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linear macromolecule A macromolecule, other (for example, they satisfy a generating rela-
2
the structure of which essentially comprises tion like y = x or are all employees in a com-
the multiple repetition in linear sequence of pany), but either may or may not have repeated
units derived, actually or conceptually, from elements. The elements of a list need not have
molecules of low relative molecular mass. any relationship to each other. Notice further that
none of these definitions requires that the mem-
linear operator See linear. bers of the collection be ordered in some way. If
they were, one would speak of an ordered set,
linear transformation See linear. bag,or list. For example, one might sort the
elements of a set alphabetically or in UNIX sort
n
Liouville’s equation Let U ⊂ R open and order. See also bag, sequence, set, and tuple.
u : U × R → R. The Liouville equation for u is
n load vector The right-hand side of a vari-
i
u − (b u) = 0. ational problem posed over the Banach space
t x i

i=1 V is an element of the dual space V .
When discretization by means of a conform-
liquid-crystal transitions A liquid crystal
ing finite element space V h is performed,
is a molecular crystal with properties that are
f has to be evaluated for the nodal basis
both solid- and liquid-like. Liquid crystals are
functions b ,i = 1, ··· ,N := dim V ,
h
i
composed predominantly of rod-like or disk-like N
of V . The resulting vector (f (b )) i=1 has
i
h
molecules, that can exhibit one or more differ-
been dubbed load vector in calculations of
ent, ordered fluid phases as well as the isotropic
linear elasticity.
fluid; the translational order is wholly or partially
destroyed but a considerable degree of orien-
logistic equation A nonlinear equation with
tational order is retained on passing from the
a x(1 − x) term. It was first motivated from
crystalline to the liquid phase in a mesomorphic
modeling biological population growth. In the
transition.
context of ordinary differential equations, the
(1) Transition to a nematic phase. A meso-
morphic transition that occurs when a molecular equation, dx/dt = ax(1 − x) can be solved.
crystal is heated to form a nematic phase in which In the context of difference equations, x n+1 =
the mean direction of the molecules is parallel or ax (1−x ). This equation exhibits a wide range
n
n
antiparallel to an axis known as the director. of interesting nonlinear phenomena: bifurcation
(2) Transition to a cholesteric phase. A meso- and periodic doubling to chaos.
morphic transition that occurs when a molecu-
lar crystal is heated to form a cholesteric phase London forces Attractive forces between
in which there is simply a spiraling of the local apolar molecules, due to their mutual polariz-
orientational order perpendicular to the long axes ability. They are also components of the forces
of the molecules. between polar molecules. Also called “disper-
(3) Transition to a smectic state. A mesomor- sion forces.”
phictransitionthatoccurswhenamolecularcrys-
tal is heated to yield a smectic state in which loop (based at x ∈ X ) A curve γ : I → X
0
there is a one-dimensional density wave which such that γ(0) = γ(1) = x and I = [0, 1].
0
produces very soft/disordered layers.
The set of all loops of X based at x ∈ X can be
0
endowed with a group product. Let λ, γ : I →
list An unordered collection of elements,
X be two loops; the product λ ∗ γ : I → X is
that may include duplicates. An enumerated list
the following loop
is delimited by brackets ([x]).
Comment: Notice the definitions of set, bag, γ(2t) 0 ≤ t ≤ 1/2
and list progressively release constraints on the λ ∗ γ(t) =
λ(2t − 1) 1/2 ≤ t ≤ 1.
elementsinthevariouscollections. Theelements
of sets and bags have some relationship to each
© 2003 by CRC Press LLC