is a perfect square, #(A) = n. The permutation group on n ele-because x 2 + 4x + 4 = (x + 2)2 . ments is the set Sn consisting of all one-to-one functions from A onto A under the group law:period matrix Let R be a compact Riemannsurface of genus g. Let ω1 , . . . , ωg be a ba- f ·g =f ◦gsis for the complex vector space of holomorphicdifferentials on R and let α1 , . . . , α2g be a ba- for f, g ∈ Sn . Here ◦ denotes the compositionsis for the 1-dimensional integral homology of of functions.R. The period matrix M is the g × 2g matrixwhose (i, j )-th entry is the integral of ωj over permutation matrix An n × n matrix P ,αi . The group generated by the 2g columns of obtained from the identity matrix In by permu-M is a lattice in Cg and the quotient yields a g- tations of the rows (or columns). It follows that adimensional complex torus called the Jacobian permutation matrix has exactly one nonzero en-variety of R. try (equal to 1) in each row and column. There are n! permutation matrices of size n × n. Theyperiod of a periodic function Let f be a are orthogonal matrices, namely, P T P = P P Tfunction deﬁned on a vector space V satisfying = In (i.e., P T = P −1 ). Multiplication fromthe relation the left (resp., right) by a permutation matrix permutes the rows (resp., columns) of a matrix, f (x + ω) = f (x) corresponding to the original permutation.for all x ∈ V and for some ω ∈ V . The number permutation representation A permutationω is called a period of f (x), and f (x) with a representation of a group G is a homomorphismperiod ω = 0 is call a periodic function. from G to the group SX of all permutations of a set X. The most common example is when X =period relation Conditions on an n×n matrix G and the permutation of G obtained from g ∈which help determine when a complex torus is G is given by x → gx (or x → xg, dependingan Abelian manifold. In Cn , let be generated on whether a product of permutations is readby (1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, . . . , right-to-left or left-to-right).0, 1), (a11 , a12 , . . . , a1n ), (a21 , a22 , . . . , a2n ),. . . (an1 , an2 , . . . , ann ). Then Cn / is an Peron-Frobenius Theorem See FrobeniusAbelian manifold if there are integers d1 , d2 , . . . , Theorem on Non-Negative Matrices.dn = 0 such that, if A = (aij ) and D = (δij di ),then (i.) AD is symmetric; and (ii.) (AD) is Perron’s Theorem of Positive Matrices Ifpositive symmetric. Conditions (i.) and (ii.) are A is a positive n × n matrix, A has a positivethe period relations. real eigenvalue λ with the following properties: (i.) λ is a simple root of the characteristic equa-permanent Given an m×n matrix A = (aij ) tion.with m ≤ n, the permanent of A is deﬁned by (ii.) λ has a positive eigenvector u. (iii.) If µ is any other eigenvalue of A, then permA = a1i1 a2i2 . . . amim , |µ| < λ.where the summation is taken over all m- Peter-Weyl theory Let G be a compact Liepermutations (i1 , i2 , . . . , im ) of the set {1, 2, group and let C(G) be the commutative asso-. . . , n}. When A is a square matrix, the per- ciative algebra of all complex valued continuousmanent therefore has an expansion similar to functions deﬁned on G. The multiplicative lawthat of the determinant, except that the factor deﬁned on C(G) is just the usual compositionc 2001 by CRC Press LLC
4. 4. of functions. Denote (1856–1941). The ﬁrst Picard theorem was   proved in 1879: An entire function which is not a   polynomial takes every value, with one possibles(G) = f ∈ C(G) : dim CLg f < ∞   exception, an inﬁnity of times. g∈G The second Picard theorem was proved inwhere Lg f = f (g·). The Peter-Weyl theory 1880: In a neighborhood of an isolated essen-tells us that the subalgebra s(G) is everywhere tial singularity, a single-valued, holomorphicdense in C(G) with respect to the uniform norm function takes every value, with one possible ex- f ∞ = maxg∈G |f (g)|. ception, an inﬁnity of times. In other words, if f (z) is holomorphic for 0 < |z − z0 | < r, andPfafﬁan differential form The name given there are two unequal numbers a, b, such thatto the expression f (z) = a, f (z) = b, for |z − z0 | < r, then z0 is not an essential singularity. n dW = Xi dxi . Picard variety Let V be a complete normal i=1 variety. The factor group of the divisors on V , algebraically equivalent to 0 modulo the group of divisors linearly equivalent to 0, has a naturalp-group A group G such that the order of canonical structure of an Abelian variety, calledG is p n , where p is a prime number and n is a the Picard variety.non-negative integer. Picard-Vessiot theory One of two main the-Picard-Lefschetz transform Let W be a lo- ories of differential rings and ﬁelds. See Galoiscal system attached to the monodromy repre- theory of differential ﬁelds. The Picard-Vessiotsentation ϕp : π1 (U, 0) → GL(H p (W, Q)). theory deals with linear homogeneous differen-For each point tj there corresponds a cycle δj of tial equations.H n−1 (W, Q) called a vanishing cycle such thatif γj is a loop based at 0 going once around tj , pi-group Let π be a set of prime numbers andwe have for each x ∈ H n−1 (W, Q), let π be the set of prime numbers not in π . A π - ϕp γj (x) = x ± x, δj δj . group is a ﬁnite group whose order is a product of primes in π . A ﬁnite group is π -solvable if every Jordan-Hölder factor is either a π -groupPicard number Let V be a complete nor- or a solvable π -group. For a π -solvable groupmal variety and let D(V ), Da (V ) be the group G, deﬁne a series of subgroupsof divisors and group of divisors algebraicallyequivalent to zero, respectively. The rank of 1 = P0 ⊆ N0 ⊂ P1 ⊂ N1 · · · ⊂ Pn ⊆ Nn = Gthe quotient group N S(V ) = D(V )/Da (V ) iscalled the Picard number of V . such that Pj /Nj −1 is a maximal normal π - subgroup of G/Pj . This is called the π -series ∗Picard scheme Let OV be the sheaf of mul- of G and n is called the π -length of G.tiplicative group of the invertible elements inOV . The group of linear equivalence classes pi-length See pi-group.of Cartier divisors can be identiﬁed withH 1 (V , OV ). From this point of view, we can pi-series See pi-group.generalize the theory of the Picard variety to thecase of schemes. The theory thus obtained is pi-solvable group A ﬁnite group G such thatcalled the theory of Picard schemes. the order of each composition factor of G is ei- ther an element of a collection, π , of prime num-Picard’s Theorem There are two important bers or mutually prime to any element of π .theorems in one complex variable proved bythe French mathematician Charles Émile Picard pivot See Gaussian elimination.c 2001 by CRC Press LLC
5. 5. pivoting See Gaussian elimination. bers Pi = (iK), (i = 2, 3, . . . ). The plurigen- era Pi , (i = 2, 3, . . . ) are the same for any twoplace A mapping φ : K → {F, ∞}, where K birationally equivalent nonsingular surfaces.and F are ﬁelds, such that, if φ(a) and φ(b) aredeﬁned, then φ(a +b) = φ(a)+φ(b), φ(ab) = plus sign The symbol “+” indicating the al-φ(a)φ(b) and φ(1) = 1. gebraic operation of addition, as in a + b.place value The value given to a digit, de- Poincaré Let R be a commutative ring withpending on that digit’s position in relation to the unit. Let U be an orientation over R of a com-units place. For example, in 239.71, 9 repre- pact n-manifold X with boundary. Then for allsents 9 units, 3 represents 30 units, 2 represents indices q and R-modules G there is an isomor- 7200 units, 7 represents 10 units and 1 represents phism 1100 units. γU : Hq (X; G) ≈ H n−q (X; G) .Plancherel formula Let G be a unimodular ˆlocally compact group and G be its quasidual. This is called Poincaré-Lefschetz duality. TheLet U be a unitary representation of G and U ∗ analogous result for a manifold X without bound-be its adjoint. For any f , g ∈ L1 (G) ∩ L2 (G), ary is called Poincaré duality.the Plancherel formula Poincaré-Birkhoff-Witt Theorem Let G be f (x)g(x) dx = ∗ t (Ug (ξ )Uf (ξ )) dµ(ξ ) a Lie algebra over a number ﬁeld K. Let X1 , . . . , G ˆ G Xn be a basis of G, and let R = K[Y1 , . . . , Yn ] be a polynomial ring on K in n indeterminatesholds, where Uf (ξ ) = G f (x)Ux (ξ )dx. The ˆ Y1 , . . . , Yn . Then there exists a unique alge-measure µ is called the Plancherel measure. bra homomorphism ψ : R → G such that ψ(1) = 1 and ω(Yj ) = Xj , j = 1, . . . , n.plane trigonometry Plane trigonometry is Moreover, ψ is bijective, and the j th homoge-related to the study of triangles, which were neous component Rj is mapped by ψ onto G j .studied long ago by the Babylonians and an- k k k Thus, the set of monomials {X11 X22 . . . Xnn },cient Greeks. The word trigonometry is derived k1 , . . . , kn ≥ 0, forms a basis of U (G) overfrom the Greek word for “the measurement of K. This is the so-called Poincaré-Birkhoff-Witttriangles.” Today trigonometry and trigonomet- Theorem. Here U (G) = T (G)/J is the quotientric functions are indispensable tools not only in associative algebra of G where J is the two-sidedmathematics, but also in many practical appli- ideal of T (G) generated by all elements of thecations, especially those involving oscillations form X ⊗ Y − Y ⊗ X − [X, Y ] and T (G) is theand rotations. tensor algebra over G.Plücker formulas Let m be the class, n thedegree, and δ, χ, i, and τ be the number of Poincaré differential invariant Let w =nodes, cups, inﬂections, tangents, and bitan- α(z − z◦ )/(1 − z◦ z) with |α| = 1 and |z◦ | <gents. Then 1, be a conformal mapping of |z| < 1 onto |w| < 1. Then the quantity |dw|/(1 − |w|2 ) = n(n − 1) = m + 2δ + 3χ |dz|/(1 − |z|2 ) is called Poincaré’s differen- tial invariant. The disk {|z| < 1} becomes m(m − 1) = n + 2τ + 3i a non-Euclidean space using any metric with 3n(n − 2) = i + 6δ + 8χ ds = |dz|/(1 − |z|2 ). 3m(m − 2) = χ + 6τ + 8i 3(m − n) = i − χ . Poincaré duality Any theorem general- izing the following: Let M be a com- pact n-dimensional manifold without bound-plurigenera For an algebraic surface S with a ary. Then, for each p, there is an isomor-canonical divisor K of S, the collection of num- phism H p (M; Z2 ) ∼ Hn−p (M; Z2 ). If, in =c 2001 by CRC Press LLC
6. 6. addition, M is assumed to be orientable, then be simple if W has no nonzero proper subco-H p (M) ∼ Hn−p (M). = algebra. The co-algebra V is called a pointed co-algebra if all of its simple subco-algebras arePoincaré-Lefschetz duality Let R be a com- one-dimensional. See coalgebra.mutative ring with unit. Let U be an orientationover R of a compact n-manifold X with bound- pointed set Denoted by (X, p), a set X whereary. Then for all indices q and R-modules G p is a member of X.there is an isomorphism polar decomposition Every n × n matrix A γU : Hq (X; G) ≈ H n−q (X; G) . with complex entries can be written as A = P U , where P is a positive semideﬁnite matrix and UThis is called Poincaré-Lefschetz duality. The is a unitary matrix. This factorization of A isanalogous result for a manifold X without bound- called the polar decomposition of the polar formary is called Poincaré duality. of A.Poincaré metric The hermitian metric polar form of a complex number Let z = 2 x + iy be a complex number. This number has ds 2 = dz ∧ dz (1 − |z|2 )2 the polar representationis called the Poincaré metric for the unit disc in z = x + iy = r(cos θ + i sin θ ) ythe complex plane. where r = x 2 + y 2 and θ = tan−1 x .Poincaré’s Complete Reducibility Theorem polarization Let A be an Abelian variety andA theorem which says that, given an Abelian let X be a divisor on A. Let X be a divisor onvariety A and an Abelian subvariety X of A, A such that m1 X ≡ m2 X for some positive in-there is an Abelian subvariety Y of A such that tegers m1 and m2 . Let X be the class of all suchA is isogenous to X × Y . divisors X . When X contains positive nonde- generate divisors, we say that X determines apoint at inﬁnity The point in the extended polarization on A.complex plane, not in the complex plane itself.More precisely, let us consider the unit sphere polarized Abelian variety Suppose that Vin R3 : is an Abelian variety. Let X be a divisor on V and let D(X) denote the class of all divisorsS = (x1 , x2 , x3 ) ∈ R3 : x1 + x2 + x3 = 1 , 2 2 2 Y on V such that mX ≡ nY , for some inte- gers m, n > 0. Further, suppose that D(X) de-which we deﬁne as the extended complex num- termines a polarization of V . Then the couplebers. Let N = (0, 0, 1); that is, N is the north (V , D(X)) is called a polarized Abelian variety.pole on S. We regard C as the plane {(x1 , x2 , 0) See also Abelian variety, divisor, polarization.∈ R3 : x1 , x2 ∈ R} so that C cuts S along theequator. Now for each point z ∈ C consider the pole Let z = a be an isolated singularity ofstraight line in R3 through z and N . This in- a complex-valued function f . We call a a poletersects the sphere in exactly one point Z = N . of f ifBy identifying Z ∈ S with z ∈ C, we have S lim |f (z)| = ∞ .identiﬁed with C ∪ {N }. If |z| > 1 then Z is in z→athe upper hemisphere and if |z| < 1 then z is in That is, for any M > 0 there is a number ε >the lower hemisphere; also, for |z| = 1, Z = z. 0, such that |f (z)| ≥ M whenever 0 < |z −Clearly Z approaches N when |z| approaches a| < ε. Usually, the function f is assumed to∞. Therefore, we may identify N and the point be holomorphic, in a punctured neighborhood∞ in the extended complex plane. 0 < |z − a| < .pointed co-algebra Let V be a co-algebra. pole divisor Suppose X is a smooth afﬁne va-A nonzero subco-algebra W of V is said to riety of dimension r and suppose Y ⊂ X is a sub-c 2001 by CRC Press LLC
7. 7. variety of dimension r − 1. Given f ∈ C(X) the usual addition and multiplication of polyno-(0), let ordY f < 0 denote the order of vanish- mials. The ring R[X] is called the polynomialing of f on Y . Then (f ) = Y (ordY f ) · Y is ring of X over R.called a pole divisor of f in Y . See also smoothafﬁne variety, subvariety, order of vanishing. polynomial ring in m variables Let R be a ring and let X1 , X2 , . . . , Xm be indetermi-polynomial If a0 , a1 , . . . , an are elements nates. The set R[X1 , X2 , . . . , Xm ] of all poly-of a ring R, and x does not belong to R, then nomials in X1 , X2 , . . . , Xm with coefﬁcients in R is a ring with respect to the usual addition and a 0 + a1 x + · · · + an x n multiplication of polynomials and is called the polynomial ring in m variables X1 , X2 , . . . , Xmis a polynomial. over R.polynomial convexity Let ⊆ Cn be a do-main (a connected open set). If E ⊆ is a Pontrjagin class Let F be a complex PLsubset, then deﬁne sheaf over a PL manifold M. The total Pontrja- gin class p([F]) ∈ H 4∗ (M; R) of a coset [F] E = {z ∈ : |p(z)| ≤ sup |p(w)| of real PL sheaves via complexiﬁcation of [F] w∈E satisﬁes these axioms: (i.) If [F] is a coset of real PL sheaves of rank for all p a polynomial} . m on a PL manifold M, then the total Pontrja- gin class p([F]) is an element 1 + p1 ([F]) +The set E is called the polynomially convex hull · · · + p[m/2] ([F]) of H ∗ (M; R) with pi ([F]) ∈of E in . If the implication E ⊂⊂ implies H 4i (M; R);E ⊂⊂ always holds, then is said to be (ii.) p( ! [F]) = ∗ p([F]) ∈ H 4∗ (N ; R) forpolynomially convex. any PL map : N → M; (iii.) p([F] ⊕ [G]) = p([G]) for any cosets [F]polynomial equation An equation P = 0 and [G] over M;where P is a polynomial function of one or more (iv.) If [F] contains a bona ﬁde real vector bun-variables. dle ξ over M, then p([F]) is the classical total Pontrjagin class p(ξ ) ∈ H 4∗ (M; R).polynomial function A function which is aﬁnite sum of terms of the form an x n , where n isa nonnegative integer and an is a real or complex Pontryagin multiplication A multiplicationnumber. h∗ : H∗ (X) ⊗ H∗ (X) → H∗ (X) .polynomial identity An equation P (X1 , X2 ,. . . , Xn ) = 0 where P is a polynomial in n (H∗ (X) are homology groups of the topologicalvariables with coefﬁcients in a ﬁeld K such that space X.)P (a1 , a2 , . . . , an ) = 0 for all ai in an algebra Aover K. Pontryagin product The result of Pontrya-polynomial in m variables A function which gin multiplication. See Pontryagin multiplica- n n n tion.is a ﬁnite sum of terms ax1 1 x2 2 . . . xmm , wheren1 , n2 , . . . , nm are nonnegative integers and a isa real or complex number. For example, 5x 2 y 3 + positive angle Given a vector v = 0 in Rn ,3x 4 z − 2z + 3xyz is a polynomial in three vari- then its direction is described completely by theables. angle α between v and i = (1, 0, . . . , 0), the unit vector in the direction of the positive x1 -axis. Ifpolynomial ring Let R be a ring. The set we measure the angle α counterclockwise, weR[X] of all polynomials in an indeterminate X say α is a positive angle. Otherwise, α is a neg-with coefﬁcients in R is a ring with respect to ative angle.c 2001 by CRC Press LLC
8. 8. positive chain complex A chain complex X positive Weyl chamber The set of λ ∈ V ∗such that the only possible non-zero terms Xn such that (β, λ) > 0 for all positive roots β,are those Xn for which n ≥ 0. where V is a vector space over a subﬁeld R of the real numbers.positive cycle An r-cycle A = ni Ai suchthat ni ≥ 0 for all i, where Ai is not in the power Let a1 , . . . , an be a ﬁnite sequence ofsingular locus of an irreducible variety V for all elements of a monoid M. We deﬁne the “prod-i. uct” of a1 , . . . , an by the following: we deﬁne 1 j =1 aj = a1 , andpositive deﬁnite function A complex val-  ued function f on a locally compact topological k+1 kgroup G such that aj =  aj  ak+1 . j =1 j =1 f (s − t)φ(s)φ(t)dsdt ≥ 0 G Then k m k+mfor every φ, continuous and compactly supported aj ak+ = aj .on G. j =1 =1 j =1 If all the aj = a, we denote a1 · a2 . . . an as a npositive deﬁnite matrix An n × n matrix A, and call this the nth power of a.such that, for all u ∈ Rn , we have power associative algebra A distributive al- (A(u), u) ≥ 0 , gebra A such that every element of A generates an associative subalgebra.with equality only when u = 0. power method of computing eigenvaluespositive divisor A divisor that has only pos- An iterative method for determining the eigen-itive coefﬁcients. value of maximum absolute value of an n × n matrix A. Let λ1 , λ2 , . . . , λn be eigenval-positive element An element g ∈ G, where ues of A such that |λ1 | > |λ2 | ≥ · · · ≥ |λn |G is an ordered group, such that g ≥ e. and let y1 be an eigenvector such that (λ1 I − A)y1 = 0. Begin with a vector x (0) such thatpositive exponent For an expression a b , the (0)exponent b if b > 0. (y1 , x (0) ) = 0 and for some i0 , xi0 = 1. De- termine θ (0) , θ (1) , . . . , θ (m) , . . . and x (1) , x (2) ,positive matrix An n × n matrix A with real . . . , x (m+1) , . . . by Ax (j ) = θ (j ) x (j +1) . Thenentries such that aj k > 0 for each j and k. See limj →∞ θ (j ) = λ1 and limj →∞ x (j ) is thealso positive deﬁnite matrix. eigenvector corresponding to λ1 .positive number A real number greater than power of a complex number Let z = x +zero. iy = r(cos θ + i sin θ ) be a complex number y with r = x 2 + y 2 and θ = tan−1 x . Let n bepositive root Let S be a basis of a root system a positive number. The nth power of z will beφ in a vector space V such that each root β can be the complex number r n (cos nθ + i sin nθ ).written as β = a∈S ma a, where the integersma have the same sign. Then β is a positive root power-residue symbol Let n be a positiveif all ma ≥ 0. integer and let K be an algebraic number ﬁeld containing the nth roots of unity. Let α ∈ K ×positive semideﬁnite matrix An n×n matrix and let ℘ be a prime ideal of the ring such thatA such that, for all u ∈ Rn , we have ℘ is relatively prime to n and α. The nth power is a positive integer and let K be an algebraic (A(u), u) ≥ 0 . number ﬁeld containing the nth roots of unity.c 2001 by CRC Press LLC
9. 9. Let α ∈ K × and let ℘ be a prime ideal of the groups (or rings, modules, etc.). There is a stan-ring of integers of K such that ℘ is relatively dard procedure for constructing a sheaf from aprime to n and α. The nth power residue symbol presheaf. α ℘ is the unique nth root of unity that is n primary Abelian group An Abelian groupcongruent to α (N℘−1)/n mod ℘. When n = 2 in which the order of every element is a powerand K = Q, this symbol is the usual quadratic of a ﬁxed prime number.residue symbol. primary component Let R be a commuta-predual Let X and Y be Banach spaces such tive ring with identity 1 and let J be an idealthat X is the dual of Y , X = Y ∗ . Then Y is of R. Assume J = I1 ∩ · · · ∩ In with each Iicalled the predual of X. primary and with n minimal among all such rep- resentations. Then each Ii is called a primarypreordered set A structure space for a non- component of J .empty set R is a nonempty collection X of non-empty proper subsets of R given the hull-kernel primary ideal Let R be a ring with identitytopology. If there exists a binary operation ∗ on 1. An ideal I of R is called primary if I = RR such that (R, ∗) is a commutative semigroup and all zero divisors of R/I are nilpotent.and the structure space X consists of prime semi-group ideals, then it is said that R has an X - primary linear programming problem Acompatible operation. For p ∈ R, let Xp = linear programming problem in which the goal{A ∈ X : p ∈ A}. A preorder (reﬂexive and / is to maximize the linear function z = cx withtransitive relation) ≤ is deﬁned on R by the rule the linear conditions n=1 aij xj = bi (i = jthat a ≤ b if and only if Xa ⊆ Xb . Then R is 1, 2, . . . , m) and x ≥ 0, where x = (x1 , x2 , . . . ,called a preordered set. xn ) is the unknown vector, c is an n × 1 vec- tor of real numbers, bi (i = 1, 2, . . . , n) andpreordered set A structure space for a non- aij (i = 1, 2, . . . , n, j = 1, 2, . . . , n) are realempty set R is a nonempty collection X of non- numbers.empty proper subsets of R given the hull-kerneltopology. If there exists a binary operation ∗ on primary ring Let R be a ring and let N beR such that (R, ∗) is a commutative semigroup the largest ideal of R containing only nilpotentand the structure space X consists of prime semi- elements. If R/N is nonzero and has no nonzerogroup ideals, then it is said that R has an X - proper ideals, R is called primary.compatible operation. For p ∈ R, let Xp ={A ∈ X : p ∈ A}. A preorder (reﬂexive and / primary submodule Let R be a commuta-transitive relation) ≤ is deﬁned on R by the rule tive ring with identity 1. Let M be an R-module.that a ≤ b if and only if Xa ⊆ Xb . Then R is A submodule N of M is called primary if when-called a preordered set. ever r ∈ R is such that there exists m ∈ M/N with m = 0 but rm = 0, then r n (M/N ) =presheaf Let X be a topological space. Sup- 0 for some integer n.pose that, for each open subset U of X, there isan Abelian group (or ring, module, etc.) F(U ). prime A positive integer greater than 1 withAssume F(φ) = 0. In addition, suppose that the property that its only divisors are 1 and itself.whenever U ⊆ V there is a homomorphism The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 27 are the ﬁrst ten primes. There are inﬁnitely many ρU V : F(V ) → F(U ) prime numbers.such that ρU U = identity and such that ρU W = prime divisor For an integer n, a prime di-ρU V ρV W whenever U ⊆ V ⊆ W . The col- visor is a prime that occurs in the prime factor-lection of Abelian groups along with the homo- ization of n. For an algebraic number ﬁeld ormorphisms ρU V is called a presheaf of Abelian for an algebraic function ﬁeld of one variable (ac 2001 by CRC Press LLC
10. 10. ﬁeld that is ﬁnitely generated and of transcen- C-characters of G, where C denotes the ﬁeld ofdence degree 1 over a ﬁeld K), a prime divi- complex numbers. Then χ ∈ Irr(G) is called asor is an equivalence class of nontrivial valua- primitive character if χ = ϕ G for any charactertions (over K in the latter case). In the number ϕ of a proper subgroup of G. See also characterﬁeld case, the prime divisors correspond to the of group, irreducible character.nonzero prime ideals of the ring of integers andthe archimedean valuations of the ﬁeld. primitive element Let E be an extension ﬁeld of the ﬁeld F (E is a ﬁeld containing Fprime element In a commutative ring R with as subﬁeld). If u is an element of E and x is anidentity 1, a prime element p is a nonunit such indeterminate, then we have the homomorphismthat if p divides a product ab with a, b ∈ R, then g(x) → g(u) of the polynomial ring F [x] intop divides at least one of a, b. When R = Z, the E, which is the identity on F and send x → u. Ifprime elements are of the form ±p for prime the kernel is 0, then F [u] ∼ F [x]. Otherwise, =numbers p. we have a monic polynomial f (x) of positive degree such that the kernel is the principal idealprime factor A prime factor of an integer n (f (x)), and then F [u] ∼ F [x]/(f (x)). Then =is a prime number p such that n is a multiple of we say E = F (u) is a simple extension of Fp. and u a primitive element (= ﬁeld generator of E/F ).prime ﬁeld The rational numbers and theﬁelds Z/pZ for prime numbers p are called primitive equation An equation f (X) = 0prime ﬁelds. Every ﬁeld contains a unique sub- such that a permutation of roots of f (X) = 0 isﬁeld isomorphic to exactly one of these prime primitive, where f (X) ∈ K[X] is a polynomial,ﬁelds. and K is a ﬁeld.prime ideal Let R be a commutative ring primitive hypercubic set A ﬁnite subgroupwith identity 1 and let I = R be an ideal of R. K of the orthogonal group O(V ) is called fullyThen I is prime if whenever a, b ∈ R are such transitive if there is a set S = {e1 , . . . , es } thatthat ab ∈ I , then at least one of a and b is in I . spans V on which K acts transitively and K has no invariant subspace in V . In this case, one canprime number A positive integer p is said choose S as eitherto be prime if (i.) the primitive hypercubic type:(i.) p > 1,(ii.) p has no positive divisors except 1 and p. S = {e1 , . . . , en } , ei , ej = δij ;The ﬁrst few prime numbers are 2, 3, 5, 7, 11,13, 17. or (ii.) the primitive hyperbolic type:prime rational divisor A divisor p = ni Pi on X over k satisfying the following S = {f1 , . . . , fn+1 } ,three conditions: (i.) p is invariant under any au- 1, i=1,...n+1,i=j ¯tomorphism σ of k/k; (ii.) for any j , there exists (fi , fj ) = 1 . σ −n, i,j =1,...,n+1,i=j ¯an automorphism σj of k/k such that Pj = P1 j ;(iii.) n1 = · · · = nt = [k(P1 ) : k]i , where X is primitive ideal Let R be a Banach algebra.a nonsingular irreducible complete curve, k is a A two-sided ideal I of R is primitive if there is asubﬁeld of the universal domain K such that X regular maximal left ideal J such that I is the setis deﬁned over k. Prime rational divisors gen- of elements r ∈ R with rR ⊆ I . The regularityerate a subgroup of the group of divisors G(X), of J means that there is an element u ∈ R suchwhich is called a group of k-rational divisors. that r − ru ∈ J for all r ∈ R.primitive character Let G be a ﬁnite group primitive idempotent element An idempo-and let Irr(G) denote the set of all irreducible tent element that cannot be expressed as a sumc 2001 by CRC Press LLC
11. 11. a + b with a and b nonzero idempotents satis- over V , and I (Q) is the two-sided ideal of T (V )fying ab = ba = 0. generated by elements x ⊗ x − Q(x) · 1 for x ∈ V . Compare with principal automorphism,primitive permutation representation Let i.e., the unique automorphism α of C(Q) suchG be a group acting as a group of permutations that α(x) = −x, for all x ∈ V .of a set X. This is called a permutation represen-tation of G. This representation is called primi- principal automorphism Let A be a com-tive if the only equivalence relations R(x, y) on mutative ring and let M be a module over A. LetX such that R(x, y) implies R(gx, gy) for all a ∈ A. The homomorphismx, y ∈ X and all g ∈ G are equality and the M x → axtrivial relation R(x, y) for all x, y ∈ X. is called the principal homomorphism associ-primitive polynomial Let f (x) be a poly- ated with a, and is denoted aM . When aM isnomial with coefﬁcients in a commutative ring one-to-one and onto, then we call aM a princi-R. When R is a unique factorization domain, pal automorphism of the module M.f (x) is called primitive if the greatest commondivisor of the coefﬁcients of f (x) is 1. For an principal divisor of functions The formalarbitrary ring, a slightly different deﬁnition is sumsometimes used: f (x) is primitive if the idealgenerated by the coefﬁcients of f (x) is R. (φ) = m1 p1 + · · · + mj pj + n1 q1 + · · · + nk qk where p1 , . . . , pj are the zeros and q1 , . . . , qkprimitive ring A ring R is called left prim- are the poles of a meromorphic function φ, miitive if there exists an irreducible, faithful left is the order of pi and ni is the order of qi .R-module, and R is called right primitive ifthere exists an irreducible, faithful right R- principal genus An ideal group of K formedmodule. See also irreducible R-module, faithful by the set of all ideals U of K relatively prime toR-module. m such that NK/k (U) belongs to H (m), where k is an algebraic number ﬁeld, m is an integralprimitive root of unity Let m be a positive divisor of k, T (m) is the multiplicative groupinteger and let R be a ring with identity 1. An of all fractional ideals of k which are relativelyelement ζ ∈ R is called a primitive mth root prime to m, S(m) is the ray modulo m, H (m)of unity if ζ m = 1 but ζ k = 1 for all positive is an ideal group modulo m (i.e., a subgroup ofintegers k < m. T (m) containing S(m)), and K/k is a Galois extension.primitive transitive permutation group LetG be a transitive group of permutations of a set principal H -series An H -series which isX. If the stabilizer of each element of X is a strictly decreasing and such that there exists nomaximal subgroup of G, then G is called prim- normal series distinct from , ﬁner than , anditive. strictly decreasing. See also H -series, normal series, ﬁner.principal adele Let K be an algebraic num-ber ﬁeld and let AK be the adeles of K. The principal ideal Let R be a commutative ringimage of the diagonal injection of K into AK is with identity 1. A principal ideal is an ideal ofthe set of principal adeles. the form aR = {ar|r ∈ R} for some a ∈ R.principal antiautmorphism A unique an- principal ideal domain An integral domaintiautomorphism β of a Clifford algebra C(Q) in which every ideal is principal. See principalsuch that β(x) = x, for all x ∈ V , where ideal.C(Q) = T (V )/I (Q), V is an n-dimensionallinear space over a ﬁeld K, and Q is a qua- principal ideal ring A ring in which everydratic form on V , T (V ) is the tensor algebra ideal is principal. See principal ideal.c 2001 by CRC Press LLC
12. 12. Principal Ideal Theorem There are at least cipal submatrix of A. Its determinant is calledtwo results having this name: a principal minor of A. For example, let (1) Let K be an algebraic number ﬁeld and  let H be the Hilbert class ﬁeld of K. Every ideal a11 a12 a13of the ring of integers of K becomes principal A = a21 a22 a23  .when lifted to an ideal of the ring of integers of a31 a32 a33H . This was proved by Furtwängler in 1930. Then, by deleting row 2 and column 2 we obtain (2) Let R be a commutative Noetherian ring the principal submatrix of Awith 1. If x ∈ R and P is minimal among theprime ideals of R containing x, then the codi- a11 a13mension of P is at most 1 (that is, there is no . a31 a33chain of prime ideals P ⊃ P1 ⊃ P2 (strict in-clusions) in R). This was proved by Krull in Notice that the diagonal entries and A itself are1928. principal submatrices of A.principal idele Let K be an algebraic number principal value (1) The principal values ofﬁeld. The multiplicative group K × injects di- arcsin, arccos, and arctan are the inverse func-agonally into the group IK of ideles. The image tions of the functions sin x, cos x, and tan x, re-is called the set of principal ideles. stricted to the domains − π ≤ x ≤ π , 0 ≤ x ≤ 2 2 π , and − π < x < π , respectively. See arc sine, 2 2principal matrix Suppose A = [Aij ] is an arc cosine, arc tangent.n × n matrix. The principal matrices associated (2) Let f (x) have a singularity at x = c,with A are A(k) = [Aij ], 1 ≤ i, j ≤ k ≤ n. with a ≤ c ≤ b. The Cauchy principal value of b a f (x) dx isprincipal minor See principal submatrix. c− bprincipal order Let K be a ﬁnite extension lim f (x) dx + f (x) dx . →0 a c+of the rational ﬁeld Q. The ring of all algebraicintegers in K is called the principal order of K. The Cauchy principal value of an improper ∞ c integral −∞ f (x)dx is limc→∞ −c f (x)dx.principal root A root with largest real part (ifthis root is unique) of the characteristic equation principle of counting constants Let X andof a differential-difference equation. Y be algebraic varieties and let C be an ir- reducible subvariety of X × Y . Let pX andprincipal series For a semisimple Lie group, pY denote the projection maps onto the fac-those unitary representations induced from ﬁnite tors of X × Y . Let a1 = dim(pX (C)) anddimensional unitary representations of a mini- a2 = dim(pY (C)). There exist a nonemptymal parabolic subgroup. open subset U1 of pX (C), contained in pX (C), and a nonempty open subset U2 of pY (C)), con-principal solution A solution F (x) of the tained in pY (C), such that all irreducible com-equation F (x)/ x = g(x), where F (x) = ponents of C(x) = {y ∈ Y : (x, y ∈ C} haveF (x + x) − F (x). Such a solution F (x) can the same dimension b2 for all x ∈ U1 and suchbe obtained by a formula in terms of integral, that all irreducible components of C −1 (y) =series, and limits. {x ∈ X : (x, y) ∈ C} have the same dimension b2 for all y ∈ U2 . These dimensions satisfyprincipal submatrix A submatrix of an m×n a1 + b2 = a2 + b1 .matrix A is an (m − k) × (n − ) matrix obtainedfrom A by deleting certain k rows (k < m) and principle of reﬂection Two complex num- columns ( < n) of A. If m = n and if the set bers z1 and z2 are said to be symmetric withof deleted rows coincides with the set of deleted respect to a circle of radius r and center z0 ifcolumns, we call the submatrix obtained a prin- (z1 − z0 )(z2 − z0 ) = r 2 . The principle ofc 2001 by CRC Press LLC
13. 13. reﬂection states that if the image of the circle the following way:under a linear fractional transformation w =(az + b)/(cz + d) is again a circle (this hap- Zp,q = Xp × Yqpens unless the image is a line), then the images ∂p,q = ∂p × 1w1 and w2 of z1 and z2 are symmetric with re- ∂ p,q = (−1)p 1 × ∂q .spect to this new circle. See also linear fractionalfunction. product formula (1) Let K be a ﬁnite exten- The Schwarz Reﬂection Principle of complex sion of the rational numbers Q. Then v |x|v =analysis deals with the analytic continuation of 1 for all x ∈ K, x = 0, where the product is overan analytic function deﬁned in an appropriate all the normalized absolute values (both p-adicset S, to the set of reﬂections of the points of S. and archimedean) of K. (2) Let K be an algebraic number ﬁeld con-product A term which includes many phe- taining the nth roots of unity, and let a and bnomena. The most common are the following: be nonzero elements of K. For a place v of K (as in (1) above), let ( a,b )n be the nth norm- v (1) The product of a set of numbers is the re- residue symbol. Then v ( a,b )n = 1. See also vsult obtained by multiplying them together. For norm-residue symbol.an inﬁnite product, this requires considerationsof convergence. proﬁnite group Any group G can be made (2) If A1 , . . . , An are sets, then the product into a topological group by deﬁning the collec-A1 × · · · × An is the set of ordered n-tuples tion of all subgroups of ﬁnite index to be a neigh-(a1 , . . . , an ) with ai ∈ Ai for all i. This deﬁni- borhood base of the identity. A group with thistion can easily be extended to inﬁnite products. topology is called a proﬁnite group. (3) Let A1 and A2 be objects in a category C. projection matrix A square matrix M suchA triple (P , π1 , π2 ) is called the product of A1 that M 2 = M.and A2 if P is an object of C, πi : P → Ai isa morphism for i = 1, 2, and if whenever X is projective algebraic variety Let K be aanother object with morphisms fi : X → Ai , ¯ ¯ ﬁeld, let K be its algebraic closure, and let Pn (K)for i = 1, 2, then there is a unique morphism ¯ be n-dimensional projective space over K. Let Sf : X → P such that πi f = fi for i = 1, 2. be a set of homogeneous polynomials in X0 , . . . , Xn . The set of common zeros Z of S in Pn (K) ¯ (4) See also bracket product, cap product, is called a projective algebraic variety. Some-crossed product, cup product, direct product, times, the deﬁnition also requires the set Z to beEuler product, free product, Kronecker prod- irreducible, in the sense that it is not the unionuct, matrix multiplication, partial product, ten- of two proper subvarieties.sor product, torsion product, wedge product. projective class Let A be a category. A pro-product complex Let C1 be a complex of jective class is a class P of objects in A suchright modules over a ring R and let C2 be a that for each A ∈ A there is a P ∈ P and acomplex of left R-modules. The tensor product P-epimorphism f : P → A.C1 ⊗R C2 gives a complex of Abelian groups, projective class group Consider left mod-called the product complex. ules over a ring R with 1. Two ﬁnitely gener- ated projective modules P1 and P2 are said toproduct double chain complex The dou- be equivalent if there are ﬁnitely generated freeble chain complex (Zp,q , ∂ , ∂ ) obtained from modules F1 and F2 such that P1 ⊕F1 P2 ⊕F2 .a chain complex X of right A-modules with The set of equivalence classes, with the opera-boundary operator ∂p and a chain complex Y tion induced from direct sums, forms a groupof left A-modules with boundary operator ∂q in called the projective class group of R.c 2001 by CRC Press LLC
14. 14. projective cover An object C in a category there is a surjection f : A → M, there is aC is the projective cover of an object A if it sat- homomorphism g : M → A such that f g is theisﬁes the following three properties: (i.) C is identity map of M.a projective object. (ii.) There is an epimor-phism e : C → A. (iii.) There is no projective projective morphism A morphism f : X →object properly between A and C. In a gen- Y of algebraic varieties over an algebraicallyeral category, this means that if g : C → A closed ﬁeld K which factors into a closed im-and f : C → C are epimorphisms and C is mersion X → Pn (K) × Y , followed by the pro-projective, then f is actually an isomorphism. jection to Y . This concept can be generalized toThus, projective covers are simply injective en- morphisms of schemes.velopes “with the arrows turned around.” Seealso epimorphism, injective envelope. projective object An object P in a category C In most familiar categories, objects are sets satisfying the following mapping property: If e :with structure (for example, groups, topologi- C → B is an epimorphism in the category, andcal spaces, etc.) and morphisms are particu- f : P → B is a morphism in the category, thenlar kinds of functions (for example, group ho- there exists a (usually not unique) morphism g :momorphisms, continuous functions, etc.), and P → C in the category such that e ◦ g = f .epimorphisms are onto functions (surjections) This is summarized in the following “universalof a particular kind. Here is an example of a mapping diagram”:projective cover in a speciﬁc category: In the Pcategory of compact Hausdorff spaces and con- f ∃gtinuous maps, the projective cover of a space X ealways exists, and is called the Gleason cover B ←− Cof the space. It may be constructed as the Stone Projectivity is simply injectivity “with the ar-space (space of maximal lattice ideals) of the rows turned around.” See also epimorphism,Boolean algebra of regular open subsets of X. injective object, projective module.A subset is regular open if it is equal to the inte- In most familiar categories, objects are setsrior of its closure. See also Gleason cover, Stone with structure (for example, groups, topologi-space. cal spaces, etc.), and morphisms are particular kinds of functions (for example, group homo-projective dimension Let R be a ring with morphisms, continuous maps, etc.), so epimor-1 and let M be an R-module. The projective phisms are onto functions (surjections) of partic-dimension of M is the length of the smallest ular kinds. Here are two examples of projectiveprojective resolution of M; that is, the projective objects in speciﬁc categories: (i.) In the cate-dimension is n if there is an exact sequence 0 → gory of Abelian groups and group homomor-Pn → · · · → P0 → M → 0, where each phisms, free groups are projective. (An AbelianPi is projective and n is minimal. If no such group G is free if it is the direct sum of copiesﬁnite resolution exists, the projective dimension of the integers Z.) (ii.) In the category of com-is inﬁnite. pact Hausdorff spaces and continuous maps, the projective objects are exactly the extremely dis-projective general linear group The quo- connected compact Hausdorff spaces. (A com-tient group deﬁned as the group of invertible pact Hausdorff space is extremely disconnectedmatrices (of a ﬁxed size) modulo the subgroup if the closure of every open set is again open.)of scalar matrices. See also compact topological space, Hausdorff space.projective limit The inverse limit. See in-verse limit. projective representation A homomorphism from a group to a projective general linear group.projective module A module M for whichthere exists a module N such that M ⊕ N is projective resolution Let B be a left R mod-free. Equivalently, M is projective if, whenever ule, where R is a ring with unit. A projectivec 2001 by CRC Press LLC
15. 15. resolution of B is an exact sequence, projective symplectic group The quotient group deﬁned as the group of symplectic ma- φ2 φ1 φ0 · · · −→ E1 −→ E0 −→ B −→ 0 , trices (of a given size) modulo the subgroup {I, −I }, where I is the identity matrix. See sym-where every Ei is a projective left R module. plectic group.(We shall deﬁne exact sequence shortly.) Thereis a companion notion for right R modules. Pro- projective unitary group The quotient groupjective resolutions are extremely important in deﬁned as the group of unitary matrices modulohomological algebra and enter into the dimen- the subgroup of unitary scalar matrices. See uni-sion theory of rings and modules. See also ﬂat tary matrix.resolution, injective resolution, projective mod-ule, projective dimension. proper component Let U and V be irre- An exact sequence is a sequence of left R ducible subvarieties of an irreducible algebraicmodules, such as the one above, where every variety X. A simple irreducible component ofφi is a left R module homomorphism (the φi U ∩ V is called proper if it has dimension equalare called “connecting homomorphisms”), such to dim U + dim V - dim X.that Im(φi+1 ) = Ker(φi ). Here Im(φi+1 ) is theimage of φi+1 , and Ker(φi ) is the kernel of φi . In proper equivalence An equivalence relationthe particular case above, because the sequence R on a topological space X such that R[K] =ends with 0, it is understood that the image of φ0 {x ∈ X : xRk for some k ∈ K} is compact foris B, that is, φ0 is onto. There is a companion all compact sets K ⊆ X.notion for right R modules. proper factor Let a and b be elements of aprojective scheme A projective scheme over commutative ring R. Then a is a proper factora scheme S is a closed subscheme of projective of b if a divides b, but a is not a unit and therespace over S. is no unit u with a = bu.projective space Let K be a ﬁeld and con- proper fraction A positive rational numbersider the set of (n + 1)-tuples (x0 , . . . , xn ) in such that the numerator is less than the denom-K n+1 with at least one coordinate nonzero. Two inator. See also improper fraction.tuples (. . . , xi , . . . ) and (. . . , xi , . . . ) are equiv-alent if there exists λ ∈ K × such that xi = proper intersection Let Y and Z be subva-λxi for all i. The set Pn (K) of all equivalence rieties of an algebraic variety X. If every irre-classes is called n-dimensional projective space ducible component of Y ∩ Z has codimensionover K. It can be identiﬁed with the set of equal to codimY +codimZ, then Y and Z are saidlines through the origin in K n+1 . The equiv- to intersect properly.alence class of (x0 , x1 , . . . , xn ) is often denoted(x0 : x1 : · · · : xn ). More generally, let R be a proper Lorentz group The group formedcommutative ring with 1. The scheme Pn (R) is by the Lorentz transformations whose matricesgiven as the set of homogeneous prime ideals of have determinants greater than zero.R[X0 , . . . , Xn ] other than (X0 , . . . , Xn ), with astructure sheaf deﬁned in terms of homogeneous proper morphism of schemes A morphismrational functions of degree 0. It is also possible of schemes f : X → Y such that f is sepa-to deﬁne projective space Pn (S) for a scheme rated and of ﬁnite type and such that for everyS by patching together the projective spaces for morphism T → Y of schemes, the induced mor-appropriate rings. phism X ×T Y → T takes closed sets to closed sets.projective special linear group The quotientgroup deﬁned as the group of matrices (of a ﬁxed proper orthogonal matrix An orthogonalsize) of determinant 1 modulo the subgroup of matrix with determinant +1. See orthogonalscalar matrices of determinant 1. matrix.c 2001 by CRC Press LLC
16. 16. proper product Let R be an integral domain ness condition for all prime ideals P is calledwith ﬁeld of quotients K and let A be an algebra pseudogeometric.over K. Let M and N be two ﬁnitely generatedR-submodules of A such that KM = KN = A. pseudovaluation A map v from a ring R intoIf {a ∈ A : Ma ⊆ M} = {a ∈ A : aN ⊆ N }, the nonnegative real numbers such that (i.) v(r)and this is a maximal order of A, then the product = 0 if and only if r = 0, (ii.) v(rs) ≤ v(r)v(s),MN is called a proper product. (iii.) v(r + s) ≤ v(r) + v(s), and (iv.) v(−r) = v(r) for all r, s ∈ R.proper transform Let T : V → W be arational mapping between irreducible varieties p-subgroup A ﬁnite group whose order isand let V be an irreducible subvariety of V . A a power of p is called a p-group. A p-groupproper transform of V by T is the union of all that is a subgroup of a larger group is called airreducible subvarieties W of W for which there p-subgroup.is an irreducible subvariety of T such that V andW correspond. p-subgroup For a ﬁnite group G and a prime integer p, a subgroup S of G such that the orderproportion A statement equating two ratios, of S is a power of p.a cb = d , sometimes denoted by a : b = c : d.The terms a and d are the extreme terms and the pure imaginary number An imaginary num-terms b and c are the mean terms. ber. See imaginary number. pure integer programming problem Aproportional A term in the proportion a = b c problem similar to the primary linear program-d . Given numbers a, b, and c, a number x which ming problem in which the solution vector x =satisﬁes a = x is a fourth proportional to a, b, b c (x1 , x2 , . . . , xn ) is a vector of integers: the prob-and c. Given numbers a and b, a number x lem is to minimize z = cx with the conditionswhich satisﬁes a = x is a third proportional to b b Ax = b, x ≥ 0, and xj (j = 1, 2, . . . , n) is an aa and b, and a number x which satisﬁes x = x b integer, where c is an n × 1 vector of real num-is a mean proportional to a and b. bers, A is an m × n matrix of real numbers, and b is an m × 1 matrix of real numbers.proportionality The state of being in pro-portion. See proportion. purely inﬁnite von Neumann algebra A von Neumann algebra A which has no semiﬁnitePrüfer domain An integral domain R such normal traces on A.that every nonzero ideal of R is invertible. Equiv-alently, the localization RM is a valuation ring purely inseparable element Let L/K be anfor every maximal ideal M. Another equivalent extension of ﬁelds of characteristic p > 0. Ifcondition is that an R-module is ﬂat if and only n α ∈ L satisﬁes α p ∈ K for some n, then α isif it is torsion-free. called a purely inseparable element over K.Prüfer ring An integral domain R such that purely inseparable extension An extensionall ﬁnitely generated ideals in R are inversible L/K of ﬁelds such that every element of L isin the ﬁeld of quotients of R. See also integral purely inseparable over K. See purely insepa-domain. rable element.pseudogeometric ring Let A be a Noethe- purely inseparable scheme Given an ir-rian integral domain with ﬁeld of fractions K. If reducible polynomial f (X) over a ﬁeld k, ifthe integral closure of A in every ﬁnite extension the formal derivative df/dX = 0, then f (X)of K is ﬁnitely generated over A, then A is said is inseparable; otherwise, f is separable. Ifto satisfy the ﬁniteness condition. A Noethe- char(k) = 0, every irreducible polynomialrian ring R such that R/P satisﬁes the ﬁnite- f (X)(= 0) is separable. If char(k) = p > 0,c 2001 by CRC Press LLC
17. 17. an irreducible polynomial f (X) is inseparable Pythagorean ﬁeld √ A ﬁeld F that containsif and only if f (X) = g(X p ). An algebraic a 2 + b2 for all a, b ∈ F .element α over k is called separable or insep-arable over k if the minimal polynomial of α Pythagorean identities The following ba-over k is separable or inseparable. An alge- sic identities involving trigonometric functions,braic extension of k is called separable if all resulting from the Pythagorean Theorem:elements of K are separable over k; otherwise,K is called inseparable. If α is inseparable, then sin2 (x) + cos2 (x) = 1k has nozero characteristic p and the minimalpolynomial f (X) of α can be decomposed as tan2 (x) + 1 = sec2 (x) r r rf (X) = (X − α1 )p (X − α2 )p . . . (X − αm )p , 1 + cot 2 (x) = csc2 (x) .r ≥ 1, where α1 , . . . , αm are distinct roots of rf (X) in its splitting ﬁeld. If α p ∈ k for some Pythagorean numbers Any combination ofr, we call α purely inseparable over k. An al- three positive integers a, b, and c such that a 2 +gebraic extension K of k is called purely in- b2 = c2 .separable if all elements of the ﬁeld are purelyinseparable over k. Let V ⊂ k n be a reduced Pythagorean ordered ﬁeld An ordered ﬁeldirreducible afﬁne algebraic variety. V is called P such that the square root of any positive ele-purely inseparable if the function ﬁeld k(V ) is ment of P is in P .purely inseparable over k. Therefore, we can de-ﬁne the same notion for reduced irreducible al- Pythagorean Theorem Consider a right tri-gebraic varieties. Since there is a natural equiva- angle with legs of length a and b and hypotenuselence between the category of algebraic varieties of length c. Then a 2 + b2 = c2 .over k and the category of reduced, separated, al-gebraic k-schemes, by identifying these two cat- Pythagorean triple A solution in positiveegories, we deﬁne purely inseparable reduced, integers x, y, z to the equation x 2 + y 2 = z2 .irreducible, algebraic k-schemes. Some examples are (3, 4, 5), (5, 12, 13), and (20, 21, 29). If x, y, z have no common divi-purely transcendental extension An exten- sor greater than 1, then there are integers a, bsion of ﬁelds L/K such that there exists a set such that x = a 2 −b2 , y = 2ab, and z = a 2 +b2of elements {xi }i∈I , algebraically independent (or the same equations with the roles of x andover K, with L = K({xi }). See algebraic inde- y interchanged). Also called Pythagorean num-pendence. bers.pure quadratic A quadratic equation of theform ax 2 + c = 0, that is, a quadratic equationwith the ﬁrst degree term bx missing.c 2001 by CRC Press LLC