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b(u, q) = g(q) ∀q ∈ W, injection A map φ : A → B which is
injective, so that whenever φ(a ) = φ(a ) then
2
1
has a unique solution. It satisfies
a = a .
1 2

1 a 1
u ≤ f + + 1 g , injective Let A and B be two sets, with A

V V W
α α β
the domain and B the codomain of a function
f . Then the function f is injective if, for any
p ≤
W
x and x ∈ A, x = x , their images f(x) =

1 a a a f(x ). Injective functions are also called one-to-

+ 1 f V + + 1 g .

W
β α α β 2 one transformations of A into B. See also into,
Similar results are known for nonsymmetric sad- onto, bijection, and surjection.
dle point problems.
inner product An inner product or scalar
product on a complex vector space V is a map
infinite set A set whose cardinality is not
<, >: V × V → C such that, for all x, y, z ∈
finite. See also cardinality, countable set, denu-
V, α ∈ C
merably infinite set, finite set, and uncountably
infinite set. (i.) <x, y >= <y, x > (the bar denotes
complex conjugate);
infix An operator written between its (ii.) <x + y, z >=<x, z > + <y, z >;
operands; thus for two operands x and y and
(iii.) <αx, y >= α< x, y >;
operator f , the syntax is xfy. See also func-
(iv.) <x, x >≥ 0;
tor, postfix, prefix, and relation.
(v.) <x, x >= 0 only if x = 0.
inherent viscosity (of a polymer) The ratio
Such a space is called an inner product space.
of the natural logarithm of the relative viscosity,
η , to the mass concentration of the polymer, c, inner product operator Let M be a mani-
r
i.e., fold, ; (M) the space of exterior k-forms on M
k
η inh ≡ η = (ln η )/c. and X a vector field on M. The inner product
r
ln
The quantity η , with which the inherent viscos- operator i is the linear map i : ; k+1 (M) →
X
X
ln
k
ity is synonymous, is the logarithmic viscosity ; (M) defined by
number. i ω(X ,...,X ) = ω(X, X ,...,X ),
3 −1
Notes: (1)Theunitmustbespecified: cm g X 1 k 1 k
is recommended. ω ∈ ; k+1 (M), X ,...X vector fields. i is an
k
1
X
(2) These quantities are neither viscosities nor antiderivative with respect to the ∧ product.
pure numbers. The terms are to be looked at
as traditional names. Any replacement by con- instance In logic and logic programming, A
sistent terminology would produce unnecessary is an instance of B if there exists a substitution θ
confusion in the polymer literature. such that A = Bθ. See also instantiation.
inhibitor A substance that diminishes the instantiation In logic and logic program-
rate of a chemical reaction; the process is called ming, the substitution of a ground term for a vari-
inhibition. Inhibitors are sometimes called nega- able to produce an instance of the variable.
tive catalysts, but since the action of an inhibitor Comment: Substitution has specialized mean-
is fundamentally different from that of a catalyst, ings in chemistry and logic programming; here
this terminology is discouraged. In contrast to a the logic programming meaning is used. See also
catalyst, an inhibitor may be consumed during instance.
the course of a reaction. In enzyme-catalyzed
reactions an inhibitor frequently acts by binding integrable distribution A distribution of
to the enzyme, in which case it may be called an subspaces such that for each point x ∈ M
enzyme inhibitor. there exists an integral manifold through x.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC