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Campbell-Hausdorff formula Also known
as Baker-Campbell-Hausdorff formula, is the
formula for the product of exponentials in
C a Lie algebra exp A exp B = exp{A + B+
1
1
1
[A, B]+
[B, [B, A]]+···+
[A, [A, B]]+
2
12
c [A[A... [A, B] ... ]]}. 12
n
candidate device A subgraph of the bio-
Calabi-Yau spaces Complex spaces with a
chemical graph with particular topological
vanishing first Chern class or, equivalently, with
properties, without necessarily having distinct
trivial canonical bundle (canonical class). They
biochemical or dynamical properties.
are used to construct possibly realistic (super)
string models. canonical bracket On a symplectic
manifold (M, ω) with canonical coordinates
1
n
calcium-induced calcium release A posi- (q ,...q ,p ,...p ) we have the canonical
1
n
tive feedback component in the calcium dynam- (Poisson) bracket between any two functions F
ics of biological cells (see autocatalysis). It and G on M as
is known from experiments that some calcium n
∂F ∂G ∂F ∂G
channels (see ion channel) which are respon- {F, G}= i − i .
∂q ∂q i ∂p ∂q
i
sible for calcium influx into cytosol are positively i=1
modulated by the calcium in the cytosol. This
canonical coordinates (Darboux’s the-
mechanism has been suggested as being respon-
orem) On any symplectic manifold (M, ω) there
sible for the widely observed calcium oscillation 1 n
exist local coordinates (q ,...q ,p ,...p ),
n
1
in cells.
called canonical coordinates, such that
n

calibration curve See calibration function. i
ω = dq ∧ dp .
i
i=1
calibration function (in analysis) The
canonical one-form On any cotangent
functional (not statistical) relationship for the
∗
manifold T M there is a unique one-form T
chemical measurement process, relating the
such that α T = α for any one form α on M.
∗
expected value of the observed (gross) signal or 1 n
In canonical coordinates (q ,...q ,p ,...p ),
1
n
response variable E(y) to the analyte amount
T is given by
x. The corresponding graphical display for a
n
single analyte is referred to as the calibration i
curve. When extended to additional variables or T = p dq .
i
analytes which occur in multicomponent analy- i=1
sis, the “curve” becomes a calibration surface or canonical symplectic form On any cotan-
hypersurface. gent manifold T M there is a unique symplectic
∗
form ; defined as ; =−dT where T is the
Callan-Symanzik equation A type of canonical one-form.In canonical coordinates
1
n
renormatization group equation in quantum field (q ,...q ,p ,...p ), ; is given by
1
n
theory which studies the variation of the Green’s n

i
function with respect to the physical mass. ; = dq ∧ dp .
i
i=1
Camassa-Holm equation A shallow water canonical transformation A smooth map
equation which is a completely integrable system f between two symplectic manifolds (M ,ω )
1
1
having peakon solutions and (M ,ω ) is called canonical or symplec-
2
2
tic if it preserves the symplectic forms, i.e.,
2
2
3
∂ u − ∂ ∂ u + 3u∂ u − 2∂ u∂ u − u∂ u = 0. f : (M ,ω ) → (M ,ω ), f ω = ω .
∗
x
x
x
x
t x
t
1 1 2 2 2 1
c
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC