Multifractals
β(q) is a decreasing function of q
Differentiating
p
i
q
r
i
β(q)
= 1
with respect to q gives
p
i
q
r
i
β(q)
(ln(p
i
) + ln(r
i
) dβ/dq) = 0
Solving for dβ/dq,
dβ/dq = -(
p
i
q
r
i
β(q)
(ln(p
i
)))/(
p
i
q
r
i
β(q)
(ln(r
i
)))
Because each
p
i
q
> 0,
r
i
β(q)
> 0,
ln(p
i
) < 0,
and
ln(r
i
) < 0,
we see
dβ/dq < 0.
When all r
i
take on a common value r, this formula leads to a simple expression for alpha:
alpha = -dβ/dq = (
p
i
q
ln(p
i
))/(ln(r)
p
i
q
)
Return to
the range of alpha values
.