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Hilbert space A vector space H which has (i.) F(0, x) = ϕ(x);
an inner product < , > (scalar product) and (ii.) F(1, x) = φ(x).
is complete with respect to the induced norm
2
x =< x, x >. The map F is called a homotopy between ϕ
and φ.
Let A ⊂ X be a subset; a homotopy F
Hodgkin-Huxley model A mathematical
between the maps ϕ and φ is a homotopy relative
model for the dynamics of electrical potential
to A ⊂ X if we have also:
and ionic currents, due to sodium and potas-
sium, across a biological cell membrane. It
(iii.) ∀t ∈ [0, 1], ∀a ∈ A, F(t, a) = φ(a) =
consists of four nonlinear ordinary differen- ϕ(a).
tial equations. The model exhibits various
interesting behavioral characteristics observed In particular one can consider homotopy of
experimentally such as threshold phenomenon loops based at x ∈ X in a topological space X.
0
and oscillation. See threshold phenomenon and Homotopy relative to A = {x } is an equivalence
0
excitability. relation on the set of all loops based at x ∈ X
0
and the quotient space π (X, x ) is called the
0 0
homotopy group of X. It is in fact a group under
H¨older inequality Let p, q, r be positive
integers satisfying p, q, r ≥ 1 and p −1 + q −1 = the compositions induced by loop composition
q
p
r −1 . If f ∈ L (X, dµ), g ∈ L (X, dµ), then (see loop); this group does not depend on the base
0
r
fg ∈ L (X, dµ) and H¨older’s inequality holds: point x ∈ X, and it is a topological invariant
of X.
See also contractible.
fg ≤ f g q
p
r
homeomorphism (between two topological horizontal lift (induced by a connection )
b
b
spaces X and Y) A map ϕ : X → Y which The unique vector N(X) ∈ projecting onto
is continuous with a continuous inverse map X. Local generators of the space are of the
b
i
µ
ϕ −1 : Y → X. form ∂ − N (x, y)∂ . If X = X ∂ is a vec-
µ
µ
µ
i
tor field over M, then the horizontal lift of X is
locally given by
homolysis (homolytic) The cleavage of a
bond (“homolytic cleavage” or “homolytic fis- µ i
N(X) = X (∂ − N (x, y))∂ ∈ b
µ
µ
i
sion”) so that each of the molecular fragments
between which the bond is broken retains one
of the bonding electrons. A unimolecular reac- hydrocarbons Compounds consisting of
tion involving homolysis of a bond (not forming carbon and hydrogen only.
part of a cyclic structure) in a molecular entity
containing an even number of (paired) electrons
+
results in the formation of two radicals. hydron ThegeneralnameforthecationH ;
It is the reverse of colligation. Homolysis is the species H − is the hydride anion and H is
also commonly a feature of bimolecular substitu- the hydro group. These are general names to
tion reactions (and of other reactions) involving be used without regard to the nuclear mass of
radicals and molecules. the hydrogen entity, either for hydrogen in its
natural abundance or where it is not desired to
distinguish between the isotopes.
homotopy Let X and Y be topological
spaces and ϕ : X → Y and φ : X → Y
be two continuous maps from X to Y. They hyperbola The conic section with equation
2
2
2
2
are homotopic if there exists a continuous map x /a − y /b = 1. See asymptote to the hyper-
F : [0, 1] × X → Y such that bola.
© 2003 by CRC Press LLC