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base for a topology A collection B of open Belousov-Zhabotinskii reaction A chem-
sets of a topological space T is a base for the ical reaction which involves the oxidation of mal-
topology of T if each open set of T is the union onic acid by bromate ions, BrO , and catalyzed
−
3
of some members of B. by cerium ions, which has two states Ce 3+ and
Ce . With appropriate dyes, the reaction can be
4+
base space Let π : E → B be a smooth
monitored from the color of the solution in a test
fiber bundle. The manifold B is called the base
tube. This is the first reaction known to exhibit
space of π.
sustained chemical oscillation. Spatial pattern

basis graph A subgraph G (V, E ) of has also been observed in BZ reaction when dif-

G(V, E) such that E ⊂ E, and that all pairs of fusion coefficients for various species are in the
nodes {v ,v }⊂ V in G and G are connected appropriate region.

i j
(i, j indices).
Benjamin-Ono equation The evolution
basis, Hamel A maximal linear independ- equation
ent subset of a vector space X. Such a basis u = Hu xx + 2uu x
t
always exists by Zorn’s lemma.
where H is the Hilbert transform
basis of a vector space A subset E of a vec-
1 +∞ f(ξ)
tor space V is called a basis of V if each vector (Hf )(x) = dξ .
π −∞ ξ − x
x ∈ V can be uniquely written in the form
n Berezin integral An integration technique

x = a e ,e ∈ E. for Fermionic fields in terms of anticommuting
i i
i
i=1 algebras. Let B = B ⊕ B be a DeWitt alge-
+ −
The numbers a , ..., a are called coordinates of bra (super algebra) and y → f(y) a supersmooth
1
n
the vector x with respect to the basis E. function from B into B. Then f(y) = f +f y
0
1
−
Example: E = (e , ..., e ) with e 1 = with f ,f in B. The Berezin integral of f on
0
1
1
n
(1, 0, ..., 0), e 2 = (0, 1, 0, ...., 0), ..., e n = B is
−
n
(0, 0, ..., 1) is the standard basis of V = R .
f(y)dy = c f
1 1
Bayes formula Suppose A and B , ..., B n B −
1
areeventsforwhichtheprobabilityP(A)isnot0, where c is a constant independent of f .

n
j
i=1 P(B ) = 1, and P(B and B ) = 0if Bergman kernel Let M be an n-dimen-
i
i = j. ThentheconditionalprobabilityP(B |A)
j sional complex manifold and H the Hilbert space
of B given that A has occurred is given by
j
of holomorphic n-forms on M. Let h ,h ,h , ...
0
1
2
P(B )P (A|B ) be a complete orthonormal basis of H and
j
j
P(B |A) = . 1 n
j n z , ..., z a local coordinate system of M. The
i=1 P(B )P (A|B )
i
i
Bergman kernel form K is defined by
beam equation u + u xxxx = 0.
t
1
n
1
n
∗
K = K dz ∧ ··· ∧ dz ∧ d¯z ∧ ··· ∧ d¯z ,
Becchi-Rouet-Stora-Tyutin (BRST) trans-
formation In nonabelian gauge theories the the function K is the Bergman kernel function
∗
effective action functional is no longer gauge on M.
invariant, but it is invariant under the BRST
transformation s Bergman metric Let M be an n-dimen-
sional complex manifold, in any complex coord-
sA = dη + [A, η], 1 n
inate system z , ..., z . The K¨ahler metric
1
sη =− [η, η] , s¯η = b, sb = 0. ds = 2 g ¯dz d¯z .
α
2
β
2 αβ
where A is the vector potential and η and ¯η are the
with
ghost and anti-ghost fields, respectively. One of 2 α β
∗
g ¯ = ∂ log K /∂z ∂ ¯z
αβ
the main properties of the BRST transformation
2
s is its nilpotency, s = 0. is called the Bergman metric of M.
© 2003 by CRC Press LLC