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the temperature increase with height which is Using the Euclidean norm and applying the
characteristic of the stratosphere begins. Lagrange multiplier rule to the subproblem
yields p from the equation
troposphere The lowest layer of the atmos-
[H (x) − uI]p =−grad (x) for some u ≥ 0.
f
f
phere, ranging from the ground to the base of
the stratosphere (tropopause) at 10–15 km of Note that for u = 0, the iteration is Newton’s
altitude depending on the latitude and meteoro- method, and for very large u, the iteration is
logical conditions. About 70% of the mass of the nearly Cauchy’s steepest ascent.
atmosphere is in the troposphere. This is where
most of the weather features occur and where the tub conformation A conformation (of sym-
chemistry of the reactive anthropogenic species metry group D ) of an eight-membered ring in
2d
released into the atmosphere takes place. which the four atoms forming one pair of diamet-
rically opposite bonds in the ring lie in one plane
Trotter product formula If A and B are and all other ring atoms lie to one side of that
plane. It is analogous to the boat conformation
self-adjoint operators and A + B is essentially
of cyclohexane.
self-adjoint, then
e
lim (e itA/n itB/n n i(A+B)t . Tung distribution (of a macromolecular assem-
) = e
n→∞ bly) A continuous distribution with the dif-
ferential mass-distribution function of the form:
true value (in analysis), τ τ τ The value that
b−1 b
characterizes a quantity perfectly in the condi- f (x)dx = abx exp(−ax )dx
w
tions that exist when that quantity is considered.
It is an ideal value which could be arrived at only where x is a parameter characterizing the chain
if all causes of measurement error were elim- length, such as relative molecular mass or degree
inated, and the entire population was sampled. of polymerization and a and b are positive
adjustable parameters.
truncated gradient Projection of the gradi- tunneling The process by which a par-
1
ent of a function, f (in C ) on a box, [a, b], to ticle or a set of particles crosses a barrier on
put zeros in coordinates if the sign of the par- its potential-energy surface without having the
tial derivative is negative at the lower bound or energy required to surmount this barrier. Since
it is positive at the upper bound. This yields a the rate of tunneling decreases with increasing
feasible direction. reduced mass, it is significant in the context of
isotope effects of hydrogen isotopes.
trustregionmethod Theiterationisdefined

as x = x + p, where p is the (complete) change tuple A collection of elements which satisfy
(no separate step size), determined by a subprob- some relation r. A tuple is delimited by paren-
lem of the form theses ((x, y)), and is usually written with the
relation as the functor of the tuple; thus r(x, y).
Max F(p) : p ≤ D, Comment: Note that a sequence is not equiv-
alent to a tuple. The functor is omitted when it
where F depends on the iterate and is an approxi- is absolutely clear what relation it satisfies. See
mation of the change in objective function value. also bag, list, relation, sequence, and set.
The particular norm and the magnitude of D
determine the set of admissible change values Turing pattern A.M. Turing was the first
(p), and this is called the trust region. A com- person who proposed a mechanism for pattern
mon choice of F is the quadratic form using the formation in a spatially homogeneous system
Taylor expansion about the current iterate, x,as to involve chemical reactions and diffusion of
two species. The pattern formation is due to

F(p) = grad (x)p + p [H (x)]p/2. diffusion-driven instability.
f f

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