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aberration The deviation of a spherical mir-
ror from perfect focusing.
A abscissa In a rectangular coordinate system
2
(Cartesian coordinates) (x, y) of the plane R , x
is called the abscissa, y the ordinate.

absolute convergence A series x is said
n
a posteriori error estimator An algorithm
to be absolute convergent if the series of absolute
for obtaining information about a discretization
values |x | converges.
n
error for a concrete discrete approximation u of
h
the continuous solution u. Two principal features absolute convergence test If |x | con-
n

are expected from such device: verges, then x converges.
n
(i.) It should be reliable: the estimated error absolute error The difference between the
(norm) must be proportional to an upper bound exact value of a number x and an approximate
for the true error (norm). Thus, discrete solutions value a is called the absolute error of the
a
that do not meet a prescribed accuracy can be approximate value, i.e., =|x − a|. The quo-
a
detected. tient δ = a is called the relative error.
a
a
(ii.) It should be efficient: the error estimator
absolute ratio test Let x be a series of
n
should provide some lower bound for the true |x n+1 |
nonnegative terms and suppose lim =
error (norm).This helps avoid rejecting a discrete n→∞ |x n |
ρ.
solution needlessly.
(i.) If ρ< 1, the series converges absolutely
In the case of a finite element discretization an (hence converges);
additional requirement is the locality of the a
(ii.) If ρ> 1, the series diverges;
posteriori error estimator. It must be possible to
extract information about the contributions from (iii.) If ρ = 1, the test is inconclusive.
individualcellsofthemeshtothetotalerror. This absolute temperature −273.15 C.
◦
is essential for the use of an a posteriori error esti-
mator in the framework of adaptive refinement. absolute value The absolute value of a real
number x, denoted by |x|, is defined by |x|= x
abacus Oldest known “computer” circa if x ≥ 0 and |x|=−x if x< 0.
1100 BC from China, a frame with sliding beads
absolute value of an operator Let A be a
for doing arithmetic.
bounded linear operator on a Hilbert space, H.
Abbe’s sine condition (Ernst Abbe 1840– Then the absolute value of A is given by |A|=
√
A A, where A is the adjoint of A.
∗
∗

1905) n l sin β = nl sin β where n, n ,β,β

are the refraction indices and refraction angles, absolutely continuous A function x(t)
respectively.
defined on [a, b] is called absolutely continuous
1
on [a, b] if there exists a function y ∈ L [a, b]
Abelian group (Niels Henrik Abel 1802– t
such that x(t) = y(s)ds + C, where C is a
1829 ) A group (G, ·) is called Abelian or com- a
constant.
mutative if a · b = b · a for all a, b ∈ G.
absorbance A logarithm of the ratio of inci-
n
Abelian theorems (1) Suppose ∞ a x dent to transmitted radiant power through a
n
n=0
converges for |x| <R and for x = R. Then the sample (excluding the effects on cell walls).
series converges uniformly on 0 ≤ x ≤ R. Depending on the base of the logarithm, decadic
(2)For n ≥ 5 the general equation of nth order or Napierian absorbance are used. Symbols:
cannot be solved by radicals. A, A ,A . This quantity is sometimes called
10 e
extinction, although the term extinction, better
x φ(ξ)
Abel’s integral equation f(x) = √
0 x−ξ called attenuance, is reserved for the quantity
1
dξ, where f(x) is C with f(0) = 0, is called which takes into account the effects of lumines-
Abel’s integral equation. cence and scattering as well.
© 2003 by CRC Press LLC