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DICTIONARY OF
ALGEBRA,
ARITHMETIC,
AND
TRIGONOMETRY
c
2001 by CRC Press LLC
Comprehensive Dictionary
of Mathematics
Douglas N. Clark
Editor-in-Chief
Stan Gibilisco
Editorial Advisor
PUBLISHED VOLUMES
Analysis, Calculus, and Differential Equations
Douglas N. Clark
Algebra, Arithmetic and Trigonometry
Steven G. Krantz
FORTHCOMING VOLUMES
Classical & Theoretical Mathematics
Catherine Cavagnaro and Will Haight
Applied Mathematics for Engineers and Scientists
Emma Previato
The Comprehensive Dictionary of Mathematics
Douglas N. Clark
c
2001 by CRC Press LLC
a Volume in the
Comprehensive Dictionary
of Mathematics
DICTIONARY OF
ALGEBRA,
ARITHMETIC,
AND
TRIGONOMETRY
Edited by
Steven G. Krantz
Boca Raton London New York Washington, D.C.
CRC Press
This book contains information obtained from authentic and highly regarded sources. Reprinted material is
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PREFACE
The second volume of the CRC Press Comprehensive Dictionary of Mathematics covers algebra,
arithmetic and trigonometry broadly, with an overlap into differential geometry, algebraic geometry,
topology and other related fields. The authorship is by well over
30
mathematicians, active in
teaching and research, including the editor.
Because it is a dictionary and not an encyclopedia, definitions are only occasionally accompanied
by
a
discussion or example. In a dictionary of mathematics,
the
primary goal is to define each term
rigorously. The derivation of a term is almost never attempted.
The dictionary is written to be a useful reference for a readership that includes students, scientists,
and engineers with a wide range of backgrounds, as well as specialists in areas of analysis and
differential equations and mathematicians in related fields. Therefore, the definitions are intended
to be accessible, as well as rigorous. To be sure, the degree of accessibility may depend upon the
individual term, in a dictionary with terms ranging from Abelian cohomology to
z
intercept.
Occasionally a term must be omitted because it is archaic. Care was taken when such circum
-
stances arose to ensure that the term was obsolete. An example of an archaic term deemed to be
obsolete, and hence not included, is “right line”. This term was used throughout a turn
-
of
-
the
-
century
analytic geometry textbook we needed to consult, but it was not defined there. Finally, reference to
a contemporary English language dictionary yielded “straight line”
as
a synonym for “right line”.
The authors are grateful to the series editor, Stanley Gibilisco, for dealing with our seemingly
endless procedural questions and
to
Nora Konopka, for always acting efficiently and cheerfully with
CRC
Press liaison matters.
Douglas
N.
Clark
Editor
-
in
-
Chief
c
2001 by CRC Press LLC
CONTRIBUTORS
Edward Aboufadel
Grand Valley State University
Allendale, Michigan
Gerardo Aladro
Florida International University
Miami, Florida
Mohammad Azarian
University of Evansville
Evansville. Indiana
Susan Barton
West Virginia Institute
of
Technology
Montgomery, West Virginia
Albert Boggess
Texas A&M University
College Station, Texas
Robert Borrelli
Harvey Mudd College
Claremont, California
Stephen
W.
Brady
Wichita State University
Wichita, Kansas
Der Chen Chang
Georgetown University
Washington, D.C.
Stephen A. Chiappari
Santa Clara University
Santa Clara. California
Joseph A. Cima
The University of North Carolina at Chapel Hill
Chapel Hill, North Carolina
Courtney
S.
Coleman
Harvey Mudd College
Claremont, California
John B. Conway
University
of
Tennessee
Knoxville, Tennessee
Neil K. Dickson
University
of
Glasgow
Glasgow, United Kingdom
David
E.
Dobbs
University of Tennessee
Knoxville, Tennessee
Marcus Feldman
Washington University
St. Louis, Missouri
Stephen Humphries
Brigham Young University
Provo, Utah
Shanyu
Ji
University of Houston
Houston, Texas
Kenneth D. Johnson
University
of
Georgia
Athens, Georgia
Bao Qin Li
Florida International University
Miami, Florida
Robert E. MacRae
University
of
Colorado
Boulder, Colorado
Charles
N.
Moore
Kansas State University
Manhattan, Kansas
Hossein Movahedi-Lankarani
Pennsylvania State University
Altoona, Pennsylvania
Shashikant B. Mulay
University of Tennessee
Knoxville, Tennessee
Judy Kenney Munshower
Avila College
Kansas City, Missouri
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2001 by CRC Press LLC
Charles W. Neville
CWN Research
Berlin, Connecticut
Daniel E. Otero
Xavier University
Cincinnati, Ohio
Josef Paldus
University
of
Waterloo
Waterloo, Ontario, Canada
Harold
R.
Parks
Oregon State University
Corvallis, Corvallis, Oregon
Gunnar Stefansson
Pennsylvania State University
Altoona, Pennsylvania
Anthony D. Thomas
University
of
Wisconsin
Platteville. Wisconsin
Michael Tsatsomeros
University
of
Regina
Regina, Saskatchewan, Canada
James
S.
Walker
University
of
Wisconsin at
Eau
Claire
Eau Claire, Wisconsin
C. Eugene Wayne
Boston University
Boston, Massachusetts
Kehe Zhu
State University
of
New
York at Albany
Albany, New York
c
2001 by CRC Press LLC
A
A-balanced mapping Let M be a right mod-
ule over the ring A, and let N be a left module
over the samering A. A mapping φ fromM ×N
to an Abelian group G is said to be A-balanced
if φ(x,·) is a group homomorphism from N to
G for each x ∈ M,ifφ(·,y) is a group homo-
morphism from M to G for each y ∈ N, and
if
φ(xa,y) = φ(x,ay)
holds for all x ∈ M, y ∈ N , and a ∈ A.
A-B-bimodule An Abelian group G that is a
left module over the ring A and a right module
over the ring B and satisfies the associative law
(ax)b = a(xb) for all a ∈ A, b ∈ B, and all
x ∈ G.
Abeliancohomology Theusualcohomology
with coefficients in an Abelian group; used if
the context requires one to distinguish between
the usual cohomology and the more exotic non-
Abelian cohomology. See cohomology.
Abeliandifferentialof thefirstkind Aholo-
morphic differential on a closed Riemann sur-
face; that is, a differential of the form ω =
a(z)dz, where a(z) is a holomorphic function.
Abelian differential of the second kind A
meromorphic differential on a closed Riemann
surface,thesingularitiesof which arealloforder
greater than or equal to 2; that is, a differential
of the form ω = a(z) dz where a(z) is a mero-
morphic function with only 0 residues.
Abelian differential of the third kind A
differential on a closed Riemann surface that is
not an Abelian differential of the first or sec-
ond kind; that is, a differential of the form ω =
a(z)dz where a(z) is meromorphic and has at
least one non-zero residue.
Abelian equation A polynomial equation
f(X) = 0 is said to be an Abelian equation if
its Galoisgroup isan Abelian group. See Galois
group. See also Abelian group.
Abelian extension A Galois extension of a
field is called an Abelian extension if its Galois
group is Abelian. See Galois extension. See
also Abelian group.
Abelian function A function f(z
1
,z
2
,z
3
,
,z
n
) meromorphic on C
n
for which there ex-
ist 2n vectors ω
k
∈ C
n
, k = 1, 2, 3, ,2n,
called period vectors, that are linearly indepen-
dent over R and are such that
f
(
z + ω
k
)
= f(z)
holds for k = 1, 2, 3, ,2n and z ∈ C
n
.
Abelian function field The set of Abelian
functions on C
n
corresponding to a given set of
period vectors forms a field called an Abelian
function field.
Abeliangroup Briefly, acommutativegroup.
More completely, a setG, together with abinary
operation, usually denoted “+,” a unary opera-
tion usually denoted “−,” and a distinguished
element usually denoted “0” satisfying the fol-
lowing axioms:
(i.) a + (b +c) = (a +b) +c for all
a, b,c ∈ G,
(ii.) a + 0 = a for all a ∈ G,
(iii.) a + (−a) = 0 for all a ∈ G,
(iv.) a + b = b +a for all a,b ∈ G.
The element 0 is called the identity, −a is
called the inverse of a, axiom (i.) is called the
associative axiom, and axiom (iv.) is called the
commutative axiom.
Abelianideal An ideal inaLiealgebrawhich
forms a commutative subalgebra.
Abelian integral of the first kind An indef-
inite integral W(p) =
p
p
0
a(z)dz on a closed
Riemann surface in which the function a(z) is
holomorphic (the differential ω(z) = a(z) dz
is said to be an Abelian differential of the first
kind).
Abelian integral of the second kind An in-
definiteintegralW(p) =
p
p
0
a(z)dzonaclosed
Riemann surface in which the function a(z) is
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2001 by CRC Press LLC
meromorphic with all its singularities of order
at least 2 (the differential a(z) dz is said to be an
Abelian differential of the second kind).
Abelian integral of the third kind An in-
definiteintegralW(p) =
p
p
0
a(z)dzonaclosed
Riemann surface in which the function a(z) is
meromorphic and has at least one non-zero resi-
due(thedifferentiala(z)dz issaidto beanAbel-
ian differential of the third kind).
Abelian Lie group A Lie group for which
the associated Lie algebra is Abelian. See also
Lie algebra.
Abelian projection operator A non-zero
projection operator E ina vonNeumann algebra
M such that the reduced von Neumann algebra
M
E
= EME is Abelian.
Abelian subvariety A subvariety of an
Abelian variety that is also a subgroup. See also
Abelian variety.
Abelian surface A two-dimensional Abelian
variety. See also Abelian variety.
Abelian variety A complete algebraic vari-
ety G that also forms a commutative algebraic
group. That is, G is a group under group oper-
ations that are regular functions. The fact that
an algebraic group is complete as an algebraic
variety implies that the group is commutative.
See also regular function.
Abel’s Theorem Niels Henrik Abel (1802-
1829) proved several results now known as
“Abel’s Theorem,” but perhaps preeminent
among these is Abel’s proof that the general
quintic equation cannot be solved algebraically.
Other theorems that may be found under the
heading “Abel’s Theorem” concern power se-
ries, Dirichlet series, and divisors on Riemann
surfaces.
absolute class field Let k be an algebraic
number field. A Galois extension K of k is an
absolute class field if it satisfies the following
property regarding prime ideals of k: A prime
ideal p of k of absolute degree 1 decomposes
as the product of prime ideals of K of absolute
degree 1 if and only if p is a principal ideal.
The term “absolute class field” is used to dis-
tinguish the Galois extensions described above,
which were introduced by Hilbert, from a more
general concept of “class field” defined by
Tagaki. See also class field.
absolute covariant A covariant of weight 0.
See also covariant.
absolute inequality An inequality involving
variables that is valid for all possible substitu-
tions of real numbers for the variables.
absolute invariant Any quantity or property
of an algebraic variety that is preserved under
birational transformations.
absolutely irreducible character The char-
acter of an absolutely irreducible representation.
A representation is absolutely irreducible if it is
irreducible and if the representation obtained by
making anextension of theground fieldremains
irreducible.
absolutely irreducible representation A
representation is absolutely irreducible if it is
irreducible and if the representation obtained by
making anextension of theground fieldremains
irreducible.
absolutely simple group A group that con-
tains no serial subgroup. The notion of an ab-
solutely simple group is a strengthening of the
concept of a simple group that is appropriate for
infinite groups. See serial subgroup.
absolutely uniserial algebra Let A be an al-
gebra over the field K, and let L be an extension
field of K. Then L ⊗
K
A can be regarded as
an algebra over L. If, for every choice of L,
L ⊗
K
A can be decomposed into a direct sum
of ideals which are primary rings, then A is an
absolutely uniserial algebra.
absolute multiple covariant A multiple co-
variant of weight 0. See also multiple covari-
ants.
c
2001 by CRC Press LLC
absolute number A specific number repre-
sented by numerals such as 2,
3
4
, or 5.67 in con-
trast with a literal number which is a number
represented by a letter.
absolute value of a complex number More
commonly called the modulus, the absolute val-
ue of the complex number z = a + ib, where a
and b are real, is denoted by |z| and equals the
non-negative real number
√
a
2
+ b
2
.
absolute value of a vector More commonly
called the magnitude, the absolute value of the
vector
−→
v =
(
v
1
,v
2
, ,v
n
)
is denoted by |
−→
v | and equals the non-negative
real number
v
2
1
+ v
2
2
+···+v
2
n
.
absolute value of real number For a real
numberr,the nonnegativerealnumber|r|,given
by
|r|=
r if r ≥ 0
−r if r<0 .
abstract algebraicvariety A set that is anal-
ogous to an ordinary algebraic variety, but de-
fined only locally and without an imbedding.
abstract function (1) In the theory of gen-
eralized almost-periodic functions, a function
mapping R to a Banach space other than the
complex numbers.
(2) A function from one Banach space to an-
other Banach space that is everywhere differen-
tiable in the sense of Fréchet.
abstract variety A generalization of the no-
tion of an algebraic variety introduced by Weil,
in analogy with the definition of a differentiable
manifold. An abstract variety (also called an
abstract algebraic variety) consists of (i.) a
family {V
α
}
α∈A
of affine algebraic sets over a
given field k, (ii.) for each α ∈ A a family of
open subsets {W
αβ
}
β∈A
of V
α
, and(iii.) for each
pair α and β in A a birational transformation be-
tween W
αβ
and W
αβ
such that the composition
of the birational transformations between sub-
sets of V
α
and V
β
and between subsets of V
β
and V
γ
are consistent with those between sub-
sets of V
α
and V
γ
.
accelerationparameter Aparameter chosen
in applying successive over-relaxation (which
is an accelerated version of the Gauss-Seidel
method)tosolveasystem of linearequationsnu-
merically. Morespecifically, onesolvesAx = b
iteratively by setting
x
n+1
= x
n
+ R
(
b − Ax
n
)
,
where
R =
L + ω
−1
D
−1
with L the lower triangular submatrix of A, D
the diagonal of A, and 0 <ω<2. Here, ω
is the acceleration parameter, also called the
relaxation parameter. Analysis is required to
choose an appropriate value of ω.
acyclic chain complex An augmented, pos-
itive chain complex
···
∂
n+1
−→ X
n
∂
n
−→ X
n−1
∂
n−1
−→
···
∂
2
−→ X
1
∂
1
−→ X
0
→ A → 0
forming an exact sequence. This in turn means
that the kernel of ∂
n
equals the image of ∂
n+1
for n ≥ 1, the kernel of equals the image of
∂
1
, and is surjective. Here the X
i
and A are
modules over a commutative unitary ring.
addend In arithmetic, a number that is to be
added to another number. In general, one of the
operands of an operation of addition. See also
addition.
addition (1) A basic arithmetic operation
that expresses the relationship between the
number of elements in each of two disjoint sets
and the numberof elementsin the union ofthose
two sets.
(2) The name of the binary operation in an
Abelian group, when the notation “+” is used
for that operation. See also Abelian group.
(3) The name of the binary operation in a
ring, under which the elements form an Abelian
group. See also Abelian group.
(4) Sometimes, thename of one of the opera-
tions in a multi-operator group, even though the
operation is not commutative.
c
2001 by CRC Press LLC
[...]... birational Birch-Swinnerton-Dyer conjecture The rank of the group of rational points of an elliptic curve E is equal to the order of the 0 of L(s, E) at s = 1 Consider the elliptic curve E : y 2 = x 3 − ax − b where a and b are integers If E(Q) = E ∩ (Q × Q), by Mordell’s Theorem E(Q) is a finitely generated Abelian group Let N be the conductor of E, and if p | N, let ap + p be the number of solutions of y 2... irreducible holomorphic representation of Gc is holomorphically induced from a one-dimensional holomorphic representation of a Borel subgroup of Gc boundary (1) (Topology.) The intersection of the complements of the interior and exterior of a set is called the boundary of the set Or, equivalently, a set’s boundary is the intersection of its closure and the closure of its complement (2) (Algebraic Topology.)... any irreducible ∗-representation Also called liminal C*-algebra center (1) Center of symmetry in Euclidean geometry The midpoint of a line, center of a triangle, circle, ellipse, regular polygon, sphere, ellipsoid, etc (2) Center of a group, ring, or Lie algebra X The set of all elements of X that commute with every element of X (3) Center of a lattice L The set of all central elements of L central extension... extension Let G, H , and K be groups such that G is an extension of K by H If H is contained in the center of G, then G is called a central extension of H centralizer Let X be a group (or a ring) and let S ⊂ X The set of all elements of X that commute with every element of S is called the centralizer of S central separable algebra An R-algebra which is central and separable Here a central R-algebra A which... inertia e group of ℘, |T | = e and σ is the Frobenius automorphism of ℘ Then L(s, ϕ) = L℘ (s, ϕ), for s > 1 ℘ K, P ai is a root of x p − x − ai = 0, L/K is Galois, and the Galois group is an Abelian group of exponent p Artin’s conjecture A conjecture of E Artin that the Artin L-function L(s, ϕ) is entire in s, whenever ϕ is irreducible and s = 1 See Artin L-function Artin’s general law of reciprocity... equivalence classes of the closed n forms modulo the differentials of (n − 1) forms.] Bezout’s Theorem If p1 (x) and p2 (x) are two polynomials of degrees n1 and n2 , respectively, having no common zeros, then there are two unique polynomials q1 (x) and q2 (x) of degrees n1 − 1 and n2 − 1, respectively, such that p1 (x)q1 (x) + p2 (x)q2 (x) = 1 biadditive mapping For A-modules M, N and L, the mapping... a group of permutations of the roots of the equation The affect of the equation is the index of the Galois group in the group of all permutations of the roots of the equation c 2001 by CRC Press LLC affectless equation A polynomial equation for which the Galois group consists of all permutations See also affect affine algebraic group group See linear algebraic affine morphism of schemes Let X and Y be... The blowing up of N at p is π : Bp (N ) → N BN-pair A pair of subgroups (B, N ) of a group G such that: (i.) B and N generate G; (ii.) B ∩ N = H N ; and (iii.) the group W = N/H has a set of generators R such that for any r ∈ R and any w ∈ W (a) rBw ⊂ BwB ∩ BrwB, (b) rBr = B Bochner’s Theorem A function, defined on R, is a Fourier-Stieltjes transform if and only if it is continuous and positive definite... Thus, the group D0 /P is a subgroup of the divisor class group Cl 0 (X) = D/P Here, D0 is the group of divisors algebraically equivalent to 0, P is the group of principal divisors, and D is the group of divisors of degree 0 The group D0 /p is exactly the subgroup of the divisor class group realized by the group of points of the Picard variety of X See algebraic family of divisors, divisor See also integral... contains a self-normalizing, nilpotent subgroup, called a Carter subgroup Cartan’s criterion of solvability Let gl(n, K) be the general linear Lie algebra of degree n over a field K and let L be a subalgebra of gl(n, K) Then L is solvable if and only if tr(AB) = 0 (tr(AB) = trace of AB), for every A ∈ L and B ∈ [L, L] Cartesian product If X and Y are sets, then the Cartesian product of X and Y , denoted . antipode which is analogous to thegeometric one just de- scribed. antisymmetric decomposition The decom- position of a compact Hausdorff space X con- sists of disjoint, closed, maximal sets of anti- symmetry. algebra classes. The product of a pair of algebra classes is defined by choos- ing an algebrafrom each class, say A and B, and letting the product of the classes be the algebra class containing. The dictionary is written to be a useful reference for a readership that includes students, scientists, and engineers with a wide range of backgrounds, as well as specialists in areas of analysis
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