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The group of all such matrices is called the each step. One of the symbols input to C refers
unitary group, and it is denoted by U(m). The to an emulation of a particular Turing machine
subgroup of unitary matrices with det U = 1 is stored in the UTM memory. This emulation is
denoted by SU(m). the set of all tuples (s , σ i,I , s i+1 , σ i,O , a ) for
i
i
that machine and forms the algorithm. The algo-
rithm may be deterministic or nondeterministic,
unitarymatrix Anonsingularmatrixwhose
but the machine executes it nonstochastically.
Hermitian adjoint equals its inverse (same as
Comment: This definition differs slightly
orthogonal for real-valued matrices). See self-
from Minsky’s (cf. M.L. Minsky, Computation:
adjoint operator.
Finite and Infinite Machines, Prentice-Hall,
Englewood Cliffs, NJ, 1967). See Post produc-
univariate optimization A mathematical tion system and von Neumann machine.
program with a single variable.
universe of discourse The nonempty set,
universal set The set containing all sets, or
U , of all possible constant terms of a program.
d
sets of interest; denoted U. See also universe of
More generally, the area of nature, thought,
discourse.
or existence described by a program or set of
Comment: In databases, one frequently
programs.
speaks of a universe of discourse, the set of
Comment: For many purposes, this is equiva-
all terms, facts, relations, and functions used in
lent to the universal set. See universal set.
reifying the database’s model of its world (its
domain model). For many purposes the two are
equivalent. unreactive Failing to react with a specified
chemical species under specified conditions. The
term should not be used in place of stable, since
universal Turing machine A universal
a relatively more stable species may neverthe-
Turing machine (UTM) is a discrete automaton
less be more reactive than some reference species
that executes a computation C. It has a read-write
toward a given reaction partner.
head that reads symbols from and writes them
to an unbounded but finite, immutable memory.
The memory stores symbols from a finite symbol unstable The opposite of stable, i.e., the
set chemical species concerned has a higher molar
K = K ∪ K ∪ K , Gibbs energy than some assumed standard. The
I O C
term should not be used in place of reactive
where K = 2, K = 1, and K are the sym-
I O C or transient, although more reactive or transient
bols internal to the computation. At each step
species are frequently also more unstable.
the automaton assumes one of a finite number
Very unstable chemical species tend to
of discrete states, s , s ∈ S. A computation
i i undergo exothermic unimolecular decomposi-
C is defined as a set of tuples each of the form
tions. Variations in the structure of the related
(s , σ i,I , s i+1 , σ i,O , a ), where s is the automa- chemical species of this kind generally affect the
i
i
i
ton’s state at the ith step, σ the symbol read
i,I energy of the transition states for these decom-
into the automaton at that step (K = K =
i,I i,O positions less than they affect the stability of the
1), s i+1 is the new state the automaton assumes decomposing chemical species. Low stability
upon completion of step i (which will be its state
may therefore parallel a relatively high rate of
as it commences step i + 1), σ is the sym-
i,O unimolecular decomposition.
bol output at step i, and a is the action altering
i
the position of the tape in the head that is per-
formed at the end of that step (move it left, right, upper semicontinuity (or upper semicontinuous
k
or nowhere). The values for each σ and ∫ persist [USC]) Suppose {x }→ x.
k
in the automaton long enough for it to execute Of a function, lim sup f(x ) = f(x).
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC