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Bernoulli equation Let f (x), g(x) be con- Betti number Let H be the pth homology
p
tinuous functions and n = 0 or 1, the Bernoulli group of a simplicial complex K. H is a finite
p
dy n
equation is + f(x)y + g(x)y = 0. dimensional vector space and the dimension of
dx
H is called the pth Betti number of K.
p
p
Bernoulli numbers The coefficients of the Let M be a manifold and H (M) the pth De
Bernoulli polynomials. Rham cohomology group. The dimension of the
p
finite dimensional vector space H (M) is called
Bernoulli polynomials the pth Betti number of M.
m Bianchi’s identities In a principal fiber bun-
m m−k
B (z) = B z . dle P(M, G) with connection 1-form ω and
m k k
k=0 curvature 2-form ; = Dω (D is the exter-
ior covariant derivative), Bianchi’s identity is
The coefficients B are called Bernoulli numbers.
k D; = 0.
The Bernoulli polynomials are solutions of the
In terms of the scalar curvature R on
equations
a Riemannian manifold, Bianchi’s identity is
R(X, Y, Z) + R(Z, X, Y) + R(Y, Z, X) = 0.
m−1
u(z + 1) − u(z) = mz ,m = 2, 3, 4, ...
bifurcation The qualitative change of a
Bessel equation The differential equation dynamical system depending on a control param-
eter.
2
d y dy 2 2
2
z 2 + z + (z − n )y = 0.
dz dz bifurcation point Let X be a vector field
λ
(dynamical system) depending on a parameter
Bessel function For n ∈ Z the nth Bessel λ ∈ R .As λ changes the dynamical system
n
n
functionJ (z)is the coefficient oft inthe expan- changes, and if a qualitative change occurs at
n
sions e z[t−1/t]/2 in powers of t and 1/t. In gen- λ = λ , then λ is called a bifurcation point
0
0
eral, of X .
λ
π
1
J (z) = cos(nt − z sin t)dt bi-Hamiltonian A vector field X is called
H
n
π 0 bi-Hamiltonian if it is Hamiltonian for two
independent symplectic structures ω ,ω , i.e.,
1
2
∞ r
(−1) z n+2r
2
1
H
= . X (F) = ω (H, F) = ω (H, F) for any func-
r!N(n + r + 1) 2 tion F.
r=0
J (z) is a solution of the Bessel equation. bijection A map φ : A → B which is at the
n
same time injective and surjective.
beta function The beta function is defined A map φ is invertible if and only if it is a
by the Euler integral bijection.
1

B(z, y) = t z−1 (1 − t) y−1 dt bijective A function is bijective if it is both
0 injective and surjective, i.e., both one-to-one
and onto. See also onto, into, injective, and
and is the solution to the differential equation
surjective.
−(z + y)u(z + 1) + zu(z) = 0.
bilateral network There are two classes of
simple neural networks, the feedforward and
The beta function satisfies
feedback (forming a loop) networks. In both
N(z) · N(y) cases, the connection between two connected
B(z, y) = . units is unidirectional. In a bilateral network,
N(z + y)
the connection between two connected units is
See gamma function. bi-directional.
© 2003 by CRC Press LLC