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element matrix Given a linear variational empty collection For sets, bags, lists, and
problem, based on the sesqui-linear form a : V × sequences, the empty collection is the corres-
V → C, and V-conforming finite element (K, ponding collection which has no elements: thus
V ,X ) with shape functions b , ··· ,b ,M = the empty set, empty list, etc. It is denoted by the
1
K
K
M
dimV , the corresponding element matrix is corresponding delimiters with nothing between
K
given by them; thus {}, [ ], and /0 for empty set, empty
list, and empty sequence, respectively.
M
A := (a(b ,b )) i,j=1 . Comment: If a bag is empty, it reduces to the
j
K
i
The bilinear form a could be replaced by a dis- empty set, also denoted ∅. See also bag, list,
crete approximation a (see variational crime). sequence, set, and tuple.
h
elementary charge Electromagnetic funda- end-group A constitutional unit that is an
mental physical constant equal to the charge extremity of a macromolecule or oligomer
of a proton and used as atomic unit of charge molecule.
e = 1.602 177 33(49) × 10 −19 C. An end-group is attached to only one con-
See electric charge. stitutional unit of a macromolecule or oligomer
molecule.
elementary forces The four elementary
forces in nature are gravitation, electromag- endomorphism (1) A map from a set to
netism, weak nuclear force, and strong nuclear itself, satisfying certain conditions depending on
force. the nature of the set. For example, f(x ∗ y) =
f(x) ∗ f(y), if the set is a group.
elementary reaction A reaction in which (2)A morphism (not necessarily invertible) of
no reaction intermediates have been detected, or an object of a category into itself.
need to be postulated in order to describe the
reaction on a molecular scale. Until evidence to energy function For a Hamiltonian system,
the contrary is discovered, an elementary reac- the Hamiltonian is also called the energy function
tion is assumed to occur in a single step and to of the system.
pass through a single transition state.
energy, kinetic In classical mechanics, that
elementary symbol See semiote. part of the energy of a body which the body pos-
sesses as a result of its motion. A particle of mass
elimination The reverse of an addition reac- 1 2
m and speed v has kinetic energy E = mv .
tion or transformation. In an elimination two 2
groups (called eliminands) are lost most often energy-momentumtensor Inclassicalfield
from two different centers (1/2/elimination or theory invariance of the Lagrangian under trans-
1/3/elimination, etc.) with concomitant forma- lations implies, via the Noether theorem, conser-
tion of an unsaturation in the molecule (double
vation of energy-momentum. Let the Lagranian
bond, triple bond) or formation of a new ring.
on space-time be given by L(φ, ∂ φ); then the
µ
conserved energy-momentum tensor T is given
elliptic equation A linear partial differen-
n
tial equation (PDE)on R of order m with con- by
j
stant coefficients is of the form a D u =
|j|≤m j µν ∂L µν
f . It is called elliptic if the equation in p, T = ∂(∂ φ ) ∂ φ − g L ,∂T = 0.
ν i
µν
j µ i
a p = 0 has no real solution p = 0.
|j|=m j
In general relativity the energy-momentum
ellipticity A sesqui-linear form a : V × tensor is defined as the conserved current
V → C on a Banach space V is said to be (via Noether’s theorem) given by varying the
V-elliptic, if metric. And in Yang-Mills theory (including
2
|Ra(u, v)|≥ α u , ∀u ∈ V Maxwell’s equation) the energy-momentum ten-
V
sor is defined as conserved current obtained by
with a constant α> 0. An inf-sup condition for varying the gauge invariant Lagrangian with
a is an immediate consequence. respect to the Yang-Mills gauge invariant frame.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC