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unimodal function A function which has
one mode (usually a maximum, but could mean a
minimum, depending on context). If f is defined
U on the interval [a, b], let x be its mode. Then,
∗
∗
f strictly increases from a to x and strictly
∗
decreases from x to b (reverse the monotonicity
∗
on each side of x if the mode is a minimum).
unbounded mathematical program The (For line search methods, like Fibonacci, the
objective is not bounded on the feasible region mode could occur in an interval, [a , b ], where
∗
∗
(from above, if maximizing; from below, if min- f strictly increases from a to a , is constant (at
∗
imizing). Equivalently, there exists a sequence its global max value) on [a , b ], then strictly
∗
∗
k k
of feasible points, say {x } for which {f (x )} decreases on [b , b].)
∗
diverges to infinity (minus infinity, if minimiz-
ing).
unimodular matrix A nonsingular matrix
whose determinant has magnitude 1. A square
unconstrained mathematical program One
matrix is totally unimodular if every nonsingular
with no constraints (can still have X be a proper submatrix from it is unimodular. This arises in
n
subset of R , such as requiring x to be integer-
(linear) integer programming because it implies
valued.)
a basic solution to the LP relaxation is integer-
valued (given integer-valued right-hand sides),
unconstrained optimization Taken liter-
thus obtaining a solution simply by a simplex
ally, this is an unconstrained mathematical
method. An example of a totally unimodu-
program. However, this phrase is also used in
lar matrix is the node-arc incidence matrix of
a context that X could contain the strict inte-
a network, so basic solutions of network flows
rior, with constraints of the form g(x) < 0,
are integer-valued (given integer-valued supplies
but the mathematical program behaves as uncon-
and demands).
strained. This arises in the context of some algo-
rithm design, as the solution is known to lie in the
interior of X, such as with the barrier function. unimolecular See molecularity.
uncountably infinite set An infinite set unisolvence A set of functionals on a finite
which is not denumerably so. See also cardinal- dimensional vector space V is called unisolvent,
h
ity, countable set, denumerably infinite set, finite if it provides a basis of the dual space of V .
h
set, and infinite set. Unisolvence is an essential property of degrees
of freedom in the finite element method.
undirected edge See edge.
unit An identity element.
unified atomic mass unit Non-SI unit of
mass (equal to the atomic mass constant), defined
as one twelfth of the mass of a carbon-12 atom unit circle A circle of radius 1. Usually, the
in its ground state and used to express masses term refers to the circle of radius 1 and center 0
in the complex plane ({z : |z|= 1}).
of atomic particles, u = 1.660 5402(10) ×
10 −27 kg.
unitary group An automorphism α :
uniformly bounded Referring to a family F C → C is called unitary if, once a basis E has
m
m
i
of functions such that the same bound holds for been chosen in C , the matrix U representing
j
m
i
all functions in F. For example, j
the automorphism by α(E ) = U E satisfies
j
j
i
†
f (x) ≤ µ U −1 = U , i.e., if the inverse of U coincides
with the transpose of the complex conjugated
with the same µ, for all f ∈ F. matrix.
c
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC