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inverse inequality Inequalities bounding ion pair A pair of oppositely charged ions
stronger norms (in the sense that they contain held together by Coulomb attraction without for-
higher derivatives) of functions in finite dimen- mation of a covalent bond. Experimentally, an
sional spaces by weaker norms. An impor- ion pair behaves as one unit in determining con-
1
tant example are bounds for the H (;)-norm of ductivity, kinetic behavior, osmotic properties,
2
finite element functions in terms of their L (;)- etc.
norm. The constants in these inequalities are Following Bjerrum, oppositely charged ions
invariably for families of approximating spaces with their centers closer together than a distance
that are asymptotically dense. For instance, for
6 + −
q = 8.36 × 10 z z /(" T)pm
the above pair of norms and h-version families r
1
of H (;)-conforming finite element spaces on
areconsideredtoconstituteanionpair(“Bjerrum
quasi-uniformandshaperegularmeshesthecon- ion pair”). [z and z are the charge numbers of
−
+
−1
stant will grow like O(h ) where h is the mesh
the ions, and " is the relative permittivity (or
r
width. dielectric constant) of the medium.]
An ion pair, the constituent ions of which are
inverse relation If R ⊆ A × B is a rela- in direct contact (and not separated by an inter-
tion from A to B, then the set R −1 ={(b, a) | vening solvent or other neutral molecule) is des-
(a, b) ∈ R} is a subset of B × A. R −1 is called ignated as a “tight ion pair” (or “intimate” or
“contact ion pair”). A tight ion pair of X and
+
the inverse of R.
Y is symbolically represented as X Y .
−
−
+
By contrast, an ion pair whose constituent
invertible A map φ : A → B is invertible if ions are separated by one or several solvents or
there exists a map φ −1 : B → A such that other neutral molecules is described as a “loose
ion pair,” symbolically represented as X ||Y .
−
+
−1 −1
φ ◦ φ = id φ ◦ φ = id The members of a loose ion pair can readilyinter-
A B
change with other free or loosely paired ions in
The map φ −1 is called the inverse of φ. the solution. This interchange may be detectable
(e.g., by isotopic labeling) and thus affords an
experimental distinction between tight and loose
involution of functions Two functions f
ion pairs.
and g on a symplectic manifold are in involu-
A further conceptual distinction has some-
tion when their Poisson bracket vanishes, i.e.,
times been made between two types of loose
{f, g}= 0.
ion pairs. In “solvent-shared ion pairs” the
ionic constituents of the pair are separated by
involutive distribution A distribution of only a single solvent molecule, whereas in
subspaces on M with generators X such that “solvent-separated ion pairs” more than one sol-
A
[X ,X ] ∈ where [ , ]isthe commutator.A vent’s molecule intervenes. However, the term
A
B
distribution is involutive if and only if it is inte- “solvent-separated ion pair” must be used and
grable (Frobenius theorem). interpreted with care since it has also widely been
used as a less specific term for “loose” ion pair.
ion channel In biochemistry, a protein, ionizing radiation Anyradiationconsisting
embedded in cell membrane, which conducts of directly or indirectly ionizing particles or a
ionic current of specific ions. An ion channel mixture of both, or photons with energy higher
can be either passive or modulated by the electri- than the energy of photons of ultraviolet light or
cal voltage across the cell membrane. The latter a mixture of both such particles and photons.
is often called voltage-gated ion channel. (cf. B.
Hille, Ionic Channels of Excitable Membranes, irradiance, E E E Radiant power received by
2nd ed., Sinauer Associates, Sunderland, MA, a surface divided by the area of that surface.
1992). For collimated beams this quantity is sometimes

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC