Page 136 -

where (iii.) The intersection of the closures of any
two cells is either empty or a vertex, edge, face,

1 j follows city i in tour etc., of both.
x =
ij
0 otherwise
This means that a triangulation constitutes

n−1
QAP:min c x x + x x : a nondegenerate cellular decomposition of ;.
ij ij k=1 ik jk+1 in j1
Special types of meshes are simplicial meshes,
x ∈ P, x ∈{0, 1}, for which all the cells are n-simplices. In two
ij
dimensions the cells of quadrilateral meshes
where have four, possibly curved, edges each. Their
three-dimensional counterparts are hexaedral

1 if tour has city i in position j meshes, whose cells are bricks (with curved faces
x =
ij
0 otherwise and edges). In a straightforward fashion the con-
cept can be generalized to the notion of a triangu-
In each formulation, P is the assignment poly- lation of a compact piecewise smooth manifold
tope. The subtour elimination constraints in ILP (with or without boundary). In the case of adap-
eliminate assignments that create cycles. tive refinement it is often desirable to relax the
For example, in subtours (1 → 2 → 3 → 1) above requirements by admitting hanging nodes.
(length 3) and (4 → 5 → 4) (length 2), the first These are vertices of some cells that lie in the
subtour is eliminated by V ={1, 2, 3}, which
interior of edges of other cells.
requires x 12 + x 23 + x 31 ≤ 2. The second sub-
tour is eliminated by V ={4, 5}, which requires
x 45 + x 54 ≤ 1. triple point The point in a one-component
system at which the temperature and pressure of
tree A connected graph containing no three phases are in equilibrium. If there are p
cycles. possible phases, there are p!/(p − 3)!3! triple
points. Example: In the sulfur system four pos-
triangle inequality A property of a distance sible triple points (one metastable) exist for the
function: f (x, y) ≤ f(x, z) + f(z, y) for all four phases comprising rhombic S (solid), mono-
x, y, z. clinic S (solid), S (liquid), and S (vapor).

triangular matrix A square matrix A,is triplet state A state having a total electron
called upper triangular if all elements are zero spin quantum number of 1.
below the main diagonal, i.e., A(i, j) = 0 for
i> j. It is called lower triangular if its
triprismo- An affix used in names to denote
transpose is upper triangular. We sometimes call
six atoms bound into a triangular prism.
a matrix triangular if it is either lower or upper
triangular.
trivial bundle A bundle (B,M,π; F)
triangulation/mesh Given a bounded which has a global trivialization so that the total
n
domain ; ⊂ R with piecewise smooth bound- space is diffeomorphic to the Cartesian product
ary a triangulation/mesh ; of ; is a finite set B - M × F. Trivial bundles always allow
h
M
{K } ,M ∈ N, of piecewise smooth open global sections, and they are the local model of
i i=1
subsets of ;, called cells, such that all bundles.
(i.) the interior of the closure of each cell is
the cell itself. tropopause The region of the atmosphere
!
(ii.) the union K coincides with ¯ ; which joins the troposphere and stratosphere,
¯
K∈; h
and K ∩ K =∅,if i = j (open partition prop- and where the decreasing temperature with alti-
i j
erty). tude,characteristic of the troposphere ceases, and

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