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Boussinesq equation bridge An edge spanning two connected
components of a graph.
2
u − u xx + 3(u ) xx − u xxxx = 0. Comment: The mathematical word accurately
tt
conveys the structure: two subgraphs joined by
Bramble-Hilbert lemma A crucial tool for
a single edge.
proving local interpolation estimates for para-
metric equivalent finite elements. Given a Brouwer’s fixed point theorem Every con-
bounded domain ; ⊂ R n with Lipschitz- n
tinuous map from the closed ball in R into itself
continuous boundary, it states that, for k ∈ N , has at least one fixed point.
0
0 <s ≤ k + 1,
Brownian dynamics To circumvent the dif-
inf u − p ≤ C|u|
s
s
p∈P k (;) H (;) H (;) ficulties of MD (see molecular dynamics), one
s
∀u ∈ H (;), approach is to introduce a larger time step in
the simulation. This naturally leads to motion
where C = C(s, k, ;) > 0 and P (;) stands being stochastic and hence the dynamic of the
k
for the space of multivariate polynomials of total
molecule is Brownian motion–like. One essen-
degree ≤ k on ;. The proof of this lemma tial difference between Brownian dynamics and
s
relies on the compact embedding of H (;) in MD is that the water molecules are explicit in the
2
L (;), a fact that is known as Rellich’s lemma.
latter, whereas they contribute as a random force
The Bramble-Hilbert lemma can be extended to
and a viscous medium in the former. Brownian
spaces of polynomials with separate degree ≤ k
dynamics have the advantage of large time steps.
in each independent variable and to non-standard
Its diffculty lies in the uncertainty about the inter-
anisotropic Sobolev spaces.
atomic interaction on the large time step and in
implicit water, known as coarse-graining.
branch point (in polymers) A point on a
chain at which a branch is attached.
Brownian motion A stochastic process
Notes: (1)A branch point from which f linear
{X(t) : t ≥ 0} is a Brownian motion process
chains emanate may be termed an f -functional
(or Wiener process)if
branch point, e.g., five-functional branch point.
(i.) X(0) = 0;
Alternatively, the terms trifunctional, tetrafunc-
tional, pentafunctional, etc. may be used, e.g., (ii.) for each t, X(t) is a normal random vari-
pentafunctional branch point. able with zero mean;
(2)A branch point in a network may be termed (iii.) if a< b ≤ c< d, the random variables
a junction point. X(b) − X(a) and X(d) − X(c) are independent
and have the same distributions whenever b −
branched chain (in polymers) A chain with
a = d − c.
at least one branch point intermediate between
the boundary units. BRST Named after Becchi, Rouet, Stora,
and Tyutin, BRST quantization is a method of
branching process A stochastic process,
quantizing gauge theories. After the introduc-
X , models the number of individuals in the nth
n tion of ghost fields the effective Lagrangian is
generation. Usually both X and n take inte-
no longer gauge invariant, but has a new global
ger values, and X is Markovian. Let Z be the symmetry, called BRST symmetry. The BRST
n
random variable representing the number of off-
operator s is defined on the algebra of local oper-
spring in the next generation of a single indi-
ators making it into a differential graded algebra.
vidual. Assuming all individuals are identical,
The induced coboundary operator of the asso-
then X is the sum of the X values of Z. This
n+1 n ciated cohomology (called BRST cohomology)is
sum of a random number of identically, inde- 2
the BRST operator s, satisfying s = 0. The clas-
pendent random variables can be analytically
sical BRST transformations of the vector poten-
obtained using the method of the generating
tial A and the ghost field η are
function: Q n+1 (s) = Q (R(s)) where generat-
n

∞ k 1
ing functions Q (s) = k=0 Prob{X = i}s
n
n
∞ k sA = dη + [A, η] , sη =− [η, η].
and R(s) = Prob{Z = k}s . 2
k=0
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